SAT II Mathmatics level 2: Designed to get a perfect score on the exam.
This book is designed for the students to get a perfect score on the exam. In this third edition, many tips are added an
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Citation preview
Dr. John Chung’s
SAT II
Mathematics Level 2
Good Luck!
Dr. John Chung's SAT II Math Level 2
1
Dear Beloved Students, With this SAT II Subject Test Math Level 2 Third Edition, I like to thank all students who sent me email to encourage me to revise my books. As I said, while creating this series of math tests has brought great pleasure to my career, my only wish is that these books will help the many students who are preparing for college entrance. I have had the honor and the pleasure of working with numerous students and realized the need for prep books that can simply explain the fundamentals of mathematics. Most importantly, the questions in these books focus on building a solid understanding of basic mathematical concepts. Without understanding these solid foundations, it will be difficult to score well on the exams. These book emphasize that any difficult math question can be completely solved with a solid understanding of basic concepts. As the old proverb says, “Where there is a will, there is a way.” I still remember vividly on fifth‐ grader who was last in his class who eventually ended up at Harvard University seven years later. I cannot stress enough how such perseverance of the endless quest to master mathematical concepts and problems will yield fruitful results. You may sometimes find that the explanations in these books might not be sufficient. In such a case, you can email me at [email protected] and I will do my best to provide a more detailed explanation. Additionally, as you work on these books, please notify me if you encounter any grammatical or typographical errors so that I can provide an update version. It is my great wish that all students who work on these books can reach their ultimate goals and enter the college of their dreams. Thank you. Sincerely, Dr. John Chung
2
Contents 61 Tips
TIP 01 Identical Equation
Page 06
Tip 02 Remainder Theorem
Page 07
Tip 03 Factor Theorem
Page 08
Tip 04 Sum and Product of the roots
Page 09
Tip 05 Complex Number
Page 11
Tip 06 Conjugate Roots
Page 12
Tip 07 Linear Function
Page 13
Tip 08 Distance from a Point to a line
Page 14
Tip 09 Distance from a Point to a Plane
Page 15
Tip 10 Quadratic Function
Page 16
Tip 11 Discriminant
Page 17
Tip 12 Circle
Page 18
Tip 13 Ellipse
Page 19
Tip 14 Parabola
Page 21
Tip 15 Hyperbola
Page 23
Tip 16 Function
Page 25
Tip 17 Domain and Range of a Composite Function
Page 26
Tip 18 Piecewise‐Defined Function
Page 28
Tip 19 Odd and Even Function
Page 29
Tip 20 Combinations of functions
Page 30
Tip 21 Periodic Functions
Page 31
Tip 22 Inverse Functions
Page 32
Tip 23 The Existence of an Inverse Function
Page 33
Tip 24 Leading Coefficient Test (Behavior of Graph)
Page 34
Dr. John Chung's SAT II Math Level 2
3
4
Tip 25 Arithmetic Sequence
Page 35
Tip 26 Geometric Sequence
Page 36
Tip 27 Exponential Functions
Page 37
Tip 28 Logarithmic Functions
Page 39
Tip 29 Basic Trigonometric Identities
Page 41
Tip 30 Circle(Trigonometry)
Page 43
TIP 31 Reference Angles and Cofunctions
Page 44
Tip 32 Trigonometric Graphs
Page 45
Tip 33 Inverse Trigonometric Functions
Page 46
Tip 34 Sum and difference of angles
Page 48
Tip 35 Double Angle Formula
Page 49
Tip 36 Half Angle Formula
Page 50
Tip 37 Trigonometric Equation
Page 51
Tip 38 The Law of Sines
Page 58
Tip 39 The Law of Cosines
Page 54
Tip 40 Permutation
Page 55
Tip 41 Combination
Page 56
Tip 42 Dividing Group
Page 57
Tip 43 Binomial Expansion Theorem
Page 58
Tip 44 Sum of Coefficients of a Binomial Expansion
Page 59
Tip 45 Binomial Probability
Page 60
Tip 46 Probability with Combinations
Page 61
Tip 47 Heron’s Formula
Page 62
Tip 48 Vectors in the Plane
Page 63
Tip 49 Interchange of Inputs
Page 65
Tip 50 Polynomial Inequalities
Page 67
Tip 51 Rational Inequalities
Page 69
Tip 52 Limits
Page 71
Tip 53 Rational Function and Asymptote
Page 73
Tip 54 Parametric Equations
Page 76
Tip 55 Polar Coordinates
Page 77
Tip 56 Matrix
Page 79
Tip 57 Inclination Angle
Page 81
Tip 58 Angle between Two Lines
Page 82
Tip 59 Intermediate Value Theorem
Page 83
Tip 60 Rational Zero Test
Page84
Tip 61 Descartes Rule of Sign
Page 85
Practice Tests
Test 01 Test 02 Test 03 Test 04 Test 05 Test 06 Test 07 Test 08 Test 09 Test 10 Test 11 Test 12
Page 087 Page 113 Page 141 Page 167 Page 195 Page 223 Page 249 Page 277 Page 303 Page 329 Page 355 Page 381
Dr. John Chung's SAT II Math Level 2
5
Tips TIP 01
Identical Equation
An identical equation is an equation which is true for all values of the variable. 10 x 5 x 15 x is an identical equation because it is always true for all real x.
10 x 5 15 is an algebraic equation because it is true for x 1 only.
Identical equation has infinitely many solutions. In an identical equation, the expressions of both sides are exactly same. Example 1: 5 x 5 5 x 5 ax b 0x 0 Example 2: ax b 0 for all real values of x
a 0 and b 0
PRACTICE 1. If ax b 3 x 2 is always true for all real x , what are the values of a and b ?
2. If a(x 1) b( x 1) x 9 is true for all real values of x , what are the values of a and b ?
3. If x2 2 x 6 a( x 1)2 b( x 1) c is true for all real x , where a, b, and c are constants, what are the values of a, b, and c ?
EXPLANATION 1. The coefficients must be equal. 2. ax a bx b (a b)x a b , Coefficients must be equal. a b 1 and a b 9 Therefore, a 5 , b 4 .
(a b)x (a b) x 9
3. Since x2 2 x 6 ax2 (b 2a) x a b c , then a 1, b 2a 2, and a b c 6 . Therefore, a 1 , b 4 , and c 1.
Answer: 1. a 3, b 2
6
2. a 5, b 4
3. a 1, b 4, c 1
Tips Remainder Theorem
TIP 02
When polynomial P(x) is divided by (x a) , the remainder R is equal to P(a) . Polynomial P(x) can be expressed as follows.
P(x) ( x a)Q( x) R The identical equation is true for any value of x , especially x a . Therefore, P(a) R Example: If P(2) 5, then you can say that “When polynomial P(x) is divided by (x 2) , the remainder is 5.
PRACTICE 2 1. If a polynomial f ( x) 2 x 3x 5 is divided by (x 1) , what is the remainder?
3 2 2. If a polynomial g ( x) x 2 x 2 x 3 is divided by ( x 1)( x 2) , then what is the remainder?
EXPLANATION
2 1. R f (1) 2(1) 3(1) 5 4
2.
g( x) (x 1)( x 2)Q( x) ax b
When divided by degree 2 polynomial, the remainder is represented by ax b
At x 1 At x 2
g(1) 8 a b g(2) 23 2a b
Therefore, a 15 and b 7 .
The remainder is 15 x 7
Also the remainder can be obtained using long-division or synthetic division.
Dr. John Chung's SAT II Math Level 2
7
Tips Factor Theorem
TIP 03
If p(a) 0, then p( x) has a factor of ( x a). Polynomial p( x) can be expressed with a factor of (x a) as follows.
p(x) (x a)Q(x), where Q(x) is quotient. If (x a) is a factor of p( x) , then the remainder after division should be 0. Example: If p(5) 0, then p( x) has a factor of ( x 5).
PRACTICE 1.
If a polynomial P( x) x 2 kx 8 has a factor of (x 2) , then what is the value of constant k ?
2.
If a polynomial f ( x) x3 ax2 bx 1 has a factor of ( x 2 1) , what are the values of a and b ?
EXPLANATION 1.
Using the factor theorem,
P(2) 0 2.
2 2 2k 8 0
k 2
Since x3 ax 2 bx 1 ( x 1)( x 1)Q( x) ,
f (1) a b 2 0 and f (1) a b 0 Therefore, a 1 and b 1 .
Answer: 1. k 2
8
2. a 1 and b 1
Tips TIP 04
Sum and Product of the Roots
For a polynomial P( x) an xn an1 xn1 an2 xn2 a1 x ao 0 Sum of the roots
an 1 an
Product of the roots
ao 1n , where n is the degree of the polynomial an
Example 1: P( x) ax2 bx c 0 , r and s are the roots of the quadratic equation. b a
Sum of the roots: r s
Difference of the roots: (r s)2 (r s)2 4r s If r and s are real and r s , then
r s (r s )2 4rs
rs
Product of the roots: r s
b2 4c b 2 4ac a a2 a
c c ( 1) 2 a a
Example 2: P( x) Ax3 Bx2 Cx D 0 Sum of the three roots
B A
Product of the three roots =
D D 13 A A
PRACTICE 1. If the roots of a quadratic equation 2 x 2 5 x 4 0 are and , what is the value of
1
1
?
2. What is the sum of all zeros of a polynomial function P( x) 2 x7 3x3 5x2 4 ?
3. What is the product of all zeros of g ( x) 3x7 5x3 3x2 x 2 ?
4. If one of the roots of a quadratic equation is 2 i , what is the equation?
Dr. John Chung's SAT II Math Level 2
9
Tips EXPLANATION
1. Since
1 1 5 5 4 . and 2 , therefore 4 2 2
2. Because the coefficient of x6 is 0, the sum of the roots
3. Because the product of all zeros is
an 1 0 0 an 2
ao 2 2 (1) n 17 an 3 3
b c x 0. a a b b The sum of the roots (2 i ) (2 i ) 4 4 a a c c The product of the roots (2 i )(2 i ) 5 5 a a
4. The quadratic equation can be defined by x 2
Therefore, the equation is x2 4x 5 0. Proof Sum of the two roots:
b b 2 4ac b b 2 4ac 2b b 2a 2a 2a a Product of the two roots:
2 b b 2 4ac b b 2 4ac b b 4ac 2a 2a 4a 2 4ac c 4a 2 a 2
Answer: 1.
10
5 4
2. 0
3.
2 3
4. x2 4 x 5 0
2
Tips TIP 05 i 1 5 i i
Complex Number i 2 1 6
i 1
i3 i 7
i i
Im
i 4 1 i8 1
z a bi
b
z
Complex number = Real numbers + Imaginary numbers 3 4i a
Pure Imaginary number = only Imaginary numbers (4i) Complex plane
1. a bi is complex number, where a and b are real numbers. (Standard form) 2. a bi is the conjugate of a bi .
a bi a bi
3. If a bi c di , then a c and b d . (Equality of Complex Numbers) 4. a bi a 2 b2 (Distance from the origin)
PRACTICE 1. What are the additive inverse and multiplicative inverse of the complex number 3 i ?
2. What is the value of 3 4i ?
3. If a b ( a b )i 6 4 i , what are the values of a and b ?
EXPLANATION 1. Additive inverse: (3 i ) ( a bi ) 0 3 i. 2.
Multiplicative inverse:
1 3 i
3 i 3 i
3i 10
3 4i 32 (4)2 25 5
3. a b 6 and a b 4 . Therefore, a 1and b 5. Answer: 1. 3 i ,
3 1 i 10 10
Dr. John Chung's SAT II Math Level 2
2. 5
3. a 1, b 5
11
Re
Tips TIP 06
Conjugate Roots
If a polynomial function P ( x ) has one variable with real coefficients, and a bi is a root with a and b real numbers, then its conjugate a bi is also a root of P ( x ) . 1. If x2 x 1 0 , the roots are
1 i 3 1 i 3 and . 2 2 2 2
2. A polynomial x 2 5 0 has the roots of i 5 and i 5 .
PRACTICE 1. If one of the roots of a quadratic equation f ( x ) 0 is 3 2i , what is the quadratic equation?
2. If one of the roots of mx 2 (4m 1) x k 0 is 1 2i , where m and k are real numbers, then what is the value of k ?
EXPLANATION 1. Using the roots
x (3 2i) x (3 2i) 0
x 3 2i ( x 3 2i ) 0
x 3 2 2i 2 0
Therefore, x2 6 x 13 0 . b c x 0 a a b c Sum: 3 2i ( 3 2i ) 6 and Product: 3 2i 3 2i 13. a a
Using sum and product of the roots.
x2
The equation is x2 6 x 13 0 (4 m 1) 1 1 2i ( 1 2i ) 2 , m 2 m k 5 Product: ( 1 2i )( 1 2i ) 5 , k 5m . Therefore, k . 2 m
2. Sum:
Answer: 1. x2 6 x 13 0
12
2. k
5 2
Tips TIP 07
Linear Function
For two linear functions:
y m1 x b1 and y m2 x b2 1. If m1 m2 and b1 b2 , then these two lines are parallel. (Inconsistent) 2. If m1 m2 and b1 b2 , then these two lines coincide. (Dependent) 3. If m1 m2 1 , then these two lines are perpendicular. 4. If m1 m2 , these two lines are intersecting. (Consistent)
PRACTICE 1. What is the equation of the line which is equidistant from two points A (4, 0 ) and B (0, 2) ?
2. If the two lines 2 x 3 y 2 0 and 3 x ky 1 0 are perpendicular, then k
3. If the two lines 2 x ay 1 and ax ( a 4) y 2 are parallel, then a
EXPLANATION 1. The midpoint 2, 1 and the slope of AB
1 . The perpendicular line has slope of 2 and passes through 2
(2, 1). Therefore, the equation is y 2 x 3. 2. The product of the slopes is 1. 3. The slopes are equal.
2 3 1 3 k
k 2
2 a a 4 a 2 0 a a4
a 4 or 2. But the y -intercepts are
1 2 . At a 4 , the two y-intercepts are equal (coincide). Therefore, a 2 and a a4
Answer: 1. y 2 x 3
Dr. John Chung's SAT II Math Level 2
2. k 2
3. a 2
13
Tips Distance from a Point to a Line
TIP 08
Distance D from a point x1 , y1 to a line ax by c 0 :
(x , y ) 1
D
ax1 by1 c a 2 b2
Distance between two points D ( x2 x1 )2 ( y2 y1 )2
1
D
ax by c 0
Distance between two straight lines is the minimum distance which is perpendicular to the lines.
PRACTICE 1. What is the distance from a point 7, 9 to a line 12 x 5 y 0 ?
2. What is the distance from the origin to a line 3 x 4 y 8 ?
3. What is the distance between two parallel lines 3 x y 12 and mx 2 y 4 ?
EXPLANATION
1. D
12(7) 5(9) 122 (5)2
39 3 13
2. Origin 0, 0 , and the equation of the line is 3 x 4 y 8 0 . Therefore, D
3(0) 4(0) 8 2
2
8 . 5
3 4 3. Since they are parallel, m 6 . Choose any point on 3 x y 12 . (4, 0) Distance between (4, 0) and 6 x 2 y 4 0 is D
6(4) 2(0) 4 62 22
Answer: 1. 3
14
2.
10 8 5
3.
10
Tips Distance from a Point to a Plane
TIP 09
Distance from point A( x2 , y2, z2 ) to point B( x1 , y1 , z1 ) in space :
D ( x2 x1 )2 ( y2 y1 )2 ( z2 z1 )2 Distance from a point x1 , y1 , z1 to a plane ax by cz d 0 :
D
ax1 by1 cz1 d a 2 b2 c 2
Distance from the origin to a point ( a , b , c ) is D a 2 b 2 c 2 .
PRACTICE 1. What is the distance from a point (1, 2, 3) to a plane 3 x 4 y 12 z 2 ?
2. What is the length of the diagonal of a rectangular solid with dimensions 3, 4, and 12?
3. What is the distance between point A (1, 1, 2) and point B (3, 4, 1) ?
EXPLANATION 1. D
3(1) 4(2) 12( 3) 2 2
2
3 4 12
2
39 3 13
2. D 32 42 122 13 3. D
3 12 4 1
Answer: 1. 3
2
1 2 30
2. 13
Dr. John Chung's SAT II Math Level 2
2
3.
30
15
Tips Quadratic Function
TIP 10
Polynomial function of x with degree n is defined as follows.
P( x) an xn an1 x n1 an2 xn2 a2 x2 a1 x ao 1. If f ( x) ao , then f ( x) is a constant function. 2. If f ( x) ax 2 bx c, then f ( x) is a quadratic function. If a 0, f has a minimum at x
b . 2a
If a 0, f has a maximum at x
b . 2a
3. f ( x) ax 2 bx c : standard form. 4. f ( x) a( x x1 )( x x2 ) : factored form, where x1 and x2 are the roots of f ( x) 0 5. f ( x ) a x h k : vertex form 2
Axis of symmetry: x h ,
Vertex: (h, k )
PRACTICE 1. If f ( x) x2 6 x 8 , what are the coordinates of the vertex?
2. If a manufacturer of game computers has daily production costs of C (n) 1200 24n 0.5n 2 , where C is the total cost, in dollars, and n is the number of units produced, how many game computers should be produced each day to minimize cost?
EXPLANATION 1. Axis of symmetry x
b 6 3 , y f (3) 32 6(3) 8 1 2a 2(1)
2. Axis of symmetry n
24 2(0.5)
24 , C has a minimum at n 24 .
Therefore, the minimum cost C (24) 1200 24(24) 0.5(242 ) 912 Answer: 1.
16
3, 1
2. 24
Tips Discriminant
TIP 11
Discriminant determines the nature of the roots of a quadratic equation ax2 bx c 0. Roots
b b 2 4 ac , Discriminant D b 2 4 ac 2a
1. If D 0 , then the roots are real and unequal. 2. If D 0 , then the roots are real and equal. 3. If D 0 , then the roots are imaginary. (No real roots)
PRACTICE 1. If a quadratic equation 2 x2 kx 3 0 have imaginary roots, what is the value of k ?
2. If the roots of x2 (k 1) x 4 0 are real and equal, what is the value of k ?
3. If y 3x 2 2 x k is positive for all x , then what is the smallest integral value of k ?
Explanation 1. D k 2 24 0
k 2 6 k 2 6 0 . Therefore, 2
2. D k 1 16 0 ( k 5)( k 3) 0 . 2
6x2 6 .
k 5 or 3.
3. That means “no x-intersections,” or “imaginary roots.” D 4 12 k 0
12 k 4 k
Answer: 1. 2 6 k 2 6
Dr. John Chung's SAT II Math Level 2
1 , The smallest integer k is 1. 3
2. k 3 or 5
3. k 1
17
Tips Circle
TIP 12
A circle is the locus of points equidistant from a given point, known as the center. The standard equation of a circle whose center is at the point h, k is
x h 2 y k 2 r 2 ,
r radius
PRACTICE 1. What is the area of a circle whose equation is x 2 4 x y 2 2 y 11 ?
2. The graph of the equation x 2 y 2 2ax 4 y 2a 2 0 represents a circle. What is the greatest possible integer value of a ?
3. What is the circumference of a circle whose equation is x 2 y 2 6 y 16 ?
EXPLANATION 1. The standard equation is x 2 y 1 16 , 2
2
r 2 16
Therefore, the area is r 2 16 . 2. The standard equation is x a y 2 a 2 4 2
To form a circle, r 2 a2 4 0
2
a 2 4 0 a 2 a 2 0
Since 2 a 2 , the greatest integer a is 1. 3. The standard equation is x 2 y 3 25 . r 5 . 2
Therefore, the circumference of the circle is 2 r 10 . Answer: 1. 16
18
2. 1
3. 10
Tips Ellipse
TIP 13
An ellipse is the set of all points x, y in a plane, the sum of whose distances from two distinct fixed point (foci) is constant. The standard equation of an ellipse with center h, k is If a b ,
x h
2
2
y k
2
1
2
b
: Major axis is horizontal
a b Vertex 1. The center of the ellipse is at point (h, k ) 2. Length of major axis is 2 a. Length of minor axis is 2b.
Vertex
If a b ,
x h 2
y k 2
focus
2
1
: Major axis is vertical
b
c
2.
a b The center of the ellipse is at point (h, k) Length of major axis is 2b. Length of minor axis is 2 a.
3.
If c is the length from the center to the focus, then c 2 b2 a 2 .
1.
Vertex c focus
3. If c is the length from the center to the focus, then c 2 a 2 b2 .
2
h, k
a
a
h, k
a
Vertex
PRACTICE 1. What is the center of an ellipse whose equation is x2 4 y 2 6 x 8 y 9 0 ?
2. What is the length of the major axis of the ellipse whose equation is
3.
x2 y 2 1? 5 27
The ellipse is given by 4 x2 y 2 36 . What are the coordinates of the foci?
4. If a line y x k is tangent to an ellipse whose equation is x 2
Dr. John Chung's SAT II Math Level 2
y2 1 , what is the value of k ? 4
19
Tips EXPLANATION
1. The standard equation is x 3 4 y 1 4 2
2
x 32 y 12 4
1
1.
Center (3, 1) 2. a 2 27 a 3 3 ,
Therefore major axis is 2a 6 3.
y
x2 y 2 1 . (Major axis is vertical) 9 36
3. The standard equation is
f
a 2 36 and b2 9 2
c
2
c b a 27 3 3
O
Therefore the coordinates of foci is f 0, 3 3 . 4. Substitute. 4 x 2 x k 4 2
5 x 2 2 kx k 2 4 0
f
Since the line is tangent to the ellipse, its discriminant must be 0.
D 4k 2 4(5)(k 2 4) 0 16k 2 80 k 2 5
y
Therefore, k 5 .
y x 5
y x 5 x
O
Answer: 1.
20
3, 1
2. 6 3
3. 0, 3 3
4. 5
b a
x
Tips Parabola
TIP 14
A parabola is the set of all points in a plane that are equidistant from a fixed line, the directrix, and a fixed point, the focus. ( p is the distance from the vertex to the focus.) 1.
The vertex form of the equation with vertex at the origin (0, 0) is Vertical axis 1 2 y x 4p
y
y
1 2 x 4p
y
Directrix y p
Focus (0, p )
( x, y )
O
x
x
O
( x, y ) Focus (0, p )
Directrix y p
The vertex form of the equation of a parabola with vertex at h, k are as follows. 1 x h 2 k (Graph opens upward) 4p
y
2.
y
1 x h 2 k (Graph opens downward) 4p
Horizontal axis 1 2 y 4p
x
x
1 2 y 4p
Directrix y x p
O
y Focus ( p, 0)
( x, y )
x
Focus ( p , 0)
Directrix x p
O
x
( x, y )
The vertex form of the equation of a parabola with vertex at h, k are as follows.
x
1 1 y k 2 h (Graph opens to the right) x y k 2 h (Graph opens to the left) 4p 4p
Dr. John Chung's SAT II Math Level 2
21
Tips PRACTICE 1. What are the focus and directrix of the parabola whose equation is given by y
1 4
2. Find the focus of the parabola given by y x 2 x
1 2 x ? 16
1 . 4
3. Find the vertex form of the equation of the parabola with the vertex at (1,0) and the focus at (2, 0).
EXPLANATION 1.
2.
y
1 2 x 4 4
focus (0,4) and directrix y 4
1 2 1 x 4x , the vertex form is 4 4 1 5 5 2 y x 2 p 1 and vertex 2, , graph opens downward. 4 4 1 4 y
5 1 Therefore, the focus h, k p 2, 1 2, 4 4
y
3. Because the axis of symmetry is horizontal, passing through 1, 0 and 2, 0 .
directrix x0
The vertex form of the equation is 1 2 x y 0 1 , where the vertex is at 1,0 and p 2 1 1. 4p Therefore the equation is 1 x y2 1 4
(1, 0) O
Answer: 1. f 0, 4 , y 4 2.
22
2, 0.25
3. x
1 2 y 1 4
(2, 0)
Focus
x
Tips TIP 15
Hyperbola
A hyperbola is the set of points in a plane, the difference of whose distances from two distinct foci is constant.
The standard form of the equation with the center at 0, 0 is x2 y2 1 a 2 b2 Transverse axis is horizontal
y 2 x2 1 b2 a 2 Transverse axis is vertical
y
y
c
( c, 0)
b a
(c, 0)
(0, c ) a b c x
x
(0, c ) Focus ( c,0) : c 2 a 2 b2 Asymptotes: y
Focus (0, c) : c 2 a 2 b2
b x a
Asymptotes: y
b x a
The standard equation of a hyperbola with center at h, k is
x h
2
a2
y k
2
b2
Dr. John Chung's SAT II Math Level 2
y k
2
b2
x h a2
1
Transverse axis is horizontal
1
Transverse axis is vertical
2
23
Tips PRACTICE 1. If the equation of a hyperbola is given by
x2 y2 1 , what are the equations of its asymptotes? 25 16
2. Find the asymptotes and foci of the hyperbola whose equation is
y 2 x2 1. 16 4
3. The equation of a hyperbola is defined by 9 x2 y 2 36 x 6 y 18 0 . Find the center.
EXPLANATION 1. Asymptotes: y
b 4 x . Since a 5 and b 4 , the asymptotes is y x . a 5
b x. a
2. Asymptotes: y
a 2 and b 4 . Therefore, the asymptotes is y
4 x 2
y 2 x.
For the foci (0, c) :
c a 2 b2 16 4 2 5
3. Perfect squared form: 9 x 2 4 x 4 y 2 6 y 9 18 36 9
9 x 2 y 3 9 2
2
Center at 2, 3 4 Answer: 1. y x 5
24
x 2 2 y 3 2 1
9
1
2. y 2x , f 0, 2 5
3.
x 2 2 y 3 2 1
9
1 , Center 2, 3
Tips Function
TIP 16
A function f from a set X to a set Y is a relation that assigns to each element in set X exactly one element in the set Y . Domain is the set of X (input).
Range is the set of Y (output).
1. x is the independent variable and y is the dependent variable. Input x
Input x
Function f
Function f
Output f ( x)
Output f ( x)
PRACTICE 1. What are the domain and range of the function f ( x ) 16 x 2 ?
2. What is the domain of the function f ( x )
x 10 ? x 15
3. What is the range of the function f ( x) 3 x 5 4 ?
EXPLANATION 1. 16 x 2 0
x 4 x 4 0 4 x 4 x 15 10 x 15 x 15 or
2. x 10 0 but
interval notation: 10,15 15,
3. Minimum of f is 4. You can a graphing calculator Answer: 1. Domain : 4 x 4 , Range : 0 y 4
2. 10 x 15 15 x
3. y 4
Dr. John Chung's SAT II Math Level 2
25
Tips TIP 17
Domain and Range of a Composite Function
Domain of a composite function is the intersection of domains of the starting and final function. Range of a composite function is the range of final function restricted by starting function. Example: f ( x)
1 x : , g ( x) x2 x3
g ( x) starting function,
f ( x) second function
f g ( x) final function
Domain of starting function g ( x) is x 3. (All real numbers except 3)
f g ( x)
1
x3 Domain of final function is x 2. 3( x 2)
x 2 x3 Therefore, the domain of f g ( x) is all real numbers x except 2 and 3. Now let’s find the range. First find the range of the final function. From the function, we can find two asymptotes.
x 2 : Vertical asymptote x3 1 : Horizontal asymptote y xlim 3( x 2) 3 y
But when x 3, the function is undefined.
O
3
1 y x 3
x2
Therefore, the range of f g ( x) is , 0 0,
26
1 1 1 , . All real numbers y except 0 and 3 3 3
Tips PRACTICE 1. If f ( x) x and g ( x) x 1, what is the domain and range of g f ( x)?
2. If f ( x )
1 1 and g ( x ) , what is the domain and range of g f ( x)? x x 1
EXPLANATION 1. Domain of f ( x) is x 0. x 1 : Domain of g f ( x) is x 0. There, the domain of the final function is x 0.
g f ( x)
The range of the final function is y 1. y
x
1 2. Domain of f ( x) is x 0 . (All real numbers except 0)
g f ( x) 1
1
x 1 1 x
x Domain of g f ( x) is x 1. Therefore, actual domain is all real numbers except 0 and 1. Range of g f ( x) :
y
Vertical asymptote: x 1 x 1 Horizontal asynptote: y xlim 1 x From the first function f ( x) ,
x0
g f (0)
x y 1
is undefined.
Therefore, the range of g f ( x) is all real numbers except y 1 and y 0.
Dr. John Chung's SAT II Math Level 2
x 1
27
Tips TIP 18
Piecewise-Defined Function
A piecewise-defined function is a function that is defined by two or more equations over a specified domain.
f ( x) x is a piecewise-defined function as follows. x, when x 0 f ( x) x x, when x 0
PRACTICE 1. What are the domain and range of the piecewise-defined function as follows?
2 x 1, f ( x) x 1,
x0 x0
EXPLANATION
1. The graph shows that the domain is all real x and the range is y 1. y
y x2 1
y x 1( x 0)
(0,1) x
O
(0, 1) Answer: 1. Domain: All real x , Range: y 1
28
Interval notation:Domain , , Range[1, )
Tips Odd and Even Functions
TIP 19
A function f is even if f ( x) f ( x) .
A function f is odd if f ( x) f ( x) . y
y
x, y
x, y
x, y
x
x
O
O
x, y Symmetric about the y-axis
Symmetric about the origin
n 1. For f ( x) x , if n even , then f ( x) is even.
if n odd , then f ( x) is odd.
PRACTICE Determine whether each function is even, odd, or neither. 1.
f ( x) x3 2 x
2. g ( x) x 4 2 x2 5 3. h( x) x3 1
EXPLANATION 1. Since f ( x) f ( x) , f ( x) is odd. 2. Since g ( x) g ( x) , g ( x) is even. 3. h( x) is neither , because h( x) h( x) or h( x) h( x) . Answer: 1. Odd
2. Even
Dr. John Chung's SAT II Math Level 2
3. Neither
29
Tips Combinations of Functions
TIP 20 Sum:
f
g ( x) f ( x) g ( x)
Difference:
f
g ( x ) f ( x ) g ( x)
Product:
fg ( x) f ( x) g ( x)
Quotient:
f f ( x) , g ( x) 0 ( x) g ( x) g
Compositions:
f g ( x) f g ( x)
PRACTICE 1. If f ( x) 2x 3 and g ( x) 2x 3 , then fg (4) 2. If f ( x) log3 x 3 and g ( x) log3 x 3 , then f g (6)
3. If f ( x) log 2 x 2 3x 2 and g ( x) log 2 x 2 , then f g (9) 4. If f ( x) e x and g ( x) 3ln( x 3) , then f g (5)
EXPLANATION 1. Since fg 2 x 3 2 x 3 4 x 2 9 , fg (4) 55 . Or, f (4) 11 and g (4) 5 . f (4) g (4) 55.
2. Since f g log3 x 3 x 3 log3 x 2 9 ,
f
g (6) log3 36 9 3log3 3 3 .
3. f g log 2 x 2 x 1 log 2 x 2 log 2
x 2 x 1 log 2 x 1 x 2
Therefore, f g (9) log 2 9 1 3log 2 2 3 4. Since f g ( x ) e 3ln( x 3) ( x 3) 3 , f g (5) (5 3) 3 8 . Answer: 1. 55
30
2. 3
3. 3
4. 8
Tips Periodic Functions
TIP 21
If a function f is periodic if there exists a number p such that
f ( x p) f ( x) for all number x. 1.
The smallest period is called the fundamental period of the function.
2.
If a periodic function f has period p , then
1) y cf ( x) still has period p.
y
Period
x
p 2) y f cx has period . c
x
x p
3. The smallest period is simply called the period.
PRACTICE 1. If a function f ( x) sin x has period 2 , then what is the period of the function
f ( x) 3sin3x ?
2. What is the period of the function y 2 cos x 5 ? 12
3. If a function is defined by f ( x) f ( x 2 ) , what is the period of the function?
EXPLANATION 1. p
2 2 c 3
2.
Dr. John Chung's SAT II Math Level 2
p
2
2
3. p 2
31
Tips Inverse Functions
TIP 22
An inverse function is a function that reverses function f . If f is a function mapping x to y, then the inverse function of f maps y back to x.
x
f ( x) x 5
y
f 1 ( x) x 5
y x
: (1, 6), (2, 7), (3, 8), (4, 9)
If
f ( x) x 5
then
f 1 ( x) x 5 : (6, 1), (7, 2), (8, 3), (9, 4)
In order to form inverse function of f just interchange x and y coordinates and express y in terms of x. If the function g is the inverse function of f , 1.
f g ( x) x and g f ( x) x.
2.
The domain of f is the range of g and the range of f is the domain of g.
3.
f 1 ( x) is a reflection of the graph of f in the line y x.
4.
If point (a, b) lies on graph of f , then point (b, a) must lie on the graph of f 1.
PRACTICE 1. What is the inverse function of f ( x ) 2. If f (4) 35, then f 1 35
EXPLANATION 1. x
3y 5 , 2
f 1 : y
2x 5 3
2. f : 4, 35 f 1 : (35, 4) Answer: 1. y
32
2x 5 3
2. 4
3x 5 ? 2
Tips The Existence of an Inverse Function
TIP 23
If a function f is one-to-one, then its inverse is a function. 1. If f is increasing on its entire domain, then f is one-to-one. 2. If f is decreasing on its entire domain, then f is one-to-one. 3. If f is increasing on its entire domain, then f 1 is a function. 4. If f is decreasing on its entire domain, then f 1 is a function. To check one-to-one, the horizontal line test can be used. y
y f ( x)
By the horizontal line test, f does not have an inverse function.
x
O
PRACTICE 1. Does the function f ( x) x 2 3 have an inverse function?
2. Does the function g ( x) x 3 have an inverse function?
EXPLANATION y
y 1.
2.
(2,3)
O
f is one - to - one. Its inverse is a function.
Dr. John Chung's SAT II Math Level 2
x
(3, 0) O
x
g is not one- to- one Its inverse is not a function.
33
Tips TIP 24
Leading Coefficient Test (Behavior of Graph)
Whether the graph of a polynomial rises or falls can be determined by the Leading Coefficient Test as follows.
P( x) an x n an 1 x n 1 a1 x ao
: an is the leading coefficient.
1. When n is odd and an is positive, the graph falls to the left and rises to the right. 2. When n is odd and an is negative, the graph rises to the left and falls to the right. 3. When n is even and an is positive, the graph rises to the left and right. 4. When n is even and an is negative, the graph falls to the left and right.
PRACTICE 1. What are the right-hand and left-hand behaviors of the graph of f ( x) x5 2 x3 3x 5 ?
EXPLANATION y
1. Using the test, leading coefficient is positive and n is odd.
lim x5 and lim x5
x
x
x O Answer: The graph falls to the left and rises to the right.
34
Tips TIP 25
Arithmetic Sequences
A sequence is arithmetic if the differences between consecutive terms are the same. If a1 a and d is the common difference, then the nth term of an arithmetic sequence is
an a (n 1)d . And the sum of a finite arithmetic sequence with n terms is Sn
n ( a1 an ) 2
PRACTICE 1. If the first term of a sequence an is a1 3 and an1 an 4, what is a25 ?
2. If three numbers m, a , and k form an arithmetic sequence in that order, the sum of the numbers is 21, and the product of the numbers is 315, what is the greatest number in the sequence?
3. If the first term of an arithmetic sequence is 5 and the common difference is 3, what is the sum of the first 100 terms?
EXPLANATION 1. Since a1 3, an1 an 4, and d 4, a25 a n 1 d 3 24 4 99 . 2. Let three numbers be a d , a, a d . Then a d a a d 3a 21 In an arithmetic sequence, the middle number (median) is equal to the average.
a7
(7 d ) 7 (7 d ) 315 49 d 2 45 d 2 4 d 2 Therefore, three numbers are 5, 7, 9 3. Since a1 5 and d 3 , then a100 5 (100 1) 3 302 and number of terms is 100. Therefore, S100 Answer: 1. 99
100 5 302 2 2. 9
Dr. John Chung's SAT II Math Level 2
15,350 3. 15,350
35
Tips Geometric Sequences
TIP 26
A sequence is geometric if the ratios of consecutive terms are the same. If a1 a and the common ratio is r , then 1) the n th term of the geometric sequence is an ar n 1 and
a(1 r n ) . 1 r
2) the sum of the sequence is Sn
a(1 r n ) . n 1 r
3) The sum of the infinite series is given by S lim Sn lim n
If r 1 , then the sum of the infinite series is as follows. S
a 1 r
PRACTICE 1. The first term of a geometric sequence an is 2 and an1 2an . What is the value of a10 ?
2. In a geometric sequence, the second term is 3 and fifth term is 24. What is the 8th term?
3. What is the sum of the infinite series 15 3
3 3 ? 5 25
EXPLANATION 1. Since a1 2 and r 2, then a10 2 2 210 . 9
2. a2 a1r 3 and a5 a1r 4 24 ,
a5 a1r 4 24 3 a2 a1r
r3 8 .
3 3 7 . The 8th term a8 2 192 2 2 a 15 1 3. Since r , S 12.5 1 r 1 1 5 5
Therefore r 2 and a1
Answer: 1. 1024
36
2.
192
3.
12.5
Tips Exponential Functions
TIP 27
A. Laws of Exponents: If a, b, and c are positive integers. 1. xa xb xa b
: Multiplication Law
2. x a xb x a b
: Division Law
3.
x
x ab
: Power Law
4.
xy a
xa y a
: Power of a Product Law
a b
a
x xa 5. a y y
: Power of a Quotient Law
6. xo 1 ( x 0)
: Zero Exponent
a 7. x a
a
1 1 ( x 0) : Negative Exponent xa x
b
8. x b x a
x b
a
: Fractional Exponent
B. Exponential Function f with base a :
f ( x) a x , where a 0, a 1 , and x is any real number. C. Graphs of Exponential Functions 1) a 1 y
2) 0 a 1
x
y
O
O
x
The graph of y 2 x is the reflection in the y-axis of the graph of y 2 x . Domain: , , Range: 0, , Horizontal asymptote: y 0 D. The natural base e e 2.718281828
1 e lim 1 x x
Dr. John Chung's SAT II Math Level 2
x
or
1
e lim 1 x x x0
37
Tips PRACTICE 1. If a total of $10,000 is invested at an annual interest rate of 5%, compounded annually, what is the balance in the account after 5 years?
2. If $5,000 is invested at an annual interest rate of 6%, compounded quarterly, what is the amount of the balance after 5 years?
3. If 32 x 3x 72 0 , what is the value of x ?
2
2 x 1 16 , then x 4. If 2 x 1
EXPLANATION 1.
A 10, 000 1 0.05 12762.82
2.
0.06 A 5,000 1 4
5
4(5)
nt
r 6734.28 , because A P 1 . n
3. Since 3x 9 3x 8 0 , then 3x 8 and 3x 9 . Therefore, x 2. 2
2 x 1 x2 x 2 24 , then x2 x 2 4 x 2 x 2 0 . 4. Since x 1 2 2 x 2 x 1 0 , x 2 or 1 . Answer: 1. $12762.82
38
2. $6734.28
3. 2
4. x 2 or x 1
Tips TIP 28
Logarithmic Functions
The function is given by
y loga x
x a , y
where x 0, a 0, and a 1 .
Properties of Logarithms: 1.
loga 1 0
2.
loga a 1
3.
log a a x x log a a x
4. 5.
aloga x x If loga x loga y , then x y.
6.
log10 x log x
: (Common logarithms)
7.
loge x ln x
: (Natural logarithms)
8.
log a x
9.
loga xy log a x loga y
log b x log b a
: (Change of base)
x log a x log a y y
10.
log a
11.
log a x n n log a x
12.
loga x log an x
13.
log a x
: (Product property) : (Quotient property) : (Power property)
n
: (All real n except 0)
1 log x a
: (Reciprocal property)
Graphs of Logarithmic Functions: y log a x
y
a 1
y log 2 x
x
O
y
0 a 1
y log1 2 x
O
x
The graph of y log 1 x is the reflection in the x -axis of the graph of y log 2 x . 2
Domain: 0, , Range: , , Vertical asymptote: x 0
Dr. John Chung's SAT II Math Level 2
39
Tips PRACTICE 1. If ln x 2 ln 2 x 3 2ln x , then x
2. $10,000 is invested in an account at an interest rate of 5%, compounded continuously. The value A , of the
investment at any time is given by the equation A Pert , where t represents the number of years. How long will it take the balance to double?
3. If log3 x log3 ( x 8) 2 , then x
4. If y 3e x 2 , what is its inverse function?
EXPLANATION
1. ln 2 x 2 7 x 6 ln x 2 2 x2 7 x 6 x 2 x 6 x 1 0
Since x 2 , the answer is x 6 only. 2. 20, 000 10, 000e 0.05t 0.05t ln 2 t
3. log3 x2 8x 2 x2 8x 32
ln 2 13.86 0.05
x 9 x 1 0
, x 9 or 1
Since domain is x 8 , x 1 cannot be the solution. 4. x 3e y 2 e y
Answer: 1. x 6
40
x2 x2 f 1 : y ln 3 3
2. 13.86
3. 9
x2 4. f 1 ( x ) ln 3
Tips TIP 29
Basic Trigonometric Identities
Functions
Domain
Range
Period
sin
All real numbers
1 sin 1
2 ,360
cos
All real numbers
1 cos 1
2 ,360
All real numbers
,180
tan
All real numbers except
180 n 90 csc
All real numbers except
csc 1
2 ,360
sec 1
2 ,360
All real numbers
,180
180 n sec
All real numbers except 90 180 n
cot
All real numbers except
180 n Reciprocal Identities sec
1 cos
Quotient Identities sin tan cos
csc
1 sin
cot
cos sin
cot
1 tan
Pythagorean Identities
sin 2 cos2 1
Dr. John Chung's SAT II Math Level 2
1 tan 2 sec2
1 cot 2 csc2
41
Tips PRACTICE 1. If sin cos
1 , then sin cos 2
2. If sin cos
1 , what is the value of tan cot ? 4
3. If cos
4 and 90o 180o , what is the value of tan ? 5
4. If the roots of the equation 3x2 kx 1 0 are sin and cos , what is the positive value of k ?
EXPLANATION 1.
sin cos 2
1 4
sin 2 cos 2 2 sin cos
1 1 1 2 sin cos 4 4
3 8
Therefore, sin cos . 2.
sin cos 2 tan cot
3. Since sec
1 16
1 2 sin cos
1 16
sin cos
15 32
sin cos sin 2 cos2 1 32 cos sin cos sin cos sin 15
1 5 , cos 4 2
9 5 1 tan 2 sec2 tan 2 sec2 1 tan 2 1 . 16 4 9 3 3 . In the second quadrant, tan 0 . Therefore, tan . 16 4 4 1 k 4. Sum and product of the roots. sin cos and sin cos . 3 3 tan
sin cos 2 1 2sin cos Therefore, 1
2 k2 3 9
Answer: 1.
3 8
42
2.
k2 9
1 k2 3 9
k2 3 . k 3
32 15
3.
3 4
4.
3
Tips Circle (Trigonometry)
TIP 30 a 0 c b cos 0 c b tan 0 a
sin
b 0 a a sin 0 c b cos 0 c tan
a 0 c b cos 0 c b tan 0 a
sin
c
a
c
b a
a b
c
a
c
is the terminal angle from initial side,
b 0 c a sin 0 c b tan 0 a cos
PRACTICE 1. If cos
4 and tan 0 , then what is the value of sin ? 5
2. In the interval 0 x 360 , sin x cos x where x is
3. If tan x
3 and cos x 0 , then what is the value of x ? 2
EXPLANATION 1. Since the angle terminates in Quadrant III, sin
3 . 5
2. They are equal in Quadrant I and III. 3. The angle is in Quadrant II. 3 x tan 1 56.3 2
Therefore, x 180 56.3 123.7
Answer: 1.
3 5
2. 45 , 225
Dr. John Chung's SAT II Math Level 2
3. 123.7
43
Tips Reference Angle and Cofunction
TIP 31
If is the terminal angle, its reference angle is the acute angle formed by the terminal side of and the horizontal axis.
Initial Side
Terminal Side Cofunction: Any trigonometric function of an acute angle is equal to the Cofunction of its complement.
If A B 90 , then sin A cos B , sec A csc B , and
tan A cot B .
PRACTICE 1. Rewrite cos 258 as a function of a positive acute angle.
2. Write an expression equivalent to cot 118 as a function of an acute angle whose measure is less than 45 .
3. If is an acute angle and csc 15 sec 45 , then sin
EXPLANATION 1. 258 180 78 ,
Using reference angle Using cofunction cos 258 cos 78 sin12
2. 180 118 62 ,
cot(118 ) cot 62 tan 28
3. Cofunction: 15 45 90 30 sin 30
Answer: 1. cos 78 or sin12
44
2. tan 28
1 2
3.
1 2
Tips TIP 32
Trigonometric Graphs
1. y a sin bx c d and y a cos bx c d .
Amplitude : a Period : p
2 b
Middle line : y d 2. y tan(bx c) d Amplitude : Does not exist.
Period : p
b Middle line : y d
PRACTICE 1. Find the period and amplitude of the function f (t ) 5 cos
t 4. 12
2. What is the period and amplitude of y 4sin x cos x 1 ? 3. If the height is given by h 30 sin
15
t 25 15 , what are the maximum and minimum
values of the height?
EXPLANATION 1. p
2 24 and amp 5 5 . 12
1.
2. y 4sin x cos x 1 2sin 2 x 1 2 Therefore, amp is 2 and p .
p 24
0
2
3. Since amplitude is 30, maximum 30 15 45 and minimum 15 30 15.
5
y 1
24 y 4 middle line y 9
Answer: 1. p 24 and amplitude 5
Dr. John Chung's SAT II Math Level 2
2. p , amplitude 2
3. maximum 45 , minimum 15
45
Tips TIP 33
Inverse Trigonometric Functions y
1, 2
Inverse Sine Function y
1
2 , 1
y sin x
y arcsin x
x
, 1 2
1
1
y sin x has an inverse function only on this interval.
1, 2
Inverse Cosine Function
y
1,
y
0, 1
x
1
y cos x x
y arccos x
, 1 y cos x has an inverse function only on this interval.
1, 0
1
Inverse Tangent Function
y
y
y tan x
2
2
2
x
y tan x has an inverse function only on this interval.
46
x
y arctan x
x
2
Tips The ranges of the inverse functions 1. arcsin x 2. 0 arccos x
3.
Compositions of Functions: 1. sin(arcsin x) x
3. tan(arctan x) x
2
2
2. cos(arccos x) x
2
arctan x
2
PRACTICE 1. What is the value of tan arctan(3) ?
5 2. What is the value of arcsin sin 3 3.
?
2 tan arccos 3
3 5
4. cos arcsin
EXPLANATION 1.
Let arctan(3) x.
2. sin
5 3 , 3 2
3. arccos
tan x 3 . Therefore tan arctan(3) tan x 3 . 5 3 arcsin sin arcsin 3 2 3 2 5 Therefore, tan arccos tan x . 3 2
2 2 x cos x 3 3
3 4 3 3 Therefore, cos arcsin cos x . 4. arcsin x sin x 5 5 5 5
Answer: 1. 3
2.
Dr. John Chung's SAT II Math Level 2
3
3.
5 2
4.
4 5
47
Tips Sum and Difference of Angles
TIP 34
Functions of the Sum of Two Angles
Functions of the Difference of Two Angles
sin( A B) sin A cos B cos Asin B
sin( A B) sin A cos B cos Asin B
cos( A B) cos A cos B sin Asin B
cos( A B) cos A cos B sin Asin B
tan( A B )
tan A tan B 1 tan A tan B
tan( A B )
tan A tan B 1 tan A tan B
PRACTICE
1. If sin x
1 1 and cos y , where x and y are positive acute angles, what is the value of cos( x y) ? 2 3
2. tan(180 y)
3. If sin A 0.6, sin B 0.8,
2
A , and B
3 , what is the value of sin( A B)? 2
EXPLANATION
1. cos( x y ) cos x cos y sin x sin y 2
3 1 1 2 2 32 2 . 2 3 2 3 6 3 y
1
x
1
3
2. tan(180 y )
tan180 tan y tan y 1 tan180 tan y
3. sin( A B) sin A cos B cos Asin B (0.6)(0.6) (0.8)(0.8) 0.28.
Answer: 1.
48
32 2 6
2. tan y
3. 0.28
2 2
Tips Double Angle Formulas
TIP 35
Functions of the double angle
cos 2 A cos2 A sin 2 A
sin 2 A 2sin A cos A
tan 2 A
2 tan A 1 tan 2 A
cos 2 A 1 2sin 2 A cos 2 A 2cos2 A 1
PRACTICE 1. If sin
4 and , what is the value of sin 2 ? 2 5
2. If sin x
5 , then cos 2 x 13
3. If cos
1 and , then tan 2 2 2
EXPLANATION 24 3 4 3 1. In Quadrant II, cos . sin 2 2 sin cos 2 25 5 5 5 2
50 119 5 2 2. cos 2 x 1 2sin x 1 2 1 169 169 13 3. In Quadrant II, tan 3 . Therefore,
tan 2
2 3 2 tan 1 tan 2 1 3
Answer: 1.
24 25
Dr. John Chung's SAT II Math Level 2
2.
2
3.
119 169
3.
3
49
Tips Half Angle formulas
TIP 36
Half Angle Formulas
sin
A 1 cos A 2 2
cos
A 1 cos A 2 2
tan
A 1 cos A 2 1 cos A
PRACTICE 1. If sin A
4 3 1 and A , what is the value of cos A ? 2 5 2
2. If cos x
5 x and x , then what is the value of cos ? 2 2 13
3. If y is a positive acute angle and sin
y 1 , then what is the value of y ? 2 2
EXPLANATION A 1 cos A A 3 3 1. Since (Quadrant II), then cos A and cos 2 2 2 2 4 5
x x 1 cos x , cos 2. Since 2 2 4 2 2 3.
y 30 2
y 60 .
Answer: 1.
50
5 1 13 2 13 . 2 13
5 5
2.
2 13 13
3. 60
3 5 5 . 2 5
1
Tips TIP 37
Trigonometric Equation
PRACTICE 1. In the interval 0 x 360 , what is the value of x that satisfies the equation 4sin 2 x 4cos x 5 0 ?
2. In the interval 0 x 360 , what is the value of x that satisfies the equation cos2 x 2sin x 1 0 ?
3. In the interval 0 x 2 , what is the value of x that satisfies the equation 2sec2 x 3tan x 1 0 ?
4. What is the measure of the positive acute angle that satisfies the equation 1 sin x 2cos2 x ?
EXPLANATION 1. 4(1 cos2 x) 4cos x 5 0 4cos2 x 4cos x 1 0 (2cos x 1)2 0
1 x 60 , 300 2 2 2. (1 sin x) 2sin x 1 0 sin 2 x 2sin x 0 sin x(sin x 2) 0 cos x
sin x 0 or sin x 2 (reject)
Therefore, x 0 , 180
3. Since sec2 x 1 tan 2 x , 2(1 tan 2 x) 3tan x 1 0 2 tan 2 x 3tan x 1 0
2 tan x 1 tan x 1 0
tan x
1 or tan x 1 2
If tan x 0.5 , then x tan 1 (0.5) 0.46 or x 0.46 3.61. 1
If tan x 1 , then x tan 1 0.79 or x 0.79 3.93. Therefore, x 0.46, 3.60 , 0.79, 3.93 4. 1 sin x 2(1 sin 2 x ) sin x
2 sin 2 x sin x 1 0
Remember:For these questions, You can use a calculator. Graph and find the zeros.
2 sin x 1 sin x 1 0
1 or sin x 1 . Therefore, the positive acute angle is 30 . 2
Answer: 1. 60 , 300
Dr. John Chung's SAT II Math Level 2
2. 0 , 180
3. 0.46, 0.79, 3.61, 3.93
4. 30
51
Tips The Law of Sines
TIP 38
Area of a Triangle:
If ABC is a triangle with sides a, b, and c, then Area
bc sin A ab sin C ac sin B . 2 2 2
C
ch c b sin A bc sin A 2 2 2 h b sin A
a
b
Area
h
A
B c
Law of Sines:
If ABC is a triangle with sides a, b, and c, then a b c sin A sin B sin C
PRACTICE 1. In ABC , what is the ratio a : b if A 30 and B 45o ?
2. How many possible triangles can be constructed if a 10, b 12, and B 20 ?
52
Tips 3. How many distinct triangles can be constructed if the measures of two sides are 4 and 6 and the measure of the
angle opposite the smaller side is 30 ?
EXPLANATION 1. Law of sines: a b sin 30 sin 45
2.
10 12 sin A sin 20
1 a b a 2 2 12 b 2 2 2 2 2 10sin 20 sin A 0.28 : A lies in Quadrant I or II. 12
A sin 1 0.28 16 or 164o
10sin 20 3.4 Or
A
B
C
16
20
164
20
144 No triangle
4 6 6sin 30 B 0.75 : sin 4 sin 30 sin B
3.4 20
B
Therefore, only one triangle can be constructed. 3.
a 10
Since b 12 is greater than 10, one triangle.
B lies in Quadrant I or II.
A sin 1 0.75 49 or 131
A
B
C
30
49
101
30
131
6sin 30 3 Or 6 4
19
4
3
30
Therefore, two triangles can be constructed.
Since 3 4 6, two triangles.
Answer: 1.
2 2
2. One triangle
Dr. John Chung's SAT II Math Level 2
3. Two triangles
53
Tips The Law of Cosines
TIP 39
Law of Cosines:
a 2 b 2 c 2 2bc cos A
cos A
b2 c2 a2 2bc
b2 a2 c2 2ac cos B
cos B
a 2 c 2 b2 2ac
c2 a2 b2 2ab cos C
cos C
a 2 b2 c 2 2ab
PRACTICE 1. In ABC , if a 3 and b 1 , and C 30 , then c
2. If two forces of 30 pounds and 40 pounds act on a body with an angle of 120 between them, what is the magnitude of the resultant?
EXPLANATION 1. The Law of cosines:
c2
3
2
1 2 2
3 (1) cos 30
c2 1 c 1
2. In a parallelogram, two consecutive angles are supplementary and opposite sides are .
30
120
30
R
60 40
R 2 302 402 2(30)(40) cos 60 1300 , Answer: 1. c 1
54
2. 36.06 lb
R 1300 36.06lb
Tips Permutation
TIP 40
A permutation of a set of values is an arrangement where order is important.
The number of ways of obtaining an ordered r elements from n elements is given by n
Pr
n! n r !
PRACTICE
1. In how many ways can a class with 15 students choose a president, a vice-president, and a treasurer?
2. In how many different orders can the program for a music concert be arranged if 6 students are to perform?
3. How many 8-letter arrangements can be made from the letters in the word PARALLEL?
EXPLANATION 1.
15 P3
2.
6 P6
15! 15 14 13 2730 12!
6! 720
3. Permutations with repetition: This is a permutation of 8 letters taken at a time when 2A and 3L are identical.
Therefore,
8! 8 P8 3360 2!3! 2!3!
Answer: 1. 2730
Dr. John Chung's SAT II Math Level 2
2. 720
3. 3360
55
Tips Combination
TIP 41
A selection in which order is not important is called a combination. The number of combinations of n things taken r at a time is
1.
n Cn
n Cr
n Cr
1
n Pr
r!
or
n n Pr r r!
n! n r ! r !
2.
n Co
1
3.
n Cr
n Cnr
PRACTICE 1. There are 10 novels and 6 biographies in a book reading list. If a student chose 5 novels and 3 biographies to read, how many different combinations can be chosen?
2. There are 6 boys and 5 girls in a chess club. In how many ways can 2 boys and 3 girls be selected to attend the regional tournament?
n 3. If n Pr 120 and 20 , then what is the value of r ? r
EXPLANATION 10 6 1. 5, 040 5 3 6 5 2. 150 2 3 n P 120 20 3. n r r! r r! Answer: 1. 5040
56
2.
Therefore, r ! 6 r 3 . 150
3. 3
Tips Dividing Group
TIP 42
When we divide group into several groups, 1) Each group has different number of people How many ways to divide group of 10 people into one group of 7 people and the other group f 3 people?
The number of ways 10 C7 3 C3 120 2) Some groups has same number of people How many ways to divide of 10 people into two groups of 3 people and one group of 4 people? The number of ways
10 C 3
7 C3 4 C 4 2100 2!
Because two groups has the same number of people,
10 C3 7 C3 4 C4
should be divided by 2!.
If the number of people n a a a b b c , then the number of ways divides into the groups of
a, a, a, b, b and c is n n a n 2a n 3a n 3a b n 3a 2b a a a b b c The number of ways (3!)(2!)
PRACTICE 1. How many ways are there to divide 8 people into two groups of three people and one group of two people?
2. Eight people are divided into one groups of 5 people and the other group of 3 people. How many ways are there?
EXPLANATION 8 5 2 1 (56)(10)(1) 280 1. 2 3 3 2 2! 8 3 2. 56 5 3
Dr. John Chung's SAT II Math Level 2
57
Tips Binomial Expansion Theorem
TIP 43
Formula for expanding x y for positive integers n is n
x y n n Co x n y 0 n C1 x n 1 y 1 n C2 x n 2 y 2 n Cn 1 x 1 y n 1 n Cn x 0 y n 1. For any binomial expansion of
x y n , there are
n 1 terms.
2. The general term of the expansion is n Cr
x n r y r
or
n nr r x y , where r 0, 1, 2, 3 n. r
PRACTICE 1. What is the third term of the expansion of a 2b ? 4
2. What is the middle term of the expansion of x 2 y ? 6
10
1 3. In the expansion of x 2 2 , what is the value of the constant term? x
EXPLANATION 4 2 2 1. The third term is a 2b 6(2)2 a 2b2 24a 2b2 . 2 6 3 3 2. The middle term is x 2 y 160 x3 y 3 . 3 10 3. x 2 r
x 10 r
2 r
10 10 r r 1 x 20 2 r 2 r 1 x 20 4 r , 20 4r 0 r 5 r r
10 5 Therefore, the constant term is 1 252 . 5 Answer: 1. 24a 2b2
58
2.
160x3 y 3
3.
252
Tips TIP 44
Sum of Coefficients of a Binomial Expansion
From Binomial Expansion Theorem below
x y n n Co x n y 0 n C1 x n 1 y 1 n C2 x n 2 y 2 n Cn 1 x 1 y n 1 n Cn x 0 y n , Sum of Binomial Coefficients (SBC) is n Co
n C1 n C2 n C3 n C4 n Cn1 n Cn .
Since Binomial expansion is true for all real values of x and y, when you put x 1 and y 1 in the expansion,
1 1n n Co 1n 10 n C1 1n 1 11 n C2 1n 2 12 n Cn 1 11 1n 1 n Cn 10 1n or
2n n Co n C1 n C2 n Cn 1 n Cn . Therefore, SBC 2n.
Example 1: What is the sum of coefficients in the binomial expansion of
2 x 3 y 3
?
3 Putting x 1 and y 1 , SBC 2 3 1.
Check:
2 x 3 y 3 8 x 3 36 x 2 y 54 xy 2 27 y 3
SBC 8 ( 36) 54 ( 27) 1
PRACTICE 1. What is the sum of coefficients in the binomial expansion of 1 2 x ? 8
2. What is the value of
n Co
n C1 n C2 n C3 n C4 n Cn1 n Cn ?
EXPLANATION 1. Putting x 1: SBC 1 2 1 1 8
2.
8
x y n n Co x n y 0 n C1 x n 1 y 1 n C2 x n 2 y 2 n Cn 1 x 1 y n 1 n Cn x 0 y n Putting x 1 and y 1: SBC 1 1 2 n n
Answer: 1. 1
Dr. John Chung's SAT II Math Level 2
2.
2n
59
Tips Binomial Probability
TIP 45
If the probability of success is p and the probability of failure is 1 p q , then the probability of exactly r successes in n independent trials is:
P n Cr p r q n r
PRACTICE 1. If a fair coin is tossed 10 times, what is the probability of rolling a head exactly 7 times?
2. If a fair coin is tosses 5 times, what is the probability of obtaining at most 3 heads?
3. A coin is loaded so that the probability of heads on a single throw is three times the probability of tails. What is the probability of at most 3 heads when the coin is tossed 6 times?
EXPLANATION 7
3
15 1 1 0.12 1. P 10 C7 2 2 128 0
5
1
4
2
3
3
2
26 1 1 1 1 1 1 1 1 0.8125 2. P 5 C0 5 C1 5 C2 5 C3 2 2 2 2 2 2 2 2 32 Or, binomcdf (5, 0.5, 3) 0.8125 (Graphing Calculator) 3. Since P(H ) 0.75 and P(T ) 0.25 , the probability is: 6 C0
0.75 0 0.25 6 6 C1 0.75 1 0.25 5 6 C2 0.75 2 0.25 4 6 C3 0.75 3 0.25 3
= binomcdf (6, 0.75, 3) = 0.16943359 0.17 Answer: 1.
60
15 128
2.
13 16
or 0.8125
3. 0.17
Tips Probability with Combinations
TIP 46
If 5 cards are drawn at random from a standard deck, what is the probability that all 5 cards are hearts? Method 1:
Since there are 13 hearts in the 52-card deck, the probability that the first card drawn is a heart is probability that the second card drawn is a heart is P (all hearts)
13 and the 52
12 . Therefore, continuing in this way, the probability is 51
13 12 11 10 9 0.000495 52 51 50 49 48
Method 2:
Using combinations:
P (all hearts)
13 C5 52 C5
0.000495
PRACTICE 1. If three marbles are picked at random from a bag containing 4 red marbles and 5 white marbles, what is the probability that exactly 2 marbles are red?
EXPLANATION
1. Method 1: There are three ways to pick: RRW ,
The probability of RRW :
RWR, WRR
4 3 5 5 5 5 . Each has the same probability. Therefore, 3 9 8 7 42 42 14
Method 2: Using combinations. P
5 C1 30 5 84 14 9 C3
4 C2
Answer: 1.
5 14
Dr. John Chung's SAT II Math Level 2
61
Tips TIP 47
Heron’s Formula
Given any triangle with sides a, b, and c , the area of the triangle is given by Area
, where s
s ( s a )( s b )( s c )
abc . 2
PRACTICE 1. Find the area of a triangle having sides of lengths 5, 12, and 15.
EXPLANATION 1.
Using Heron’s Formula: s
5 12 15 16 , 2
Area 16(16 5)(16 12)(16 15) 26.53
Using the law of cosines: In the figure, find angle using the law of cosines. cos
52 152 122 106 2(5)(15) 150
The area of the triangle
62
106 45.036 150
cos 1
ab sin 2
(5)(15) sin 45.036 26.53 2
Answer: 1. 26.53
15
5 12
Tips TIP 48
Vectors in the Plane
For a vector V AB A
Directed Line Segment
Initial Point
B Terminal Point
1. If the vectors have the same magnitude and direction, then they are equivalent. 2. Magnitude of a vector is the length of the segment: Distance formula. If A (0, 0) and B (3,4) , then AB (3 0) 2 (4 0) 2 5 . 3. If a vector has an initial point in the origin, then the vector is in standard position. V (v1 , v2 ) is the component form in standard position and V v12 v2 2 .
4. Unit vector u has a magnitude of 1 and the same direction as V . V u : Divide a vector by its length. V
PRACTICE 1. Find the unit vector in the direction of V (3, 4) .
2. If A is represented by the directed line segment from P (2,3) to Q (2, 8) , what is the magnitude of vector A ?
Dr. John Chung's SAT II Math Level 2
63
Tips 3. If a (2,3) and b (4,6) , then what is the value of 2a b ?
EXPLANATION
1. V is in standard position and the initial point is at (0, 0). V (3)2 42 5 V ( 3, 4) Therefore, u 0.6, 0.8 5 V 2.
PQ (2, 8) (2, 3) ( 4, 5) PQ (4)2 52 41 Or, use distance formula:
D (2 2)2 (3 8)2 41 3. 2a b 2(2,3) (4,6) (0,12) Therefore, 2a b 02 122 12
Answer: 1.
64
0.6, 0.8
2.
41
3. 12
Tips Interchange of Inputs
TIP 49
Example 1. If f ( x) 5x 2, what is f ( x 5)? x
y y f ( x)
Input
x5
Output
y y f ( x 5)
Input
Output
Interchange of input means replacement x with x 5. If f ( x) 5x 2, then f ( x 5) 5( x 5) 2 5x 27. The best way to interchange the input is as follows.
Step 1: Change the input variable of original function: f (k ) 5k 2 Step 2: Find relationship between two inputs: k x 5 Step 3: Replace with a new input and simplify: f ( x 5) 5( x 5) 2 5x 27 x3 , then f ( x 1) 5 k 3 Step 1: f ( k 5) Step 2: k 5 x 1 k x 4 5 ( x 4) 3 x 1 Step 3: f ( x 1) 5 5
Example 2. If f ( x 5)
PRACTICE
x 1. If f (2 x) x2 5, what is f ? 2
x 1 2. If f 2 x 3, then f ( x) 4
Dr. John Chung's SAT II Math Level 2
65
Tips x 3. If f 5 x 2 1, then 2
f (2 x 1)
EXPLANATION 1. Step 1: f (2 x) x 2 5
Step 2: 2 k
f (2k ) k 2 5
x x k 2 4 2
x x Step 3: f 5 2 4
2 x x f 5 2 16
k 1 2. Step 1: f 2k 3 4
Step 2:
k 1 x k 1 4x k 4x 1 4
Step 3: f ( x) 2(4x 1) 3
x 3. Step 1: f 5 x 2 1 2
Step 2:
k 5 2x 1 2
2
66
k f 5 k2 1 2
k 2x 4 2
Step 3: f (2 x 1) 4 x 8 1
Answer: 1.
f ( x) 8x 5
2 x x f 5 2 16
2.
k 4x 8
f (2 x 1) 16 x 2 64 x 63
f ( x) 8x 5
3.
f (2 x 1) 16 x2 64 x 63
Tips Polynomial Inequalities
TIP 50
Graphical Solution: 1. Solve
( x 4)( x 2)( x 2) 0 .
Step 1. Find critical points. ( x 4)( x 2)( x 2) 0 x 4, 2, 2 Step 2. Graph y ( x 4)( x 2)( x 2) . y0
y0 x
y0
2
2
y0
4
Step 3) From the graph above, find the intervals for y 0 . The solution is 2 x 2 or x 4 . 2 2. Solve ( x 4)( x 2) ( x 2) 0 .
Step 1. Find the critical points. ( x 4)( x 2)2 ( x 2) 0 x 4,
2, 2
2
Step 2. Graph y ( x 4)( x 2) ( x 2) . Zeros: x 4, x 2(two equal roots), and x 2. y0
y0 2
y0
2
y0
4
Step 3. Find the intervals for y 0 . The solution is 2 x 2 or 2 x 4 .
Dr. John Chung's SAT II Math Level 2
67
Tips PRACTICE 1. Solve the inequality ( x 1)( x 2)2 ( x 1)( x 2)2 0 .
2. What is the solution set of
2 x3 3x2 24 x 12 ?
EXPLANATION 1. Graphical solution: use graphing utility. y
y0
2
y0
1
O
1
From the graph, the solution set is 2. By using a graphing utility,
y0
y0 x
2
, 1 1,
y
y0
y0 3.06
x
O 0.48
4.08
The solution set is 3.06, 0.48 4.08, . Answer: 1.
68
x 1 or x 1
2.
3.06,
0.48 4.08,
Tips TIP 51
Rational Inequalities
Graphic Solution: 1. Solve the rational inequality
( x 1)( x 2) 0. x2
Method 1. Test value using critical points
Critical points from the equation:
( x 1)( x 2) 0 x 1 and x 2 (closed) x2 x 2 (open)
Critical points from undefined: There are four possible solution set
2
1
2
Test value
Results
x 3
( 3 1)( 3 2) 0 3 2
No
x0
(0 1)(0 2) 0 02
Satisfied
x 1.5
(1.5 1)(1.5 2) 0 1.5 2
No
x3
(3 1)(3 2) 0 32
Satisfied
Solution: 2 x 1 or
x2
Method 2. Using the graph
Since
x 2 2
x 2 2
0 , multiply both sides by x 2 . 2
( x 1)( x 2) 2 0 x 2 x2
2
1
x 2 x 1 x 2 0
2
From the graph above, the solution set is 2, 1 2, .
Dr. John Chung's SAT II Math Level 2
69
Tips PRACTICE
1. Solve the inequality
( x 2)2 ( x 3) 0. ( x 1)( x 2)
EXPLANATION 1. Method 1) Multiply both sides by ( x 1) 2 ( x 2)2 which is positive.
( x 1)( x 2)( x 2) 2 ( x 3) 0 and x 1 , x 2 y y ( x 1)( x 2)( x 2)2 ( x 3)
1 2 3
2
x
Therefore, the solution set is x 2 1 x 3 .
Method 2) Use a graphing utility.
y
2
1
3
From the graph, the solution set is x 2 1 x 3 .
70
x
Tips Limit
TIP 52
lim f ( x) L x c
The limit of f ( x) is L as x approaches c . 1. If f ( x) is a polynomial function, then lim f ( x) f (c) : direct substitution. x c
2. If f ( x) is a rational function given by
lim f ( x) lim x c
x c
3. If f ( x )
N ( x) such that D(c) 0 , then D( x)
N ( x ) N (c ) D( x) D(c)
1 1 , then lim or : the limit does not exist. x 0 x x
x3 x x ( x 2 1) lim lim x 1 x 1 x 2 1 x 1 x 2 1 x 1
4. lim
1
1 5. e lim(1 x ) x or e lim 1 x x0 x
x
PRACTICE
1.
lim
x 1
2. lim
x 2
3. lim
x 1
4. lim
x 3
x2 x 6 = x2
x2 x 6 x2
x2 2 x 3 x 1
x3 x2 9
Dr. John Chung's SAT II Math Level 2
71
Tips x4 2 x
5. lim
x 0
EXPLANATION x2 x 6 1 1 6 6 : (direct substitution) x 1 x2 1 2
1. Since D(1) 0 , lim
2. Since D(2) 0 and N (2) 0 , lim
x 2
( x 3)( x 2) lim( x 3) 5 x 2 ( x 2)
3. Rationalizing Skill: Since D(1) 0 and N (1) 0 , then lim
x 3 x 1
x 1
x 1
lim x 3 x 1
x 1
x 1
lim x 3
x 1
x 1
x 1
x 1
x 1 8 .
x 3 1 1 lim . x 3 x 3 x 3 x 3 x 3 6
4. Since D(3) 0 and N (3) 0 , then lim 5. Rationalizing Skill: D(0) 0 and N (0) 0
lim
x 0
x4 2 x
x42
x4 2
Answer: 1. 6
72
2. 5
lim x 0
x
x4
3. 8
2
22
x42
4.
1 6
lim
x 0
x
x42
5.
1 4
x
lim
x 0
1 x42
1 4
Tips TIP 53
Rational Function and Asymptote
A rational function can be written in the form R ( x)
N ( x) , where N ( x) and D(x) are polynomials D( x)
and D( x) 0. 1. The line x a is a vertical asymptote of the graph if D(a) 0 . 2. The line y b is a horizontal asymptote of the graph if R( x) b as x or x . y 1 Example 1: R ( x ) x Domain: , 0 (0, ) or x 0
Range: , 0 0, or y 0 Vertical asymptote: x 0 Horizontal asymptote: y 0 2x x 1 D( x) 0 , vertical asymptote: x 1
Example 2: f ( x)
2x 2 , horizontal asymptote: y 2 x 1 Domain: x 1 Range: y 2 lim
x
O
y
2
x
Example 3: g ( x)
1
2x x 1
x
2
y
D( x) 0 x 2 1 0 , no vertical asymptote. 2x 0 , horizontal asymptote: y 0 . x x 1 Domain: all real x Range: 1 y 1 (using graphing utility) lim
1
2
O
x
1
3. Slant asymptote (or, oblique asymptote): If the degree in the numerator is greater than the degree in denominator, the original function should be rearranged by long division. 2 x 2 3x 1 6 f ( x) 2x 5 The slant asymptote is y 2x 5 . x 1 x 1
Dr. John Chung's SAT II Math Level 2
73
Tips y Example 4: h( x)
h( x )
2
x 2x 3 x2 1
x 3 x 1 x 3 x 1 x 1 x 1
D( x ) 0 y lim
x
(1, 2) y 1
x 1:vertical asymptote
O
x
x3 1 y 1:horizontal asymptote x 1
f (1) is still undefined. Domain: (, 1) (1, 1) (1, )
x 1
Range : (, 1) (1, 2) (2, )
Example 5: f ( x)
x2 x x 1
f ( x) x 2
D( x) 0
2 (Use long division) x 1
y lim ( x 2) x
x 1 : vertical asymptote
2 ( x 2) x 1
y x 2 : slant asymptote
Domain: (, 1) (1, )
y
Range: (, 5.82] [0.17, ) (using graphing utility)
y x2 O
x 1
74
x
Tips PRACTICE 1. Find all asymptotes of the rational function f ( x)
2. If g ( x )
2 x2 1 ? x
x2 x 2 , then its vertical asymptote(s) is(are) x2 2 x 3
3. If the vertical asymptote of the rational function R ( x )
x 2 3x b is x 1 , what is the value of b ? ( x 1)( x 1)
EXPLANATION 1. Since f ( x) 2 x
2. g ( x)
1 , vertical asymptote: x 0 , and slant asymptote: y 2x . x
( x 2) ( x 1) ( x 3) ( x 1)
( x 2) ( x 3)
vertical asymptote: D( x) 0 x 3
3. Since x 1 is not asymptote, the numerator must have a factor of ( x 1) . Therefore,
f (1) 0
Answer: 1. x 0 ,
1 3 b 0
y 2x
2.
Dr. John Chung's SAT II Math Level 2
b2 x 3
3.
b2
75
Tips Parametric Equations
TIP 54
Parametric equations define a relation using parameters. Conversion from two parametric equations to a single equation: Eliminating the parameter from the simultaneous equations.
Parametric Solve for t in Substitute in Rectangular equations one equation second equation equation Example 1: If x a cos t and y b sin t , what is the graph of the parametric equations?
cos t
x a
and sin t
y b
To eliminate the parameter t ,
cos2 t sin 2 t 1
x2 y 2 1 : Ellipse a2 b2
PRACTICE 1. What is the curve given by the parametric equations x t and y t 2 ?
2. What is the graph given by the parametric equations x sec and y tan ?
EXPLANATION 1. t x2 y x2 2, x 0 (Domain is restricted because x t ) 2. Since 1 tan 2 sec2 , 1 y 2 x 2 x 2 y 2 1: hyperbola y
y x 2 2, ( x 0)
2 O
76
x Answer: 1. Parabola
2. Hyperbola
Tips Polar Coordinates
TIP 55 y
(r , )
Polar Coordinates and Rectangular Coordinates Conversions x r cos
r
O
y r sin y tan x
y
x
x
tan 1
y x
r 2 x2 y 2 r x2 y 2
PRACTICE 1. If the coordinates of a polar point are 5, , what are its rectangular coordinates? 2
2. If the rectangular coordinates are ( 3, 1) , what are its polar coordinates?
Equation Conversion: 3. What is the graph of the polar equation r 5 ?
4. What is the graph of the polar equation r sec ?
5. Convert the polar equation r
6 to rectangular form. 2 3sin
6. Convert the rectangular equation 3x 6 y 2 0 to polar form.
Dr. John Chung's SAT II Math Level 2
77
Tips 7. Convert the rectangular equation x2 y 2 6 x 0 to polar form.
EXPLANATION
1. x r cos 5cos
3
2. r
2
2
0 and y r sin x 5sin
2
5. Therefore, (0,5) .
1 30 . Therefore, 2, . 6 6 3
tan 1
12 2 ,
3. r x 2 y 2 5 x 2 y 2 25 4. r sec
Therefore, r
5. r
sec
1 x , cos cos r
sec
r x
r x 1. x
6 6 r 2r 3 y 6 2r 3 y 6 y 2 3sin 23 r
2 x2 y 2 3 y 6 4 x2 y 2 3 y 6
2
4 x 2 4 y 2 9 y 2 36 y 36
Therefore,
4 x 2 5 y 2 36 y 36 0. Hyperbola 6. 3x 6 y 2 0 3(r cos ) 6(r sin ) 2 0 r (3cos 6sin ) 2
r
2 3cos 6sin
7. x2 y 2 r 2 and x r cos . Therefore, r 2 6r cos r 6cos .
Answer: 1. (0, 5)
2. 2, or 6
11 2, 6
5. 4 x2 5 y 2 36 y 36 0 , Hyperbola
78
3. x2 y 2 25 , circle 6. r
2 3cos 6sin
4. x 1 7. r 6 cos
Tips Matrix
TIP 56
A. Order of Matrix: An m n ( m by n ) matrix is a rectangular array of numbers arranged in m rows (horizontal lines) and n columns (vertical lines). 5 2 2 1 1 5
This matrix has three rows and two columns. The order of the matrix is 3 2 (3 by 2).
B. Addition of Matrices: If two matrices have the same order, then you can add two matrices by adding their corresponding entries.
1 2
3 2 2 1 2 4 1 3 2 1
3 2 1 4 3 3
1 1
C. Scalar Multiplication:
1 3 3 3 4 6 2
9 12
D. Scalar Multiplication and Matrix Subtraction:
1 If A 2
2 2 4 and B , then 2 A 3 B is 3 3 2
1 2 2 4 2 2 3 2 3 3 2 4
4 6 12 4 8 6 9 6 13 0
E. Multiplication of Matrices: Row by Column Multiplication The number of columns of the first matrix must equal to the number of rows of the second matrix.
2 3 1 2 1 (3) 3 7 1 2 3 (1) 1 2 3 5
Dr. John Chung's SAT II Math Level 2
79
Tips
A m n Order of A
B
n p Order of B
AB m p Order of AB
F. Determinant of a 2 2 matrix: a c
b ad bc d
PRACTICE 2 3 1. What is the determinant of the matrix A ? 2 1
2. If matrices A, B, C, and D have orders of 2 3, 2 3, 3 2, and 2 2 , respectively, what are the orders of following operations?
II) D( A 3B)
I) A(2B)
III) ( BC D) A
EXPLANATION
1.
2 3 2 2 (3)(1) 4 3 1 1 2
2. 1) A(2 B )
2 3 2 3 :
2) D( A 3B)
Invalid
2 2 2 3 2 3
3. BC D A BC :
2 3 3 2 2 2 , BC D : ( BC D) A : 2 2 (2 3) (2 3)
80
2 2
Tips TIP 57
Inclination Angle
Inclination angle is the angle measured counterclockwise from the x-axis to the line.
y y mx b m tan
Inclination angle O
Inclination Angle:
x
Horizontal line: 0 Vertical line : 90 Positive slope : Acute angle Negative slope: Obtuse angle
PRACTICE 1. If a line has an equation of 2 x 3 y 5 , what is the inclination angle of the line?
EXPLANATION 2 5 2 x m 3 3 3 2 2 tan tan 1 33.69006753 33.7 3 3
1. 2 x 3 y 5 y
Answer: 1. 33.7o
Dr. John Chung's SAT II Math Level 2
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Tips Angle between Two Lines
TIP 58
y
y m2 x b2 y m1 x b1
2 1 x
O
1. 2 1
2. tan 1 m1
tan tan 2 1
3. tan 2 m2
tan 2 tan 1 m m1 2 1 tan 2 tan 1 1 m2 m1
m m1 Therefore, tan 1 2 . 1 m2 m1
PRACTICE 1.
Find the angle between the two lines y x 2 and y 2 x 1 .
EXPLANATION y
y x 2
y 2x 1
1 O
2
m2 tan 2 1 and m1 tan 1 2
2 1 x
tan
m2 m1 1 2 3 1 m1m2 1 (1)(2)
tan 1 3 71.56505118 71.6 Or, find the angles directly. 2 tan 1 (1) 135 and 1 tan 1 (2) 63.4 135 63.4 71.6
82
Tips TIP 59
Intermediate Value Theorem
If f is continuous on a closed interval [ a, b] and k is any number between f (a) and f (b), then there is at least one number c in a, b such that f (c) k. 1. If f (a) 0 and f (b) 0 on a closed interval a, b , then there are at least one real zeros somewhere between a and b .
PRACTICE 1. If P( x) x3 4 x2 6 , what are the intervals needed to guarantee real zeros?
(Use the Intermediate Value Theorem to find intervals of length 1.)
PRACTICE 1. Using a graphic utility,
Since f (2) 18 0 and f (1) 1 0 , there is a zero between 2 and 1.
f (1) 3 0 and f (2) 2 0 , there is a zero between 1and 2. f (3) 3 0 and f (4) 6 0 , there is a zero between 3 and 4.
Answer: 1. 2, 1 , 1, 2 , 3, 4
Dr. John Chung's SAT II Math Level 2
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Tips TIP 60
Rational Zero Test
If the polynomial f ( x) an x n an 1 x n 1 a1 x ao has integer coefficients, the possible rational zeros of f are
Possible Rational Zeros =
factors of constant term factors of leading coefficient
PRACTICE 1. What is all possible rational zeros of f ( x) x3 x 1 ?
2. Find the possible rational zeros of f ( x) 2 x3 3x 2 8 x 3 .
PRACTICE
1.
Factors of constant term 1 1 Factors of leading coefficients 1 By testing these zeros, you can see that neither works. f (1) 1 and f (1) 1 Therefore, you can conclude that the polynomial function has no rational zeros. The real zero is x 1.324718 irrational root.
2.
1, 3 1 3 1, 3, , 2 2 1, 2 By using a graphing utility, you can find the real rational zeros of f are 1 x 1, x , and x 3 . 2 Answer: 1. 1
84
1 3 2. 1, 3, , 2 2
Tips TIP 61
Descartes Rule of Sign
If f ( x) an x n an 1 x n 1 a1 x ao with real coefficients and a0 0, 1. The number of positive zeros of f is either equal to the number of variations
in sign of f ( x) or less than the number by an even integer. 2. The number of negative zeros of f is either equal to the number of variations
in sign of f ( x) or less than the number by an even integer.
PRACTICE 1. Find the possible real zeros of f ( x) 4 x3 6 x 2 3x 3 ?
PRACTICE 3 2 1. Check the number of variations in sign of f ( x) 4 x 6 x 3x 3
to to to 4 6 6 3 3 3 Three variations in sign.
3 2 Check the number of variations in sign of f ( x) 4 x 6 x 3x 3 No variations in sign.
Conclusion: 1) Three positive zeros and no negative zeros. or 2) One positive real zero and no negative zeros. (One positive real zero and two imaginary roots)
Answer: 1. Three positive real zeros or one positive real zero and has no negative real zeros.
Dr. John Chung's SAT II Math Level 2
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Test 1 Dr. John Chung’s SAT II Mathematics Level 2
Dr. John Chung's SAT II Math Level 2 Test 1
87
2
2
MATHEMATICS LEVEL 2 TEST REFERENCE INFORMATION
THE FOLLOWING INFORMATION IS FOR YOUR REFERENCE IN ANSWERING SOME OF THE QUESTIONS IN THIS TEST 1 Volume of a right circular cone with radius r and height h : V r 2 h 3 1 Lateral Area of a right circular cone with circumference of the base c and slant height : S c 2
Volume of a sphere with radius r : V 4 r 3 3 Surface Area of a sphere with radius r : S 4 r 2 1 Volume of a pyramid with base area B and height h : V 3 Bh
88
2
2 Dr. John Chung’s SAT II Math Level 2
Answer Sheet 01 02
A
B
C
D
E
A
B
C
D
E
03
A
B
C
D
E
B
C
D
04 05
A A
B
C
06
A
B
07
A
08
26 27
A
B
C
D
E
A
B
C
D
E
A
B
C
D
E
E
28 29
A
B
C
D
E
D
E
30
A
B
C
D
E
C
D
E
A
B
C
D
E
B
C
D
E
31 32
A
B
C
D
E
A
B
C
D
E
33
A
B
C
D
E
09
A
B
C
D
E
34
A
B
C
D
E
10 11
A
B
C
D
E
A
B
C
D
E
A
B
C
D
E
35 36
A
B
C
D
E
12
A
B
C
D
E
37
A
B
C
D
E
13 14
A
B
C
D
E
B
C
D
E
A
B
C
D
E
38 39
A
A
B
C
D
E
15
A
B
C
D
E
40
A
B
C
D
E
16 17
A
B
C
D
E
A
B
C
D
E
A
B
C
D
E
41 42
A
B
C
D
E
18
A
B
C
D
E
B
C
D
E
19 20
B
C
D
E
43 44
A
A
A
B
C
D
E
A
B
C
D
E
A
B
C
D
E
A
B
C
D
E
A
B
C
D
E
21 22
45 46
A
B
C
D
E
47
A
B
C
D
E
23
A
B
C
D
E
48
A
B
C
D
E
24
A
B
C
D
E
49
A
B
C
D
E
25
A
B
C
D
E
50
A
B
C
D
E
The number of right answers:
The number of wrong answers:
# of correct
Dr. John Chung's SAT II Math Level 2 Test 1
1 4 # of wrong
Raw score
89
2
2 Score Conversion Table 6
Scaled Score 480
630
5
470
26
620
4
470
800
25
620
3
460
46
800
24
610
2
460
45
800
23
610
1
450
44
800
22
600
0
450
43
800
21
600
42
800
20
590
41
800
19
590
40
780
18
580
39
760
17
570
38
750
16
560
37
740
15
550
36
720
14
540
35
710
13
530
34
700
12
520
33
690
11
510
32
680
10
500
31
670
9
490
30
660
8
490
29
650
7
480
28
Scaled Score 640
800
27
48
800
47
50
Scaled Score 800
49
Raw Score
90
Raw Score
Raw Score
2
2
MATHEMATICS LEVEL 2 TEST For each of the following problems, decide which is the BEST of the choices given. If the exact numerical value is not one of the choices, select the choice that best approximates this value. Then fill in the corresponding circle on the answer sheet Note: (1) A scientific or graphing calculator will be necessary for answering some (but not all) of the questions in this test. For each question you will have to decide whether or not you should use a calculator. (2) For some questions in this test you may have to decide whether your calculator should be in the radian mode or the degree mode. (3) Figures that accompany problems in this test are intended to provide information useful in solving the problems. They are drawn as accurately as possible EXCEPT when it is stated in a specific problem that its figure is not drawn to scale. All figures lie in a plane unless otherwise indicated. (4) Unless otherwise specified, the domain of any function f is assumed to be the set of all real numbers x for which f ( x) is a real number. The range of f is assumed to be the set of all real numbers f ( x) , where x
is in the domain of f . (5) Reference information that may be useful in answering the questions in this test can be found on the page preceding Question 1. USE THIS SPACE FOR SCRATCHWORK 1.
If a( x 2) b( x 1) 3 for all x , then a (A) 1
2.
(C) 1
(D) 2
(E) 3
If a b 2 and ab 1 , then a 2 b 2 (A) 4
3.
(B) 0
(B) 5
(C) 6
(D) 8
(E) 10
If the graph of 3x 4 y 5 is perpendicular to the graph of kx 2 y 5 , then k (A) 2 (B) 2.67 (C) 2.15 (D) 3.20 (E) 4
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Dr. John Chung's SAT II Math Level 2 Test 1
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2
2 MATHEMATICS LEVEL 2 TEST - Continued USE THIS SPACE FOR SCRATCHWORK.
4.
If K
(A) (B) (C) (D) (E)
5.
AB , then B A B
A 1 A AK A K AK KA Ak A A K AK
If log 3 a , then log 90 (A) 1 2a (B) (C) (D) (E)
10a 2 10 2a 30a 10 3a
` 6.
If f ( x) 3ln x and g ( x) e x , then g f ( x) (A) 3x (B) e x (C) e2x (D) x3 (E) x 2 1 20 cm
7.
In Figure 1, the slant height of a regular circular cone is 20 cm and the radius of the base is 10 cm . Find the volume of the cone? (A) 1813.8 cm3
(B) 3000.5 cm3 (C) 4120.4 cm3
(D) 7024.8 cm3
(D) 7046.6 cm3
10 cm Figure 1
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2
2 MATHEMATICS LEVEL 2 TEST - Continued USE THIS SPACE FOR SCRATCHWORK.
8.
If 2 i is one of the zeros of the polynomial p ( x ) , then a factor of p ( x) could be (A) x 2 2 (B) x 2 4 (C) x 2 4 x 4 (D) x 2 4 x 5 (E) x 2 4 x 3
9.
When a polynomial function f ( x) x 2 5 x k is divided by ( x 2) , the remainder is 5. What is the value of k ? (A) (B) (C) (D) (E)
10.
19 18 16 10 9
Figure 2 shows a cube with edge of length 6, what is
P
the length of diagonal PQ ? (A) 18 (B) 15
Q
(C) 6 6 (D) 6 3 Figure 2
(E) 6 2
y
11.
An equation of line in figure 3 is (A) 3x 4 y 4 0 (B) 3x 4 y 4 0
4
(C) 4 x 3 y 4 0 (D) 4 x 3 y 12 0 (E) 4 x 3 y 12 0
O
3
x
Figure 3
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2
2 MATHEMATICS LEVEL 2 TEST - Continued USE THIS SPACE FOR SCRATCHWORK.
12.
The mean score of 10 students of an algebra class was 85. When two new students enrolled, the mean increased to 86. What was the average of the new students? (A) (B) (C) (D) (E)
13.
88 89 90 91 92
If sin cos
1 , then tan cot 2
(A) 4.12 (B) 2.67 (C) 1.35 (D) 2.67 (E) 4.12 14.
3x What is the period of the function f , where f ( x) 5 cos 2 (A) 3 2 (B) 3 4 (C) 3
?
(D) 2 (E)
15.
3 2
Find all of the asymptotes of y
x 2 3x 2 . x2 1
(A) x 1 (B) x 1, x 1 (C) x 1, y 1 (D) x 1, y 1 (E) x 1, x 1, and y 1
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2
2 MATHEMATICS LEVEL 2 TEST - Continued USE THIS SPACE FOR SCRATCHWORK.
16.
If f (2 x) (A)
17.
2x 3
x , then f ( x) 3 x x (B) (C) 3 6
(D) 3x (E) 6x
If three numbers log a, log b, and log c in that order form an arithmetic progression, which of the following is true? (A) b ac (B) c ab (C) b
ac 2
(D) b 2 ac (E) b 2 a c 18.
The point of set ( x, y ) such that x 2 y 2 0 is (A) (B) (C) (D) (E)
19.
What is the sum of the geometric series 5 5 5 10 5 ? 2 4 8 (A) (B) (C) (D) (E)
20.
A circle An ellipse A hyperbola A point Two lines
30 25 20 19 18
Which of the following is an equation whose graph is the set of points equidistant from the point (0, 4) and (2, 2) ? (A) y 2 (B) x 2 (C) y x (D) y x 2 (E) y x 2 GO ON TO THE NEXT PAGE
Dr. John Chung's SAT II Math Level 2 Test 1
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2
2 MATHEMATICS LEVEL 2 TEST - Continued USE THIS SPACE FOR SCRATCHWORK.
21.
The solution set of 3x 5 10 is (A)
x 5
(B)
x 5
5 (C) x 5 3 5 (D) x 5 or x 3 (E) x 5 or x 15 y
22.
In the graph in Figure 4, which of the following could be the equation of the graph? (A) y sin x 3 (B) y sin 2 x
3
(C) y 3sin x 3 (D) y 3sin 2 x 3 1 (E) y 3sin x 3 2
23.
O
4
x
Figure 4
What is the range of f ( x) x2 5x 6 ? (A) y 0 (B) y 0.25 (C) 2 y 3 (D) y 5 (E) All real numbers
24.
If sin( x 10) cos(2 x 28) , then which of the following could be the value of x ? (A) (B) (C) (D) (E)
0 10 20 24 30
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2
2 MATHEMATICS LEVEL 2 TEST - Continued USE THIS SPACE FOR SCRATCHWORK.
25.
Which of the following is the equation of the inverse of
y 3x 1 ? (A) y log x 1 3 (B) y log( x) 1 x (C) y log3 3 (D) y log3 x 1
(E) y log 3 ( x 1)
26.
If f ( x) x2 kx 9 is always greater than 0 for all real x , which of the following could be the value of k ? (A) 10
27.
(B) 5
(C) 10
(D) 12
(E) 13
Which of the following is the center of a circle
x2 y 2 10 x 6 y 10 ? (A) (B) (C) (D) (E)
28.
5, 3 5,3 10,6 10, 6 10, 10
What is the domain of the function defined by 2 x 10 ? f ( x) 2 x 3x 2 (A) (B) (C) (D) (E)
All real numbers except 1 All real number except 2 All real numbers except 1 and 2 x5 x5
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2
2 MATHEMATICS LEVEL 2 TEST - Continued USE THIS SPACE FOR SCRATCHWORK.
29.
If x2 y 2 20 and x2 y , then x (A) (B) (C) (D) (E)
30.
3 2, 2 4 10 4,10
If f ( x) log 2 x 1 2 , then f 1 x (A) 2 x1 (B) x 2 1 (C) 2 x 2 (D) 2 x 2 1 (E) 22 x 2 x 1
31.
Figure 5 shows a right triangle. If the length of AD is 8 and
B
the length of BD is 4, then the length of AC is (A) (B) (C) (D) (E)
32.
D
8 A
C
Figure 5
If complex number z 4 6i , then z 2 (A) (B) (C) (D) (E)
33.
15.23 16.42 17.89 18.44 20.25
4
5.87 6.32 6.38 7.21 7.31
The coefficient of the middle term of the expansion of
x 3 y 6
is
(A) 540 (B) 270 (C) 135 (D) 270 (E) 540
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2
2 MATHEMATICS LEVEL 2 TEST - Continued USE THIS SPACE FOR SCRATCHWORK.
34.
If
x2 6 x 9 x , which of the following is the solution?
(A) x (B) x
1 2
(C) x (D) x
3 2
1 4
1 4
(E) No solution 35.
A sequence is defined as a1 1 and an 1 an 3 . Which of the following is the n th term of this sequence? (A) 3n 2 (B) 4 n 3 (C) 5n 4 (D) 6n 5 (E) n 2 n 1
36.
If f ( x) x2 2 x 4 for x 0 , what is the value of f 1 4 ?
(A) (B) (C) (D) (E) 37.
2 4 5 8 10
In triangle ABC , a 6, b 6 3 , and A 30 . What is the measure of C ? (A) 30 only (B) 90 only (C) 30 or 90 (D) 120 only (E) 60 or 120
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2
2 MATHEMATICS LEVEL 2 TEST - Continued USE THIS SPACE FOR SCRATCHWORK.
38.
If log 2 5 a and log 2 3 b , then log 2 75 (A)
2a b
(B) 2a b 1 a b 2 1 (D) a 2b 2 1 (E) a b 2
(C)
39.
Which of the following could be the graph of the parametric equations represented by x 3sin and y 4cos ? (A)
y
O
(C)
x
y
O
(E)
(B)
O
(D)
x
y
x
y
O
x
y
O
x
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2
2 MATHEMATICS LEVEL 2 TEST - Continued USE THIS SPACE FOR SCRATCHWORK.
40.
If the three sides of a triangle are 4, 5, and 7, then the area of the triangle is (A) 9.80
41.
If
6 7 8 10 12
50 68 70 75 80
If the 11th term of an arithmetic sequence is 30 and the 21st term is 0, what is the 10th term of the sequence? (A) (B) (C) (D) (E)
44.
n?
The radius of the base of a right circular cone is 5 and the slant height of the cone is 10. What is the surface area of the cone? (A) (B) (C) (D) (E)
43.
(D) 11.43 (E) 11.56
n 1! 56 , what is the value of n 1!
(A) (B) (C) (D) (E)
42.
(B) 9.92 (C) 10.2
lim
30 33 36 39 42
n2 1
n 1
(A) 0
n 1
(B) 1
(C) 2 (D) 4 (E) Limit does not exist.
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2
2 MATHEMATICS LEVEL 2 TEST - Continued USE THIS SPACE FOR SCRATCHWORK.
45.
If one of the roots of a polynomial function f ( x) 0 is 1
3 i , which of the following could be f ( x) ? 2
(A) 2 x 2 4 x 6 (B) x 2 2 x 10 (C) 3 x 2 6 x 4 (D) 4 x 2 8 x 7 (E) 5 x 2 10 x 7
46.
If p( x) x3 2 x2 mx n is divisible by x 2 3 x 2 , what is the value of m ? (A) (B) (C) (D) (E)
47.
2 1 0 1 2
If f ( x) cos x and g ( x) x2 3 , which of the following is not true? (A) f ( x) g ( x) 2 is an even function. (B) f ( x) g ( x) is an even function. (C) g f ( x) is an even function. (D) f ( x) g ( x) 2 is an even function. (E) f ( x 1) g ( x 1) is an even function
48.
Under which of the following conditions is x 0 ? x 1 x 2 (A) x 0 or x 2 (B) x 2 or x 1 (C) x 1 or x 0 (D) x 2 or 0 x 1 (E) 2 x 0 or x 1
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2
2 MATHEMATICS LEVEL 2 TEST - Continued USE THIS SPACE FOR SCRATCHWORK.
49.
What are the asymptotes of the hyperbola whose equation is
16 x2 25 y 2 400 0 ? 16 x 25 25 y x 16 4 y x 5 5 y x 4 16 y x 5
(A) y (B) (C) (D) (E)
50.
If kx 2 4 x 2 0 have two different real roots, which of the following is the values of k ? (A)
k 2
(B)
k 2
(C) 2 k 0 or k 0 (D) 2 k 2 (E) 1 k 3
STOP IF YOU FINISH BEFORE TIME IS CALLED, YOU MAY CHECK YOUR WORK ON THIS TEST ONLY. DO NOT TURN TO ANY OTHER TEST IN THIS BOOK.
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2
2 S ANSWERS
TEST 1 # 1 2 3 4 5 6 7 8 9 10
answer C C B B A D A D E D
# 11 12 13 14 15 16 17 18 19 20
answer E D B C D C D E C D
# 21 22 23 24 25 26 27 28 29 30
answer C E B D C B A E B D
# 31 32 33 34 35 36 37 38 39 40
answer C B A E A A C E D A
# 41 42 43 44 45 46 47 48 49 50
answer B D B D D B E D C C
Explanations: Test 1 1. (C)
a( x 2) b( x 1) 3 (a b) x 2a b 3 To be equal for all numbers x : a b 0 and 2a b 3. Therefore, ab0
2a b 3 3a 3 2. (C)
a 1
(a b) 2 a 2 b 2 2ab 22 and ab 1 . Therefore, a 2 b 2 4 2ab 4 2(1) 6.
3.
(B)
4.
(B)
5. (A)
3 5 k 5 3x 4 y 5 y x and kx 2 y 5 y x . 4 4 2 2 To be perpendicular, the product of the two slopes should be 1. 8 3 k 3k 2.67. 1 . k Therefore, 3 4 2 8
AB , KA KB AB or KA AB KB ( A K ) B. A B AK . Therefore, B A K Since K
Since log 90 log 3 3 10 log 3 log 3 log10 , log 3 a and log10 1, log 90 a a 1 2a 1.
6. (D)
3
g f ( x) e3ln x eln x x3
Dr. John Chung's SAT II Math Level 2 Test 1
105
2 7. (A)
2 202 102 300 10 3.
By Pythagorean Theorem the height of the cone is
Therefore, the volume of the cone is
r 2h
20cm
20
3
10 3 10
2
10 10 3 3
1813.8cm . 3
10 cm
Figure 1
8. (D)
Since 2 i is one of the roots, the other root must be 2 i (conjugate). Therefore, the equation with these two roots is x (2 i ) x (2 i ) ( x 2) i ( x 2) i 0. ( x 2)2 i 2 0 x 2 4 x 4 1 0 x 2 4 x 5 0
9. (E)
Remainder Theorem x 2 5 x k ( x 2)Q( x) R , where Q( x) is quotient and R 5. When x 2 , both sides have the same number. Since 22 5(2) k 5, k 9 Or, (1) use long division. x 7
x 2 x2 5x k
Therefore, k 9.
2
x 2x 7x k 7 x 14 k 14 5 (2) Synthetic Division.
2
1 5 k 2 14 1 7
Therefore, the remainder is 14 k 5 k 9 .
14 k 62 62 62 108 6 3.
10. (D)
The length of diagonal PQ is
11. (E)
If a and b are x- and y-intercept respectively, the equation of the line is Since
12. (D)
106
x y 1. a b
x y 1, the equation is 4 x 3 y 12. Therefore, 4 x 3 y 12 0. 3 4
The sum of scores of 10 students is 85 10 850. Let the average of score of two student be x .
2
2 Since
13. (B)
14. (C)
850 2 x 86 850 2 x 12(86) 2 x 182 , x 91. 12
sin cos sin 2 cos 2 1 ------- (1) cos sin sin cos sin cos 1 3 sin cos ------- (2) sin cos 2 1 2sin cos 4 8 Substitute (2) into (1) 1 8 3 3 8 tan cot
The frequency of the periodic function is
3 , because the function can be expressed as follow. 2
3 f ( x) 5 cos x , where the coefficient of x is the frequency. 2 3 2 2 4 Therefore, the period of the function is . f 32 3 15. (D)
16. (C)
( x 2)( x 1) x 2 , there is no asymptote at x 1. ( x 1)( x 1) x 1 Denominator : x 1 0 x 1 (vertical asymptote) x2 1 (horizontal asymptote) For other asymptote: y lim x x 1 Since y
Since f (2 x)
x (2 x) , 3 6
f ( x)
x . 6
Or, to avoid confusion, change the function to f (2k )
x k . Let 2k x , then k . 2 3
x x 2 . Substitute the function in terms of x . f ( x) 3 6 17. (D)
Since log a, log b, and log c are arithmetic progression, log c log b log b log a . c b c b b 2 ac. By log operation, log log b a b a
18. (E)
x 2 y 2 0 x y x y 0 y x or y x . That represent two lines.
19. (C)
a(1 r n ) , where a is the first term and r is the n 1 r a 10 10 20. . Therefore, S common ratio. When r 1 , the sum is S 1 1 1 r 1 2 2 The sum of geometric series is S lim
Dr. John Chung's SAT II Math Level 2 Test 1
107
2 20. (D)
2 Line m which is equidistant from two points must be perpendicular bisector of the line segment. 42 02 42 The midpoint is 1 , 1,3 and the slope of the segment is 02 2 2 y
m
(0, 4)
22. (E)
(2, 2) x
O
21. (C)
Therefore, the slope of line m is 1. The equation of line m is y 3 1( x 1) y x 2 .
3x 5 10
10 3 x 5 10 5 3x 15
5 x5 3
From the graph, the middle line is 3, amplitude is 3, and period is 4 . The frequency is 2 1 . 4 2 y A sin( Bx) C
A : amplitude
B : frequency
C : middle line
Period
2 B
1 Therefore, the equation of the trigonometric graph is y 3sin x 3 . 2 23. (B)
The graph of y x 2 5 x 6 is concave up which has a minimum on axis of symmetry. b 5 The axis of symmetry is x 2.5 . Therefore, the minimum of y is 2a 2 f (2.5) (2.5) 2 5(2.5) 6 0.25 . The range is y 0.25 . Or, by completing squared form, y ( x 2.5) 2 0.25 . You can use a graphing calculator.
24. (D)
Cofunction: Since sin( x 10) and cos(2 x 28) are cofunctions, x 10 2 x 28 90. Therefore, 3x 72 x 24.
25. (C)
Switch x and y , then express y in terms of x. Therefore, x 3 y 1
x y 1 log 3 x y log3 x 1 log3 x log3 3 log3 . 3
26. (B)
Since f ( x) is greater than 0 for all x , f ( x) 0 cannot have x-intercept ( zeros). Therefore, discriminant must be negative (imaginary roots). D b 2 4ac k 2 4(1)(9) 0 (k 6)(k 6) 0 6 k 6
27. (A)
Complete squared form: x 2 10 x 25 y 2 6 y 9 10 25 9 ( x 5) 2 ( y 3)2 44 Therefore, the center is (5, 3) .
108
2 28. (E)
2 Domain of the polynomial function is 2 x 10 0 and ( x 2)( x 1) 0 . Therefore, x 5 x 2,1 x 5 .
29. (B)
Substitute y x 2 into x 2 y 2 20. Then x 4 x 2 20 0 ( x 2 5)( x 2 4) 0 Since x 2 5 0 , x 2 4 0 x 2 or x 2 .
30. (D)
Switch x and y : x log 2 y 1 2 x 2 log 2 ( y 1) Therefore, y 1 2 x 2
31. (C)
33. (A)
34. (E)
y 2x2 1 .
By the formula, BD DC 82 DC AC 2 CD CB
32. (B)
64 16 4
AC 2 16 20 320 AC 320 17.89
z 2 4 6i 2 2 6i ,
2 6i 22 (6)2 40 6.32
The middle term is the 4th term. The 4th term is 6 C3 ( x)3 (3 y )3 540 x3 y 3 .
x2 6 x 9
Or, since
2
x 2 x 1.5 But x 0 . No solution.
x 2 6 x 9 ( x 3) 2 x 3 , then x 3 x .
If x 3 , x 3 x 3 0 No solution 3 3 But is not less than 3 . No solution If x 3 , x 3 x x 2 2 35. (A)
From the recursive equation an 1 an 3 , it is arithmetic sequence with common difference 3 and the first term 1. Therefore, an a1 (n 1)d 1 (n 1)(3) 3n 2.
36. (A)
From original function, you can find f 1 (4). The input of f 1 was the output of f . Since, x 2 2 x 4 4 x 2 2 x 8 0 ( x 4)( x 2) 0 , x 2 ( x 0) . This value of x is the output of f 1 . Therefore, f 1 (4) 2.
37. (C)
sin 30 sin B 6 3 sin 30 3 sin B 6 6 2 6 3 Therefore, B 60 or 120. Now find C .
The law of sines:
A : 30
30
B : 60 C : 90
120 30
Dr. John Chung's SAT II Math Level 2 Test 1
C 90 or 30.
109
2
2
38. (E)
Since log 2 5 a and log 2 3 b , 1 1 1 log 2 75 log 2 75 log 2 5 5 3 log 2 5 log 2 5 log 2 3 2 2 2 1 1 2a b a b 2 2
39. (D)
Parametric equation: x 3sin and y 4 cos From cos 2 sin 2 1 (Pythagorean identity), sin x2 y 2 1 32 42
40. (A)
41. (B)
x y , and cos 3 4
It is ellipse and major axis is on the y -axis. Graph (D) is correct.
Three sides of a triangle are given. Heron’s formula: The area of the triangle is abc A s ( s a )( s b)( s c) , where s 2 457 Therefore, s 8 and the area is 8(8 4)(8 5)(8 7) 9.80 . 2 (n 1)! (n 1)n (n 1) ! (n 1)n , (n 1)! (n 1) !
n 2 n 56 (n 8)(n 7) 0
Since n 8 , n 7. 42. (D)
10
180 o 50
10
5
25 5
Since the circumference of the base is 10 and the circumference of the lateral side is 20 , lateral side is exactly a semicircle which has a central angle 180 . Therefore, the surface area is 25 50 75 . Or, use the formula S f rs r 2 , s is a slant height. (5)(10) (52 ) 75 43. (B)
110
Since a11 a 10d 30 and a21 a 20d 0 , a 10d 30 a 20d 0 d 3 and a 60. 10d 30 Therefore, a10 a 9d 60 27 33. Or, since a10 (3) a11 , then a11 30 (3) 33 .
2
2 n2 1
44. (D)
lim
45. (D)
Since 1
x 1
n 1
lim
x 1
(n 1)(n 1)( n 1) ( n 1)( n 1)
lim
x 1
(n 1) (n 1)( n 1) (n 1)
2 2 4 1
i 3 i 3 , the other root mist be 1 (conjugate). Therefore, 2 2
i 3 i 3 i 3 i 3 i 3 2 0 x 1 x 1 x 1 x 1 x 1 2 2 2 2 2 7 When simplified, x 2 2 x 0 is equivalent to 4 x 2 8 x 7 0 . 4
2
Therefore, f ( x) 4 x 2 8 x 7. Or, 1) Use sum and product of two root. The equation is defined by ax 2 bx c 0 x 2
b c x 0. a a
b b i 3 i 3 1 1 2 2 a a 2 2 c i 3 i 3 7 Product of roots 1 1 a 2 2 4 7 When substitute, x 2 2 x 0 4 x 2 8 x 7 0. 4
Sum of roots
2) Because the coefficients of the equation is real, let x 1 x 1
i 3 i 3 ( x 1)2 2 2
2
Therefore, the equation is x 2 2 x 46. (B)
x2 2x 1
i 3 . 2
3 4
7 0 4 x 2 8 x 7 0. 4
Remainder Theorem: Let x3 2 x 2 mx n ( x 1)( x 2)Q ( x) , where Q( x) is the quotient. When x 1 , 1 2(1) m(1) n 0 m n 1 ----- (1) When x 2 , 8 2(4) m(2) n 0 2m n 0 ------ (2) (2) (1) m 1 and n 2 . Or, you can solve it using long-division and synthetic division.
Dr. John Chung's SAT II Math Level 2 Test 1
111
2
2 x 1 2
3
x 3x 2 x 2 x 2 mx n x3 3x 2 2 x x 2 (m 2) x n
Therefore, m 1 and n 2 .
2 x 2 3x (m 1) x (n 2)
47. (E)
If f ( x) and g ( x) are even, f g even, f g = even, and f g = even. (E) is not even, because it is translated to the right by 1.
48. (D)
Multiplying both sides by ( x 1) 2 ( x 2)2
b b 2 4ac , it will be simplified as follows. 2a
x 0 ( x 1) 2 ( x 2) 2 ( x 1)( x 2) y x( x 1)( x 2) 0 Using graphic solution, ( x 1) 2 ( x 2)2
2
0
1
x
From the graph, the value of y is negative for x 2 or 0 x 1 . 49. (C)
Asymptote of hyperbola: 16 x 2 25 y 2 x2 y 2 1 1 400 400 25 16 4 Since a 5 and b 4, the asymptote is y x. 5 16 x 2 25 y 2 400
50. (C)
END
112
In order to have two different real roots, discriminant D b 2 4ac 0 . Since 16 4k (2) 0 , k 2. But the leading coefficient of quadratic equation cannot be zero. k 0 . Therefore, 2 k 0 or k 0.
Test 2 Dr. John Chung’s SAT II Mathematics Level 2
Dr. John Chung's SAT II Math Level 2 Test 2
113
2
2
MATHEMATICS LEVEL 2 TEST REFERENCE INFORMATION
THE FOLLOWING INFORMATION IS FOR YOUR REFERENCE IN ANSWERING SOME OF THE QUESTIONS IN THIS TEST 1 Volume of a right circular cone with radius r and height h : V r 2 h 3 1 Lateral Area of a right circular cone with circumference of the base c and slant height : S c 2
Volume of a sphere with radius r : V 4 r 3 3 Surface Area of a sphere with radius r : S 4 r 2 1 Volume of a pyramid with base area B and height h : V 3 Bh
114
2
2 Dr. John Chung’s SAT II Math Level 2
Answer Sheet 01 02
A
B
C
D
E
A
B
C
D
E
03
A
B
C
D
E
B
C
D
04 05
A A
B
C
06
A
B
07
A
08
26 27
A
B
C
D
E
A
B
C
D
E
A
B
C
D
E
E
28 29
A
B
C
D
E
D
E
30
A
B
C
D
E
C
D
E
A
B
C
D
E
B
C
D
E
31 32
A
B
C
D
E
A
B
C
D
E
33
A
B
C
D
E
09
A
B
C
D
E
34
A
B
C
D
E
10 11
A
B
C
D
E
A
B
C
D
E
A
B
C
D
E
35 36
A
B
C
D
E
12
A
B
C
D
E
37
A
B
C
D
E
13 14
A
B
C
D
E
B
C
D
E
A
B
C
D
E
38 39
A
A
B
C
D
E
15
A
B
C
D
E
40
A
B
C
D
E
16 17
A
B
C
D
E
A
B
C
D
E
A
B
C
D
E
41 42
A
B
C
D
E
18
A
B
C
D
E
B
C
D
E
19 20
B
C
D
E
43 44
A
A
A
B
C
D
E
A
B
C
D
E
A
B
C
D
E
A
B
C
D
E
A
B
C
D
E
21 22
45 46
A
B
C
D
E
47
A
B
C
D
E
23
A
B
C
D
E
48
A
B
C
D
E
24
A
B
C
D
E
49
A
B
C
D
E
25
A
B
C
D
E
50
A
B
C
D
E
The number of right answers:
The number of wrong answers:
# of correct
Dr. John Chung's SAT II Math Level 2 Test 2
1 4 # of wrong
Raw score
115
2
2 Score Conversion Table 6
Scaled Score 480
630
5
470
26
620
4
470
800
25
620
3
460
46
800
24
610
2
460
45
800
23
610
1
450
44
800
22
600
0
450
43
800
21
600
42
800
20
590
41
800
19
590
40
780
18
580
39
760
17
570
38
750
16
560
37
740
15
550
36
720
14
540
35
710
13
530
34
700
12
520
33
690
11
510
32
680
10
500
31
670
9
490
30
660
8
490
29
650
7
480
28
Scaled Score 640
800
27
48
800
47
50
Scaled Score 800
49
Raw Score
116
Raw Score
Raw Score
2
2
MATHEMATICS LEVEL 2 TEST For each of the following problems, decide which is the BEST of the choices given. If the exact numerical value is not one of the choices, select the choice that best approximates this value. Then fill in the corresponding circle on the answer sheet Note: (1) A scientific or graphing calculator will be necessary for answering some (but not all) of the questions in this test. For each question you will have to decide whether or not you should use a calculator. (2) For some questions in this test you may have to decide whether your calculator should be in the radian mode or the degree mode. (3) Figures that accompany problems in this test are intended to provide information useful in solving the problems. They are drawn as accurately as possible EXCEPT when it is stated in a specific problem that its figure is not drawn to scale. All figures lie in a plane unless otherwise indicated. (4) Unless otherwise specified, the domain of any function f is assumed to be the set of all real numbers x for which f ( x) is a real number. The range of f is assumed to be the set of all real numbers f ( x) , where x
is in the domain of f . (5) Reference information that may be useful in answering the questions in this test can be found on the page preceding Question 1. USE THIS SPACE FOR SCRATCHWORK 1.
If
x2 1 3x 5 , then x 3 x 1
(A) 3 (B) 2 (C) 0 (D) 2 (E) 4
2.
The slope of a line which contains the points a 3, 4 1 and 6a 2, 6 is . What is the value of a ? 2
(A) 3 (B) 2 (C) 2 (D) 3 (E) 5
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Dr. John Chung's SAT II Math Level 2 Test 2
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2
2 MATHEMATICS LEVEL 2 TEST - Continued USE THIS SPACE FOR SCRATCHWORK.
3.
What is the equation of a line whose x -intercept is 5 and y -intercept is 4 ? (A) 4 x 5 y 15 (B) 5 x 4 y 20 (C) 4 x 5 y 15 (D) 4 x 5 y 20 (E) 4 x 5 y 20
4.
If c 5 4c 1 , what is the value of c ? (A) 2 (B) 12 (C) 2 or 12 (D) 2 (E) 12
5.
If (a b) (a b)i 1 5i , where a and b are real numbers, then a (A) (B) (C) (D) (E)
6.
If z1 1 2i and z2 3 5i , then z1 z2 (A) (B) (C) (D) (E)
7.
0 1 2 3 4
10 5 4 3 2
If 90 180 and sin
1 , then sin 2 2
(A) 0.87 (B) 0.94 (C) 0.60 (D) 0.87 (E) 0.94
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118
2
2 MATHEMATICS LEVEL 2 TEST - Continued USE THIS SPACE FOR SCRATCHWORK.
8.
If figure 1 shows the graph of y f ( x) , then which of the following is the graph of y f ( x ) ? y
(A)
y
y f ( x)
y
(B)
O x
O
x
x
O
Figure 1 (C)
(D)
y
x
O
x
O
y
(E)
x
O
9.
y
If f ( x) (A)
1 2
3x 4 3x3 2 x 1000 , what values does f ( x) approach as x gets infinitely larger? 5 x 4 2000 (B) 0
(C)
1 2
(D)
3 5
(E) Infinite
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119
2
2 MATHEMATICS LEVEL 2 TEST - Continued USE THIS SPACE FOR SCRATCHWORK.
10.
In the graph of Figure 2, the equation of the graph is y A cos Bx C D . What is the value of B ?
y
3
(A) 6 (B) 3 (C) (D)
O
6
6
x
3
3
(E) 6
11.
3
Figure 2
3 cos 2 (A) sin (B) cos (C) sin (D) cos (E) 2sin cos
12.
In how many ways can 15 people be divided into two groups, one group with 10 and the other with 5 people? (A) (B) (C) (D) (E)
13.
3003 6006 12000 48600 9018009
If the polar equation is r sin , which of the following represents the graph? (A) (B) (C) (D) (E)
An ellipse A circle A line A parabola A hyperbola
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120
2
2 MATHEMATICS LEVEL 2 TEST - Continued USE THIS SPACE FOR SCRATCHWORK.
14.
Figure 3 shows a parallelogram with sides 10 and 15. If
Q
Q 135 , what is the area of the parallelogram? (A) (B) (C) (D) (E)
15.
10 P
15 Figure 3
S
Figure 4 shows a rectangular solid. If the area of face I is 15, the area of face II is 20, and the area of face III is 18, then what is the volume of the solid? (A) (B) (C) (D) (E)
16.
53.03 66.02 83.33 106.07 121.67
R
135
67.48 73.48 88.98 96.76 101.44
Figure 4
If 180 270 and cos 0.707 , what is the value of tan ? 2 (A) (B) (C) (D) (E)
17.
2.414 2.414 3.424 3.424 1.414
What is the smallest positive value of x, in radian that satisfies the equation 3sin x sin 2 x 0 ? (A) (B) (C) (D) (E)
0.01 2.14 3.14 6.28 9.42
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121
2
2 MATHEMATICS LEVEL 2 TEST - Continued USE THIS SPACE FOR SCRATCHWORK.
18.
Figure 5 shows the graph of y log 2 2 x . What is the sum
y
of the areas of the three shaded rectangles? (A) (B) (C) (D) (E)
24 32 36 42 48
y log 2 2 x
O
2
4
8
12
x
Figure 5 Note: Figure not drawn to scale.
19.
The current population of Lake Pond is 50,000. The population at any time t can be calculated by the function
P(t ) Ae0.025t , where A is initial population and t is the time in years. How many years would it take for the population to reach half the present population? (A) (B) (C) (D) (E)
20.
12.4 15.5 18.4 24.6 27.7
Which of the following are asymptotes of the graph of x2 1 ? y x ( x 1) I. x 0 II. x 1 III. y 1 (A) (B) (C) (D) (E)
I only II only I and II only I and III only I, II, and III
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122
2
2 MATHEMATICS LEVEL 2 TEST - Continued USE THIS SPACE FOR SCRATCHWORK.
21.
What is the range of the function defined by
x 1 , x 1 f ( x) 2 1 x , x 1 (A) y 1 (B) y 0 (C) y 0 (D) 1 y 1 (E) All real numbers 22.
If x 3 y 2 0 and x y 2 0 , then x (A) (B) (C) (D) (E)
23.
0.5 1.0 1.5 2.0 2.5
If the 5th term of a geometric sequence is 24, and the 7th term is 144, what is the first term of the sequence? (A) 2 3 2 2 (C) 3 1 (D) 3 1 (E) 4 (B)
24.
If a cube is inscribed inside a sphere of radius 10, what is the volume of the cube? (A) 1539.60 (B) 1450.56 (C) 1300.48 (D) 1148.04 (E) 1200
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2
2 MATHEMATICS LEVEL 2 TEST - Continued USE THIS SPACE FOR SCRATCHWORK.
25.
If sin x 2cos x 0 , where 0 x 2 , then x could equal (A) (B) (C) (D) (E)
26.
1.017 1.107 2.034 2.412 3.003
Figure 6 shows a cube with edge 10. What is the area of triangle ABC ? (A) (B) (C) (D) (E)
129.90 88.83 86.60 82.37 50.00
A
C B
Figure 6 2
27.
n 1! 2 n 1!
(A) n 1 (B) n 2 (C) n 2 n (D) n2 (n 1)2 (E) (n 1)(n 1)
28.
For all , sin(90 ) cos(180 ) tan(180 ) (A) sin (B) cos (C) tan (D) sin cos (E) cos sin
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2
2 MATHEMATICS LEVEL 2 TEST - Continued USE THIS SPACE FOR SCRATCHWORK.
29.
If f (2 x 3) 4 x 2 , f ( x) (A) (B) (C) (D) (E)
30.
x4 x4 2x 4 3x 5 3x 5
If a circle is defined by the equation x 2 10 x y 2 2 y 10 , what are the coordinates of the center of the circle? (A) (B)
5, 1 5, 1
(C) (5, 1) (D) (E)
31.
If x , 12 , 3x 6 ,… are the first three terms in a geometric progression, then the 5th term could be (A) (B) (C) (D) (E)
32.
5, 1 10, 2
1.5 36.5 40.5 60 96
The multiplicative inverse of
1 i is 3i
1 2i 10 (B) 2 i
(A)
(C) 1 2i (D) 2 i 3i (E) 2
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125
2
2 MATHEMATICS LEVEL 2 TEST - Continued USE THIS SPACE FOR SCRATCHWORK.
33.
1 1 The value of sin 2cos 2 (A)
1 3
(B)
3 2
(C)
3 2
(D)
2 2 3
(E)
34.
b0 a0 b0 b0 d 0
( 1, 0)
O
(1, 0)
Figure 7
A solution for the equation cos 2 x 2sin x 7 0 , where 0 x 2 , is (A) 4
36.
y
Figure 7 shows the graph of p( x) ax3 bx 2 cx d . Which of the following must be true. (A) (B) (C) (D) (E)
35.
2 2 3
(B) 2
(C)
4
(D)
2
(E) No solution
What is the period of the graph of y 2sin 3 x 1 ? (A) 2 3 2 (C) 3
(B)
(D)
3
(E) 2 GO ON TO THE NEXT PAGE
126
x
2
2 MATHEMATICS LEVEL 2 TEST - Continued USE THIS SPACE FOR SCRATCHWORK.
37.
Which of the following could be the graph of f ( x) x x ? (A)
(B)
y
y
x
x
(C)
(D)
y
y
x
x
(D) y
x
38.
If a function f has the property of the fundamental period such that f ( x ) f ( x 2) , which of the following could be f ? (A) 2sin x 1 (B) sin 2x (C) cos 2 x 1 (D) 3tan 2x (E) 4 tan
2
x GO ON TO THE NEXT PAGE
Dr. John Chung's SAT II Math Level 2 Test 2
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2
2 MATHEMATICS LEVEL 2 TEST - Continued USE THIS SPACE FOR SCRATCHWORK.
39.
h
lim
h0
4h 2
(A) (B) (C) (D) (E)
0 2 4 Infinite Undefined
A
40.
In Figure 8, if BD 10 , what is the length of AC ? (A) (B) (C) (D) (E)
41.
20 25.12 34.64 36 36.56
B
60 D 10 Figure 8
30
C
Which of the following is the equation of the common chord of the circles with equations x 2 y 2 16 and
x 2 y 2 8 x 8 y 16 0 ? (A) y x (B) y x (C) y x 4 (D) y x 4 (E) y x 4 42.
Which of the following functions are odd? I. f ( x) x 4 5 x 2 3 II. f ( x) 3x3 5 x 1 III. f ( x) x3 x (A) (B) (C) (D) (E)
I only II only III only II and III I, II, and III
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2
2 MATHEMATICS LEVEL 2 TEST - Continued USE THIS SPACE FOR SCRATCHWORK.
43.
Figure 9 is a tetrahedron such that OB OA OC and OA 4 , OB 3 , and OC 4 . What is the area of ABC ? (A) (B) (C) (D) (E)
11.66 16.42 18.44 20.25 21.32
B
O C
A 44.
Figure 9
3log9 18 (A) 3 2 (B) 3 3 (C) 2 5 (D) 2 6 (E) 2 7
45.
If sin x t for all x in the interval
2
x ,
then sin 2x (A) (B) (C)
t 1 t2
t2 1 t2 1
t 1 t
(D) 2t 1 t 2 (E)
2t t 2 1
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129
2
2 MATHEMATICS LEVEL 2 TEST - Continued USE THIS SPACE FOR SCRATCHWORK.
46.
If a vector a (5, 12) , then which of the following is the unit vector of a ? (A) 1, 1 1 1 (B) , 5 12 (C) 0.38, 0.92 (D) 0.5, 1 (E) 0.25, 0.75 y
47.
In Figure 10, if OH is perpendicular to the line 3x 4 y 28 ,
3x 4 y 28
what is the length of OH ? (A) (B) (C) (D) (E) 48.
49.
5.6 6.5 8.0 8.5 8.7
H
If matrix A has dimension m n , matrix B has dimension p m , and matrix C has dimension n p , which of the following must be true? (A) (B) (C) (D) (E)
The product The product The product The product The product
x
O
Figure 10
AB exists. BC exists. ABC exists. CBA exists. BCA exists.
If the height of a cylinder is increased by 10 percent, by what percent must the radius of the circular base be increased so that the volume of the cylinder is increased by 25 percent? (A) (B) (C) (D) (E)
5.6% 6.2% 6.6% 7.5% 7.7% GO ON TO THE NEXT PAGE
130
2
2 MATHEMATICS LEVEL 2 TEST - Continued USE THIS SPACE FOR SCRATCHWORK.
50.
If the binomial expansion is defined by
x y n n C0 x n y o n C1 x n 1 y1 n C2 x n 2 y 2 .... n Cn xo y n , then (A)
n C1
n C2 n C3 .... n Cn
n(n 1) 2
(B) n 2 n (C) 2n1 (D) 2 n 1 (E) 2n 2n1
STOP IF YOU FINISH BEFORE TIME IS CALLED, YOU MAY CHECK YOUR WORK ON THIS TEST ONLY. DO NOT TURN TO ANY OTHER TEST IN THIS BOOK.
GO ON TO THE NEXT PAGE Dr. John Chung's SAT II Math Level 2 Test 2
131
No Test Material on This Page
132
2
2
TEST 2 # 1 2 3 4 5 6 7 8 9 10
ANSWERS
answer C A E B D B A D D D
# 11 12 13 14 15 16 17 18 19 20
Explanations: 1. (C)
Since
answer C A B D B B C B E D
# 21 22 23 24 25 26 27 28 29 30
answer E C C A C C D C C A
# 31 32 33 34 35 36 37 38 39 40
answer C C B C E C B E C C
# 41 42 43 44 45 46 47 48 49 50
answer E C A A D C A D C D
Test 2
x 2 1 ( x 1) ( x 1) x 1 , then x 1 3 x 5 x 3. x 1 x 1
Therefore, x 3 3 3 0. 2 1 6 4 10 2 , then . Therefore, a 3. a 1 2 6a 2 (a 3) 5a 5 a 1
2. (A)
Since
3. (E)
The equation is
x y 1 . Therefore, 4 x 5 y 20. 5 4
Or, y mx 4 passes through a point (4,0) . Therefore, by substitution, m 4. (B)
c 5 2
4c 1
2
4 . 5
c 2 14c 24 0 (c 2)(c 12) 0
c 2 , 12. From the equation, c 5 . Hence c 12. 5. (D)
Complex number identity: a b 1 ab 5 Hence a 3. 2a 6
6. (B)
z1 z2 1 2i 3 5i 4 3i ,
7. (A)
z1 z2 4 3i 42 (3) 2 5
1 3 2 sin . 1 in the second quadrant 90 180 , then cos 2 2 3 2 1 3 3 Therefore, sin 2 2sin cos 2 0.87. 2 2 2
Dr. John Chung's SAT II Math Level 2 Test 2
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2
2
8. (D)
y f ( x) is a reflection with y -axis . (D) is correct.
9. (D)
3 x 4 3 x 3 2 x 1000 3 x 5 5 x 4 2000
10. (D)
From the graph, period is 6. Frequency B
11. (C)
3 3 cos cos 2 2
12. (A)
15 C10
13. (B)
Since sin
lim
15 C5
3 cos sin 2
15
C5
2 2 . P(period) 6 3
sin 0 cos 1 sin sin
15 14 13 12 11 3003 5!
y y , r sin r r 2 y. r 2 x 2 y 2 in rectangular coordinates. r r
Therefore, x 2 y 2 y x 2 y 2 y 0 represents a circle. 2
1 1 The equation of the circle is x 2 y . 2 4 14. (D)
The area 10 15 sin135 106.07
15. (B)
Let the dimensions of the solid be a, b, and c. The volume of the solid is V abc. Therefore, the areas of each face are as follows. Multiply both sides and find the volume. ab 15 bc 20 ca 18 V abc 5400 73.48. (abc) 2 5400
16. (B)
cos 1 (0.707) 135 , but 225 where 180 270 . Therefore, tan
225 2.414. 2
17. (C)
3sin x sin 2 x 0 3sin x 2sin x cos x 0 sin x(3 2 cos x) 0 3 ( 1 cos x 1) . Therefore, sin x 0 x , 2 ,... and cos x 2 The smallest positive number is 3.14.
18. (B)
Sum of the areas: 2 log 2 4 4 log 2 8 4 log 2 16 2 log 2 22 4 log 2 23 4 log 2 24
4 12 16 32
134
2 19. (E)
2 Calculator will be needed. 25,000 50, 000e0.025t
20. (D)
21. (E)
1 ln 0.5 e0.025t t 27.7 2 0.025
( x 1)( x 1) x 1 , then x( x 1) x Denominator: x 0 (Vertical asymptote) x 1 y lim 1 (Horizontal asymptote). x x Since y
The range of f ( x) is all real from the graph below. Ry , y
1
y
x 1 x
O 1
y x2 1
22. (C)
If x 3 , x 3 y 2 0 x y 1 0
x y 2 0
No solution.
3 0 If x 3, x 3 y 2 0 x y 5 0 x y 2 0 Therefore, x 2 x 3 0
3 1.5. 2
23. (C)
Geometric sequence: a5 ar 4 24 a 24 2 ar 6 , 7 4 r 2 6 . Therefore, a(6)2 24 a . 6 36 3 a7 ar 144 a5 ar
24. (A)
Let the length of a side of the cube be x . Since the length of the diagonal of the cube is equal 20 to the diameter of the sphere, x 2 x 2 x 2 x 3 x 3 20 x . 3 3
20 Therefore, the volume of the cube is x 1539.60 . 3 3
25. (C)
sin x 2cos x 0 sin x 2cos x
sin x 2 tan x 2 cos x
x tan 1 (2) 1.1071 . Therefore x 2.035 or 5.176 in the interval 0, 2 . Or, use your graphing calculator.
Dr. John Chung's SAT II Math Level 2 Test 2
135
2 26. (C)
2 ABC is equilateral and the length of a side is 10 2 . Therefore, the area of ABC is 10 2 10 2 sin 60 86.60 . 2 2
27. (D)
28. (C)
n 1! 2 (n 1)n (n 1) 2 n 2 2 n 1!
Trigonometry identities: sin(90 ) cos(180 ) tan(180 ) tan180 tan sin 90 cos cos 90sin cos180 cos sin180sin 1 tan180 tan cos ( cos ) ( tan ) tan
29. (C)
Since f (2 x 3) 2(2 x 3) 4 , then f ( x) 2 x 4. Or , change variable : f (2k 3) 4k 2 and let x 2k 3 and k
x3 . Then replace 2
with x.
x 3 Therefore, f ( x) 4 2 2x 4 . 2 30. (A)
x 2 10 x y 2 2 y 10
x 5 2 y 12 36
Center is at (5, 1). 31. (C)
The common ratio between two consecutive terms of geometric progress is equal. 12 3x 6 hence . From the equation x 12 3x 2 6 x 144 0 x 2 2 x 48 0 ( x 8)( x 6) 0
x 8 or 6 .
4
3 3 and a5 8 40.5 . 2 2 12 If x 6 , then r 2 and a5 (6)(2) 4 96. 6
If x 8 , then r
1 i 3i (3 i )(1 i ) 2 4i is 1 2i . 3i 1 i (1 i)(1 i ) 2
32. (C)
The multiplicative inverse of
33. (B)
Use your calculator. 1 1 Or, cos 1 cos , where 0 . 2 2 3 1 3 3 Because sin , sin 2 2sin cos 2 . 2 2 2 2 3 . Or, 60 and sin 2 sin120 2
136
2 34. (C)
2 From the graph, a 0 and the sum of the roots is
b (1 0 1) 0 b 0 . a
d (1)(0)(1) 0 d 0. a Or, find p (0) d 0, p (1) a b c 0, and p(1) a b c 0 , and solve it.
And product of the root is
35. (E)
36. (C)
cos 2 x 2sin x 7 0 (1 sin 2 x) 2sin x 7 0 sin 2 x 2sin x 8 0 (sin x 4)(sin x 2) 0 sin x 4 or sin x 2. No solution.
1 y 2sin(3 x ) 1 y 2sin 3 x 1 3 From the equation frequency is 3 and period is P
37. (B)
2 2 . 3 3
f ( x) x x . For the following intervals,
0 x 1, f ( x) x 1 x 2, f ( x) x 1 2 x 3, f ( x) x 2 ... (B) is correct.
38. (E)
f ( x) f ( x 2) is a periodic function with the fundamental period 2. 2 2 (A) period 2 (B) period (C) period 2 2 (D) period
39. (C) 40. (C)
lim
h 0
(E) period
2
h 4h 2
lim
h 0 (
2 2
h( 4 h 2) 4 h 2)( 4 h 2)
A
lim
h 0
h ( 4 h 2) 4 h
In the figure, AD 20 and AB 10 3 . AC 20 3 34.64. 20
10 3
60o B
41. (E)
42. (C)
10
D
30o
C
From the equation x 2 y 2 8 x 8 y 16 0 and x 2 y 2 16 16 8 x 8 y 16 0 8 x 8 y 32 0 y x 4 f ( x) x3 x : odd function + odd function = odd function
Dr. John Chung's SAT II Math Level 2 Test 2
137
2 43. (A)
2 By Pythagorean Theorem, AC 5 , BC 5, and 4 2 .
5
5
The height is The area is
52 (2 2)2 17 .
4 2 17 11.66 2
4 2
Or, use Heron’s formula. A s ( s a )( s b)( s c) , where s 44. (A)
By the logarithmic operation : log a b log 3log9 18 3log3
45. (D)
18
If sin x t on
a
abc . 2
b log a 2 b 2
18 3 2
2
x , then cos x
1 t2 . 1
Therefore, sin 2 x 2sin x cos x 2(t )( 1 t 2 ) 2t 1 t 2 .
46. (C)
a Since unit vector u and a 52 (12)2 13 , then a (5, 12) 5 12 u = , (0.38, 0.92) . 13 13 13
47. (A)
The distance from (0,0) to the line 3x 4 y 28 0 is D
48. (D)
(A) AB m n p m doesn’t exist
3(0) 4(0) 28 2
3 4
2
28 5.6. 5
(B) BC p m n p doesn’t exist
(C) ABC m n p m n p doesn’t exist
(D) CBA n p p m m n exist
(E) BCA p m n p m n doesn’t exist
49. (C)
138
Let the volume be V r 2 h and new radius be R . Since h increased by 10% and R 2 1.25 h V increased by 25%, 1.25V R 2 (1.1h) 1.25 r 2 h R 2 (1.1h) 2 1.1 h r R 1.25 1.066 1 0.066 Therefore, the radius is increased by 6.6%. r 1.1
2 50. (D)
2 In order to find the sum of the coefficients of the binomial expansion, substitute x y 1 .
(1 1)n n C0 n C1 n C2 ...... n Cn 2n
1 n C1 n C2 ...... n Cn
2n 1 n C1 n C2 ...... n Cn END
Dr. John Chung's SAT II Math Level 2 Test 2
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140
Test 3 Dr. John Chung’s SAT II Mathematics Level 2
Dr. John Chung's SAT II Math Level 2 Test 3
141
2
2
MATHEMATICS LEVEL 2 TEST REFERENCE INFORMATION
THE FOLLOWING INFORMATION IS FOR YOUR REFERENCE IN ANSWERING SOME OF THE QUESTIONS IN THIS TEST 1 Volume of a right circular cone with radius r and height h : V r 2 h 3 1 Lateral Area of a right circular cone with circumference of the base c and slant height : S c 2
Volume of a sphere with radius r : V 4 r 3 3 Surface Area of a sphere with radius r : S 4 r 2 1 Volume of a pyramid with base area B and height h : V 3 Bh
142
2
2 Dr. John Chung’s SAT II Math Level 2
Answer Sheet 01 02
A
B
C
D
E
A
B
C
D
E
03
A
B
C
D
E
B
C
D
04 05
A A
B
C
06
A
B
07
A
08
26 27
A
B
C
D
E
A
B
C
D
E
A
B
C
D
E
E
28 29
A
B
C
D
E
D
E
30
A
B
C
D
E
C
D
E
A
B
C
D
E
B
C
D
E
31 32
A
B
C
D
E
A
B
C
D
E
33
A
B
C
D
E
09
A
B
C
D
E
34
A
B
C
D
E
10 11
A
B
C
D
E
A
B
C
D
E
A
B
C
D
E
35 36
A
B
C
D
E
12
A
B
C
D
E
37
A
B
C
D
E
13 14
A
B
C
D
E
B
C
D
E
A
B
C
D
E
38 39
A
A
B
C
D
E
15
A
B
C
D
E
40
A
B
C
D
E
16 17
A
B
C
D
E
A
B
C
D
E
A
B
C
D
E
41 42
A
B
C
D
E
18
A
B
C
D
E
B
C
D
E
19 20
B
C
D
E
43 44
A
A
A
B
C
D
E
A
B
C
D
E
A
B
C
D
E
A
B
C
D
E
A
B
C
D
E
21 22
45 46
A
B
C
D
E
47
A
B
C
D
E
23
A
B
C
D
E
48
A
B
C
D
E
24
A
B
C
D
E
49
A
B
C
D
E
25
A
B
C
D
E
50
A
B
C
D
E
The number of right answers:
The number of wrong answers:
# of correct
Dr. John Chung's SAT II Math Level 2 Test 3
1 4 # of wrong
Raw score
143
2
2 Score Conversion Table 6
Scaled Score 480
630
5
470
26
620
4
470
800
25
620
3
460
46
800
24
610
2
460
45
800
23
610
1
450
44
800
22
600
0
450
43
800
21
600
42
800
20
590
41
800
19
590
40
780
18
580
39
760
17
570
38
750
16
560
37
740
15
550
36
720
14
540
35
710
13
530
34
700
12
520
33
690
11
510
32
680
10
500
31
670
9
490
30
660
8
490
29
650
7
480
28
Scaled Score 640
800
27
48
800
47
50
Scaled Score 800
49
Raw Score
144
Raw Score
Raw Score
2
2
MATHEMATICS LEVEL 2 TEST For each of the following problems, decide which is the BEST of the choices given. If the exact numerical value is not one of the choices, select the choice that best approximates this value. Then fill in the corresponding circle on the answer sheet Note: (1) A scientific or graphing calculator will be necessary for answering some (but not all) of the questions in this test. For each question you will have to decide whether or not you should use a calculator. (2) For some questions in this test you may have to decide whether your calculator should be in the radian mode or the degree mode. (3) Figures that accompany problems in this test are intended to provide information useful in solving the problems. They are drawn as accurately as possible EXCEPT when it is stated in a specific problem that its figure is not drawn to scale. All figures lie in a plane unless otherwise indicated. (4) Unless otherwise specified, the domain of any function f is assumed to be the set of all real numbers
x for which f ( x) is a real number. The range of f is assumed to be the set of all real numbers f ( x) , where x is in the domain of f . (5) Reference information that may be useful in answering the questions in this test can be found on the page preceding Question 1. USE THIS SPACE FOR SCRATCHWORK 1.
If a( x 1) b( x 1) 2 0 for all real x , then a (A) (B) (C) (D) (E)
2.
2 1 0 1 2
If i 1 , which of the following is a negative integer?
(A) i 24 3.
(B) i 33 (C) i 46
(D) i 55
(E) i 72
If f (2) 0 and f (1) 0 , which of the following must be a factor of f ( x) ?
(C) x 2 x 2
(A) x 2
(B) x 1
(D) x 2 x 2
(E) None of these
GO ON TO THE NEXT PAGE
Dr. John Chung's SAT II Math Level 2 Test 3
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2
2 MATHEMATICS LEVEL 2 TEST - Continued USE THIS SPACE FOR SCRATCHWORK.
4.
5.
If
x 2 x , then the solution consists of
(A) (B) (C) (D) (E)
Zero only Positive real numbers only Negative real numbers only All real numbers No real numbers
If x
2 , then 9 x 2 4 is equivalent to 3
(A) 9 x 2 4 (B) 9 x 2 4 (C) 4 9x 2 (D) 4 9x 2 (E) 3 x 2 6.
If 1 is a root of the equation kx 2 6 x 4 0 , then the other root is (A) 4
7.
(B) 2
(C) 0.4
(D) 0.4
(E) 2.5
y
Figure 1 shows the graph of the equation 2 x 3 y 9 0 . What is the value of ? (A) 33.7
2x 3 y 9 0
(B) 34.2
O
(C) 37.8
x
(D) 38.1 (E) 40.6 8.
If f ( x) x 1 and
Figure 1
f
g ( x) x 2 3 x 2 , which of the
following is g ( x ) ? (A) x 2 (B) x 2 3 x (C) x 2 3 x 1 (D) x 2 3 x 1 (E) ( x 1)2 ( x 2)
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146
2
2 MATHEMATICS LEVEL 2 TEST - Continued USE THIS SPACE FOR SCRATCHWORK.
9.
If x 3 is a factor of 2 x 3 kx 3 , then k (A) (B) (C) (D) (E)
3 7 9 15 17
10. If f ( x) x
(A) 4
2 3
1 , then f 8
(B) 4
(C)
1 4
(D)
1 4
(E) 8
11. If the pendulum of a clock swings through an angle of 2.8 radians and the length of the arc that its tip travels is 40, then the length of the pendulum is
(A) (B) (C) (D) (E)
12.54 13.58 14.29 52 112
12. If y 10 5sin 2 x , what is the minimum value of y ?
(A) 5 (B) 10 (C) 5 (D) 10 (E) 15 13. The period of the graph of y 3cos 2 2 x is
(A)
8
(B)
4
(C)
2
(D)
(E) 2
14. If 2 7 is a root of the equation x 2 4 x 3k 0 , then k
(A) 1
(B) 1
(C) 2
(D) 2
(E) 4 GO ON TO THE NEXT PAGE
Dr. John Chung's SAT II Math Level 2 Test 3
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2
2 MATHEMATICS LEVEL 2 TEST - Continued USE THIS SPACE FOR SCRATCHWORK.
15. If ax 2 bx c 0 for all real numbers x , then which of the following must be true?
(A) a 0 and b 2 4ac 0 (B) a 0 and b 2 4ac 0 (C) a 0 and b 2 4ac 0 (D) a 0 and b 2 4ac 0 (E) a 0 and b 2 4ac 0
16. If the roots of the equation x 2 x 3 0 are and ,
then
(A)
1
1 3
1
(B)
2 3
(C)
2 3
(D)
2 5
(E)
3 5
17. If the points 3, 7 , 1, k , and 1, 1 are collinear, what
is the value of k ? (A) 2
(B) 4
(C) 5
(D) 7
18. The graph of the function f ( x)
(E) 8
x2 x 2 has a vertical x 2 3x 2
asymptote at x (A) (B) (C) (D) (E)
1 only 2 only 2 only 1 and 2 only 1, 2 , and 2
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2
2 MATHEMATICS LEVEL 2 TEST - Continued USE THIS SPACE FOR SCRATCHWORK.
19. Which of the following is the length of the radius of the
sphere x 2 2 x y 2 2 y z 2 3 (A)
3
(B) (C) (D)
5 3 5
(E) 2 5
20. Which of the following is the equation whose graph is the set of points equidistant from points 0, 4 and 2, 0 ?
(A) y x 1 (B) y 2 x 1 1 1 x 2 2 1 3 (D) y x 2 2 1 3 (E) y x 4 2
(C) y
n
21.
1 i n i 1 2 lim
(A) (B) (C) (D) (E)
0 1 5 10 Infinite
22. Which of the following is equivalent to
(A) (B) (C) (D) (E)
x 1 0? x
x0 0 x 1 x0 0 x5 x 1
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2
2 MATHEMATICS LEVEL 2 TEST - Continued USE THIS SPACE FOR SCRATCHWORK.
23.
A school committee of 5 is to be chosen from a group consisting of 5 boys and 6 girls. How many ways can the committee be made up of 3 boys and 2 girls? (A) (B) (C) (D) (E)
24.
The interquartile range of a data set is 12. If the first quartile is 65, which of the following could be the median? (A) (B) (C) (D) (E)
25.
100 150 600 1200 1800
50 64 70 78 80
What is the range of the function defined by f ( x) (A) (B) (C) (D) (E)
1 2 ? x
All real numbers All real numbers except 2 All real numbers except 0 All real numbers except 2 All real numbers between 2 and 3 nt
26.
r The formula A P 1 gives the amount A in a n savings account with initial investment P which is compounded monthly at an annual interest rate of 6 percent for t years. How many years will it take the initial investment to double?
(A) (B) (C) (D) (E)
5.3 6.5 8.3 11.6 13.1
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150
2
2 MATHEMATICS LEVEL 2 TEST - Continued USE THIS SPACE FOR SCRATCHWORK.
27.
A line has parametric equations x 5 t and y 7 2t , where t is the parameter. The slope of the line is (A)
28.
5 7
(B) 2
(C)
7 2t 5t
(D)
7 5
(E) 2
ln (n 1)2 lim n 0 n (A) (B) (C) (D) (E)
0 1 2 5 Undefined y
29. If the graph of f in Figure 2 is a polynomial of degree 7,
which of the following could be f ? (A) x3 ( x 3)2 ( x 2)( x 4)
(B) x 2 ( x 3)( x 2)( x 4)( x 1)2 (C) x 2 ( x 3)( x 2)( x 3)( x 1)2 (D) x 2 ( x 3) 2 ( x 3) 2 ( x 4)
3
O
2
4
x
Figure 2
(E) x 2 ( x 3)( x 2)( x 4)( x 2 1) 30. If a1 1 , a2 3 , and an 1
an an 2 , then what is the 2
20th term of this sequence? (A) (B) (C) (D) (E)
36 39 41 43 60
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2
2 MATHEMATICS LEVEL 2 TEST - Continued USE THIS SPACE FOR SCRATCHWORK.
31. If f ( x) 2 x3 3x 2 , which of the following statements is true?
(A) The function is increasing for all real x. (B) The function is increasing for x 0 (C) The equation f ( x) 0 have three real roots. (D) The equation f ( x) 0 have two imaginary roots. (E) The inverse of the function is also a function.
32. If f ( x) x2 1 , where x 0 , then f f 1 ( x) could
equal (B) x (C) x2 (D) x 2 1
(A) 1
(E)
1 x 1 2
33. If log a x 3 and log b x 4 , then log ab x
(A) 12
(B)
8 3
(C)
12 7
(D)
7 12
(E)
3 8
34. If a cylinder whose height is equal to the diameter of its base is inscribed in a sphere, then the ratio of the volume of the cylinder to the volume of the sphere is
(A)
1 2
(B)
2 3
(C)
2 2 3
(D)
2
(E)
3 2 8
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2
2 MATHEMATICS LEVEL 2 TEST - Continued USE THIS SPACE FOR SCRATCHWORK.
35. Which of the following is the graph of
(A)
x 2
y
(B)
y
1?
3 y
2 3
3 O
2 x
2
x
O
2
(C)
(D)
y
y 3
2 3
3
O
2
x
2
O
2
x
3
(E)
y
3 2
2 O
x
3
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2
2 MATHEMATICS LEVEL 2 TEST - Continued USE THIS SPACE FOR SCRATCHWORK.
36. Figure 3 shows the part of a circle with a radius r and central angle in radians. What is the area of the figure?
(A) r (B) r 2 r 2 (C) 2 (D) (E)
r 2 360
r
r
Figure 3
2
37. When a fair coin is tossed 4 times, what is the probability of tossing at least 3 heads?
(A) (B) (C) (D) (E)
5 16 1 2 9 16 3 4 5 8
38. If a 10 and b 18 , then which of the following could NOT be a b ? (A) (B) (C) (D) (E)
7 8 15 22 28
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2
2 MATHEMATICS LEVEL 2 TEST - Continued USE THIS SPACE FOR SCRATCHWORK.
1 39. If 2 (A) (B) (C) (D) (E)
2a
b
a 1 , what is the value of ? b 3
0.33 0.67 0.79 0.81 0.87
40. If the equation sin 2 8cos 5 0 , where
0 360 , how many solutions are there in the interval? (A) 1
(B) 2
(C) 3
(D) 4
(E) 0
41. If n ! 118n ! 240 0 , what is the value of n ? 2
(A) 2
(B) 3
(C) 4
(D) 5
(E) 6
42. If the equation of an ellipse is 18 x 2 5 y 2 90 0 , what is the length of the major axis of the ellipse?
(A) (B) (C) (D) (E)
2.24 5.00 8.49 9.00 10.0
43. If 2 x 4 x 4 2 x , then x
(A) 2 (B) 3 (C) 5 (D) log 2 3 (E) log 4 3
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2
2 MATHEMATICS LEVEL 2 TEST - Continued USE THIS SPACE FOR SCRATCHWORK.
44. Which of the following intervals contain a root of
x3 3 x 2 3 x 2 0 (A) (B) (C) (D) (E)
2 x 1 1 x 0 0 x 1 2 x3 3 x 4
45. If the statement is “If xy 0 , then x 0 or y 0 ”, an indirect proof of the statement begins with the assumption that
(A) x 0 or y 0 (B) x 0 and y 0 (C) xy 0 (D) x 0 or y 0 (E) x 0 and y 0 46. By the rational zero theorem, which of the following could not be a possible rational zero of the equation
y 2 x 4 3x 2 2 x 10 ? (A)
1 2
(B)
1 5
(C)
5 2
(D)
5 2
(E) 5
a 1 2 3 and matrix B . If AB , 47. Matrix A b 3 4 4 then a (A) 3
(B) 2
(C) 1
(D) 1
(E) 2
48. If a set A 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 , how many
subsets are there containing the elements 3, 4, and 5? (A) (B) (C) (D) (E)
64 128 256 512 1024
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156
2
2 MATHEMATICS LEVEL 2 TEST - Continued USE THIS SPACE FOR SCRATCHWORK.
49. If a cone with a slant height equal to the diameter of the base is inscribed in a sphere with a radius of 10, what is the volume of the cone?
(A) (B) (C) (D) (E)
375 300 250 200 160
50. Given the parametric equations x sec and y tan ,
which of the following is the graph of the points ( x, y ) ? (A) (B) (C) (D) (E)
Circle Ellipse Parabola Hyperbola None of these
STOP IF YOU FINISH BEFORE TIME IS CALLED, YOU MAY CHECK YOUR WORK ON THIS TEST ONLY. DO NOT TURN TO ANY OTHER TEST IN THIS BOOK.
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2
2
TEST 3 # 1 2 3 4 5 6 7 8 9 10
ANSWERS
answer D C D D C C A D E A
Explanations:
# 11 12 13 14 15 16 17 18 19 20
answer C A C B D A B C B D
# 21 22 23 24 25 26 27 28 29 30
answer B B B C B D E C E B
# 31 32 33 34 35 36 37 38 39 40
answer D B C E D C A A C B
# 41 42 43 44 45 46 47 48 49 50
answer D C D B E B E B A D
Test 3
1. (D)
a ( x 1) b( x 1) 2 0 (a b) x a b 2 0 must be true for all real x. Therefore, a b 0 and a b 2 . By addition a 1 and b 1.
2. (C)
i 46 i 44 i 2 1
3. (D)
Since ( x 2) and ( x 1) are factors of f ( x), f ( x) ( x 2)( x 1)Q( x) .
Therefore, f ( x) x 2 x 2 Q( x) . 4. (D) 5. (C)
x 2 x is always true for any value of x. 2 2 2 , then x . 3 3 3 4 For this interval: 9 x 2 9 9 x 2 4 9 x 2 4 0 . 9
Since x
Therefore, 9 x 2 4 (9 x 2 4) 4 9 x 2 . 6. (C)
The product of the roots is
4 6 and the sum of the roots is . Let the other root be r. k k
6 6 1 r k (1) k r 1 4 4 1r k (2) k r 6 4 4r 4 6r r 0.4 From (1) and (2), r 1 r 7. (A)
tan slope of the line.
Dr. John Chung's SAT II Math Level 2 Test 3
3 y 2x 9 y
2 2 x 3 The slope is . 3 3
159
2
2 Therefore, tan
2 2 tan 1 33.7o 3 3
8. (D)
Since f g ( x) g ( x) 1 x 2 3x 2 , then g ( x) x 2 3x 1 .
9. (E)
By factor theorem: f (3) 2(3)3 k (3) 3 0 51 3k k 17
10. (A)
f ( x) x
2 3
1 1 f 8 8
2 3
23
2 3
22 4. Or use a calculator.
11. (C)
The length of a arc s r , where r is a radius(the length of the pendulum) and is the central angle in radian. 40 14.29. Therefore, 40 r 2.8 r 2.8
12. (A)
Since 5 5sin(2 x) 5 , the minimum will be 10 5 5.
13. (C)
The period of y cos(2 x) is follows.
2 . Therefore, the period of y 3cos 2 (2 x) is as 2 2
y
y 3cos 2 (2 x)
O
x
y cos(2 x )
14. (B)
If 2 7 is one of the roots, then the other root is 2 7 . The product of the roots is 3k . Therefore, 3k (2 7)(2 7) (2) 2 ( 7)2 3 k 1 .
15. (D)
The graph must be as follows. Therefore, the graph is concave down and the function do not have real roots (Imaginary). x
y f ( x)
16. (A)
Since sum of the roots 1
160
1
1 1 . 3 3
1 3 3 , 1 and product of the roots 1 1
2
2
17. (B)
Because the three point on the same line, the slope between any two points are equal. k 7 k 1 k 7 k 1 k 4. 1 3 1 (1) 2 2
18. (C)
Since f ( x)
19. (B)
x 2 2 x y 2 2 y z 2 3 ( x 1)2 ( y 1) 2 z 2 5
( x 2) ( x 1) ( x 2) ( x 1)
( x 2) , f ( x) has a vertical asymptote at x 2 . ( x 2)
Therefore, the radius is r 5 . 20. (D)
The equation which pass through (0, 4) and (2, 0) : y 2 x 4 40 02 40 2 , midpoint , (1, 2) 2 02 2 Therefore, the line of equidistance is perpendicular to the line and pass through (1, 2). 1 1 3 y x b 2 (1) b b 2 2 2 1 3 The equation is y x . 2 2
slope
21. (B)
Series: S
a 1 r
1 1 1 1 1 1 lim i ... 2 1. 1 n 2 4 8 16 i 1 2 1 2 n
x 1 0 x 2 x( x 1) 0 0 x 1 x Or, you can use the test value method.
22. (B)
Because x 2 0, x 2
23. (B)
5 C3
24. (C)
Interquartile range = upper quartile – lower quartile = 12 Since Interquartile has the range 65, 77 , the median must be in this range.
25. (B)
y 2 is an asymptote.
26. (D)
Since the interest is compounded monthly, n 12.
6 C2 150.
12t
0.06 2 P p 1 12 12t log1.005 2
Dr. John Chung's SAT II Math Level 2 Test 3
2 1.00512t
t
log 2 11.6 12log1.005
161
2
2
27. (E)
t 5 x substitute y 7 2(5 x) y 2 x 17 Slope is 2.
28. (C)
Since e lim 1 n n , lim
29. (E)
At x 0 , the graph is bounced on x-axis and pass at x 3, 2, and 4. Therefore, the polynomial have factors as follows. P( x) x m ( x 3)n ( x 2) n ( x 4) n (imaginary roots ) , where m even and n odd. (E) could be the polynomial function.
30. (B)
Since 2an 1 an an 2 an 1 an an 2 an 1 , the sequence is arithmetic progress.
1
n 0
1 2 ln(n 1) lim 2 ln 1 n n 2 ln e 2 . n 0 n 0 n
a1 1 and d 2 . Therefore, a20 a1 (20 1)d 1 19 2 39. Or, find the pattern. You can find a1 1, a2 3, a3 5, a4 7,..... , which is arithmetic sequence. 31. (D)
Use your graphic calculator. The graph will be as follows.
(D) is correct. f ( x) have one real root and two imaginary roots. (E) is incorrect. Use horizontal line test to check whether it’s inverse is a function. When the function is both increasing and decreasing, it’s inverse is not a function.
32. (B)
f f 1 ( x) f 1 f ( x) x
33. (C)
log ab x
log x log x log ab log a log b log x log x 3 log a log a 3
-------- (1) and
log x log x 4 log b log b 4
Substitute into (1) log x 12 log ab x log x log x 7 3 4 Or, use the formula: log ab x
162
1 1 1 1 12 1 1 7 log x ab log x a log x b 7 3 4 12
2
2 3
2
34. (E)
d3 d Vc d 4 2
d 2 4 2 d3 2 , where the diameter of the sphere and Vs 3 3
d3
Vc 3 3 2 4 3 Vs d 2 4 2 8 3 Or, you can use a convenient number for d . Use d 2.
is d 2 . d 2
d d
35. (D)
Memorize the graph of x y 1 is as follows. In first quadrant, x 0 and y 0 , then x y 1 y x 1
1 y x 1 1
1
Therefore, the graph of
y
2 3 When x 0 , y -intercept is 3 .
1
r 2 2
x
because A r 2
36. (C)
A
37. (A)
At least 3 heads is greater than or equal to 3 heads. Therefore, 3
2
1 is (D).
.
4
4 1 5 1 1 1 P 4 C3 4 C4 . 2 2 2 16 16 16 38. (A)
If a and b are in the same direction, a b a b 10 18 28 . If a and b are in the opposite direction, then a b b a 18 10 8 . Therefore, 8 a b 28. It cannot be 7.
2a
39. (C)
b
1 b 1 b 2 3
Therefore,
2a
1 b 1 3 2
1 log 2a 1 3 log 3 log 1 1 log 2 b 3 2 log 2
log 3 a 0.79. b 2log 2
Dr. John Chung's SAT II Math Level 2 Test 3
163
2 40. (B)
2 Use graphic utility.
O
2
Or algebraically as follows. Since sin 2 1 cos 2 , the equation is 1 cos 2 8cos 5 0 cos 2 8cos 6 0 . 8 64 24 cos 8.69 or cos 0.69 . cos 8.69 and 2 cos 0.69 . Therefore, 2.3328 or 3.9503 .
Therefore, cos
41. (D)
42. (C)
The equation can be factored as follow. Since n ! 2 , then n ! 120 n ! 120 n ! 2 0 Change the equation into a standard form.
18 x 2 5 y 2 90 90 90 90
5! 120
n 5.
x2 y2 1 90 90 18 5
90 90 90 , major axis is on y -axis. Therefore, a 2 a 18 . 5 18 5 Major axis is 2a 2 18 8.49 .
Since
43. (D)
Since 2 x 22 x 4 2 x 22 x 3 2 x 0 2 x (2 x 3) 0 and 2 x 0 , the solution is 2 x 3 0 2 x 3 x log 2 3.
44. (B)
If f (a) f (b) 0 , then f ( x) 0 has at least one solution on interval (a, b) . Since f (1) 5 0 and f (0) 2 0 , then f (1) f (0) 0 . Therefore, the equation has a solution on interval 1, 0 . Or, use a graph calculator to find the zeros. The graph will be as follow.
0.44
45. (E)
164
Indirect proof is a proof in which a statement to be proved is assumed false and if the assumption leads to impossibility, then the statement assumed false has been proved true. Therefore, ( x 0 or y 0) is x 0 and y 0. In symbolic notation is as follows. ( x 0 y 0) ( x 0) ( y 0) : De Morgan’s Law
2
2
46. (B)
The possible rational zeros are obtained as follows. factors of 10 1, 2, 5, 10 factors of 2 1, 2 1 Therefore, cannot be the rational root of the equation. 5
47. (E)
From matrix equation 1 2 a 3 3 4 b 4 a 2b 3 and 3a 4b 4 3a 4b 4 2 a 4b 6 Therefore, a 2
48. (B)
Number of subsets of 1, 2,6,7,8,9,10 is 27 128 . Now add elements 3, 4, and 5 to those subsets. Therefore, number of subsets containing 3, 4, and 5 is also 128.
49. (A)
10 O 5 30o 5 3
From the figure above, the radius of the cone is 5 3 and the height is 15. (5 3)2 15 Therefore, the volume of the cone is V 375 . 3 50. (D)
Parametric equation: eliminate . Since 1 tan 2 sec2 , substitute x and y. Therefore, 1 y 2 x 2 x 2 y 2 1 , which represents hyperbola.
END
Dr. John Chung's SAT II Math Level 2 Test 3
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166
Test 4 Dr. John Chung’s SAT II Mathematics Level 2
Dr. John Chung's SAT II Math Level 2 Test 4
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2
2
MATHEMATICS LEVEL 2 TEST REFERENCE INFORMATION
THE FOLLOWING INFORMATION IS FOR YOUR REFERENCE IN ANSWERING SOME OF THE QUESTIONS IN THIS TEST 1 Volume of a right circular cone with radius r and height h : V r 2 h 3 1 Lateral Area of a right circular cone with circumference of the base c and slant height : S c 2
Volume of a sphere with radius r : V 4 r 3 3 Surface Area of a sphere with radius r : S 4 r 2 1 Volume of a pyramid with base area B and height h : V 3 Bh
168
2
2 Dr. John Chung’s SAT II Math Level 2
Answer Sheet 01 02
A
B
C
D
E
A
B
C
D
E
03
A
B
C
D
E
B
C
D
04 05
A A
B
C
06
A
B
07
A
08
26 27
A
B
C
D
E
A
B
C
D
E
A
B
C
D
E
E
28 29
A
B
C
D
E
D
E
30
A
B
C
D
E
C
D
E
A
B
C
D
E
B
C
D
E
31 32
A
B
C
D
E
A
B
C
D
E
33
A
B
C
D
E
09
A
B
C
D
E
34
A
B
C
D
E
10 11
A
B
C
D
E
A
B
C
D
E
A
B
C
D
E
35 36
A
B
C
D
E
12
A
B
C
D
E
37
A
B
C
D
E
13 14
A
B
C
D
E
B
C
D
E
A
B
C
D
E
38 39
A
A
B
C
D
E
15
A
B
C
D
E
40
A
B
C
D
E
16 17
A
B
C
D
E
A
B
C
D
E
A
B
C
D
E
41 42
A
B
C
D
E
18
A
B
C
D
E
B
C
D
E
19 20
B
C
D
E
43 44
A
A
A
B
C
D
E
A
B
C
D
E
A
B
C
D
E
A
B
C
D
E
A
B
C
D
E
21 22
45 46
A
B
C
D
E
47
A
B
C
D
E
23
A
B
C
D
E
48
A
B
C
D
E
24
A
B
C
D
E
49
A
B
C
D
E
25
A
B
C
D
E
50
A
B
C
D
E
The number of right answers:
The number of wrong answers:
# of correct
Dr. John Chung's SAT II Math Level 2 Test 4
1 4 # of wrong
Raw score
169
2
2 Score Conversion Table 6
Scaled Score 480
630
5
470
26
620
4
470
800
25
620
3
460
46
800
24
610
2
460
45
800
23
610
1
450
44
800
22
600
0
450
43
800
21
600
42
800
20
590
41
800
19
590
40
780
18
580
39
760
17
570
38
750
16
560
37
740
15
550
36
720
14
540
35
710
13
530
34
700
12
520
33
690
11
510
32
680
10
500
31
670
9
490
30
660
8
490
29
650
7
480
28
Scaled Score 640
800
27
48
800
47
50
Scaled Score 800
49
Raw Score
170
Raw Score
Raw Score
2
2
MATHEMATICS LEVEL 2 TEST For each of the following problems, decide which is the BEST of the choices given. If the exact numerical value is not one of the choices, select the choice that best approximates this value. Then fill in the corresponding circle on the answer sheet Note: (1) A scientific or graphing calculator will be necessary for answering some (but not all) of the questions in this test. For each question you will have to decide whether or not you should use a calculator. (2) For some questions in this test you may have to decide whether your calculator should be in the radian mode or the degree mode. (3) Figures that accompany problems in this test are intended to provide information useful in solving the problems. They are drawn as accurately as possible EXCEPT when it is stated in a specific problem that its figure is not drawn to scale. All figures lie in a plane unless otherwise indicated. (4) Unless otherwise specified, the domain of any function f is assumed to be the set of all real numbers
x for which f ( x) is a real number. The range of f is assumed to be the set of all real numbers f ( x) , where x is in the domain of f . (5) Reference information that may be useful in answering the questions in this test can be found on the page preceding Question 1. USE THIS SPACE FOR SCRATCHWORK 2
1.
2
1 1 a a a a (A) 4 (B) 4 (C) 2 (D) 2 (E) 2a
2.
1 If cos x for 0 x , then sin 2x 3 2 (A) (B) (C) (D) (E)
0.25 0.30 0.50 0.63 0.75
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Dr. John Chung's SAT II Math Level 2 Test 4
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2
2 MATHEMATICS LEVEL 2 TEST - Continued USE THIS SPACE FOR SCRATCHWORK.
3.
The inverse of which of the following graphs is also a function? (A)
(B) y
y
O
x
(C)
O
x
(D) y
y
O
x
O
x
(E) y
O
x
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172
2
2 MATHEMATICS LEVEL 2 TEST - Continued USE THIS SPACE FOR SCRATCHWORK.
4.
If
x 2 5 , then x
(A) 5 only (B) 5 only (C) 5 and 5 (D) 25 and 15 (E) 25 only
5.
2 2 The radius of the circle x 2 x y 4 y 9 is
(A) (B) (C) (D) (E)
6.
3.00 3.74 4.12 5.43 6.15
If f ( x) 9 x 2 x and g ( x)
(A) 7.
2 3
(B)
3 3
(C)
x 1 , then f g (2) x 1
5 3
(D)
6 3
(E)
2 2 3
If f ( x) 5 for all real numbers x , then f ( x 2) f ( x 2)
(A) 0 8.
(B) 2
(C) 5
(D) 10
(E) 20
10! 10! 9!2 100! 81! 100 (B) 81
(A)
(C) 100 1000 81 10000 (E) 9
(D)
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2
2 MATHEMATICS LEVEL 2 TEST - Continued USE THIS SPACE FOR SCRATCHWORK.
9.
The set of all real numbers of x such that x 2 1 1 consists of
x 2 (B) 2 x 0 0 x 2 (C) 0 x 2 (D) x 2 x 2 (E) x 2 (A)
10.
If the line y x k is tangent to the graph of the circle
x2 y 2 4 , then k (A) 2 2 only (B) 3 2 only (C) 4 2 only (D) 2 2 (E) 4 2
11.
x If f ( x) 2 ln x 1 and g ( x) e , then
g f ( x)
(A) e x 1 2
(B) e x 2 x 1 (C) x 1 (D) 2( x 1) (E) x 2 2 x 1 12.
3 If f ( x) x 3x 1 , then f 1 f ( x)
(A) x (B) x 2 (C) x3 3x 1 3
(D) (E)
x3 3x 1
x
3
3x 1
3
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2
2 MATHEMATICS LEVEL 2 TEST - Continued USE THIS SPACE FOR SCRATCHWORK.
13.
log16 81 log 2 3 (A) 2
14.
(C) 0
(D) 1
(E) 3
Three numbers have a sum of 36, a product of 1680, and form an arithmetic sequence. What is the largest number? (A) (B) (C) (D) (E)
15.
(B) 1
10 12 14 16 18
If a bi
3i , which of the following is true? 1 i
(A) a 1, b 2 (B) a 2, b 1 (C) a 2, b 1 (D) a 2, b 1 (E) a 2, b 1
16.
If sin( A B) 0.25 , sin A
3 , and 2
90 A B 180 , then B could be (A) 65 (B) 80.5 (C) 105.5 (D) 120.5 (E) 125.4
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2
2 MATHEMATICS LEVEL 2 TEST - Continued USE THIS SPACE FOR SCRATCHWORK.
17.
2 If the equation of a parabola is y 2 x , then the directrix of the graph is
(A) y 2 1 2 1 (C) y 8
(B) y
1 8 1 (E) y 2 (D) y
18.
Which of the following is symmetric with respect to the origin? 2 (A) y x x
(B) y x 5 5 3 (C) y x 3x x 6 4 2 (D) y x x x 7 5 (E) y x x 1
19.
What is the x-intercept of the hyperbola
x 12 y 2 2 10
4
1?
(A) (4.16, 0) and (4.16, 0) (B) (3.12, 0) and (3.12, 0) (C) (4.16, 0) and (2.16, 0) (D) (5.47, 0) and (3.47, 0) (E) (5.12, 0) and (5.12, 0)
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2
2 MATHEMATICS LEVEL 2 TEST - Continued USE THIS SPACE FOR SCRATCHWORK.
20.
sin 2 (A) sin (B) sin (C) cos (D) cos (E) sin cos
21.
Which of the following is the solution set of ( x 2)( x 1)2 0 x2 (A) (B) (C) (D) (E)
22.
There are 4 boys and 5 girls in a chess club. In how many ways could 3 boys and 3 girls be selected to attend the school tournament? (A) (B) (C) (D) (E)
23.
x 2 x0 2 x 1 2 x 2 2 x 1 or x 2
40 80 120 360 720
1 , what 10 is the probability that a package of 10 light bulbs has exactly two defective bulbs?
If the probability that a light bulb is defective is
(A) 0.01
(B) 0.10
(C) 0.19
(D) 0.25
(E) 0.33
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2
2 MATHEMATICS LEVEL 2 TEST - Continued USE THIS SPACE FOR SCRATCHWORK.
24.
A polynomial P( x) has remainder of three when divided by ( x 1) and remainder of five when divided by ( x 2) . If P( x) is divided by ( x 1)( x 2) , then the remainder is (A) (B) (C) (D) (E)
25.
8 x8 2x 1 2x 1 3x 1
In Figure 1, a triangle is inscribed in a semicircle. If BC 10 , what is the area of ABC in terms of ? (A) 50sin
B
(B) 50cos (C) 50sin cos (D) 50 tan
O
C
Figure 1
50 tan
(E)
26.
A
10
If a rectangular prism has dimensions a, b, and c , which of the following represents the length of its diagonal?
abc
(A) (B)
3
a 2 b2 c 2
(C)
a 2 b2 c 2
(D)
a3 b3 c3
(E)
3
a3 b3 c3
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2
2 MATHEMATICS LEVEL 2 TEST - Continued USE THIS SPACE FOR SCRATCHWORK.
27.
If vectors a (3, 4) and b (2, 3) , then a b (A) (B) (C) (D) (E)
28.
If 3 4i is a root of 2 x 2 ax b 0 , then b (A) (B) (C) (D) (E)
29.
5.68 7.07 8.60 9.13 10.87
25 25 50 50 It cannot be determined from the information given.
If cos
(A)
3 3 8
(B) (C)
30.
1 and 90 180 , then sin 2 equals 3
3 2 8
4 2 9
(D)
4 2 9
(E)
5 2 11
What is the distance from the plane 3x 4 y 5 z 10 0 to the point (0, 0, 0)? (A)
2
(B) 2 (C) 2 2 (D) 4 (E) 4 2
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2
2 MATHEMATICS LEVEL 2 TEST - Continued USE THIS SPACE FOR SCRATCHWORK.
31.
If x0 1 and xn 1 xn 2n , then x10 (A) (B) (C) (D) (E)
32.
20 91 162 268 381
The line ax by 4 0 forms a triangular region with the x-axis and y -axis . What is the area of the region in terms
of a and b ? 2 ab
(A)
33.
(B)
4 ab
(C)
ab
8
(D)
ab
16
(E)
8 ab
Which of the following is equivalent to the expression tan 70 tan 20 ? 1 tan 70 tan 20 (A) tan 90 (B) tan 50 tan 90 tan 50
(C)
tan 50 1 tan 50 1 tan 500 (E) tan 50o (D)
34.
In ABC , B is an obtuse angle, AB 15 , BC 20 , and the area of the triangle is 90. What is the measure of B ? (A) (B) (C) (D) (C)
0.36 0.64 2.50 5.48 5.63
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180
2
2 MATHEMATICS LEVEL 2 TEST - Continued USE THIS SPACE FOR SCRATCHWORK. 10
35.
1 The constant term of the expansion of x is x (A) 1
36.
(C) 150
(D) 210
(E) 252
The lateral surface area of a right cylinder in Figure 2 is 80. If the height of the cylinder is 10, what is the volume of the cylinder? (A) (B) (C) (D) (E)
37.
(B) 45
10
48.4 50.9 54.8 60.3 61.4
Figure 2
2 Figure 3 shows the graph of y ax bx c . Which of the
y
following could NOT be true? (A) ab 0 (B) bc 0 (C) ac 0
O
(D) b 2 4ac
Figure 3
(E) b 2 4ac 38.
x
Which of the following is an odd function? 2
(A) f ( x) x 5 (B) f ( x) x sin x (C) f ( x) x x (D) f ( x) 3 (E) f ( x)
39.
1 x 1 2
If Arc cos
(A)
1 2
1 , what is the value of sin ? 2 2
(B)
1 2
(C)
3 2
(D)
3 2
(E)
3 3
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181
2
2 MATHEMATICS LEVEL 2 TEST - Continued USE THIS SPACE FOR SCRATCHWORK.
40.
2 If f ( x) ( x 1)( x x 1) , which of the following
statements are true? I. The function f is increasing for x 1 II. The function f ( x) 0 has three real solutions. III. The domain of the function f ( x) is all real numbers. (A) (B) (C) (D) (E)
41.
I only II only I and III only II and III only I, II, and III
3 1 If A is obtuse and cos A , cos A is 5 2 3 5 1 (B) 5 1 (C) 5 (A)
42.
(D)
5 5
(E)
2 5 5
What is the length of the major axis of an ellipse whose 2 2 equation is 4 x 16 x y 4 y 16 0 ?
(A) (B) (C) (D) (E)
1 2 4 6 8
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182
2
2 MATHEMATICS LEVEL 2 TEST - Continued USE THIS SPACE FOR SCRATCHWORK.
43.
Which of the following is an even function? (A) f ( x) sin x (B) f ( x) tan x 2x (C) f ( x) e 2 (D) f ( x) 2x 3
(E) f ( x) log x 44.
What is the sum of the infinite series 2 4 8 1 ....? 3 9 27 (A) (B) (C) (D) (D)
lim
45.
n
(A) (B) (C) (D) (E) 46.
0 0.2 0.4 0.6 Infinite 2 n2 n n
0 2 4 10 Infinite
If y log5 x 2 6 x 14 , what is the minimum value of the equation ? (A) 2 (B) 1 (C) 1 (D) 2 (E) 5
47.
If
n 1 P2
(A) 5
n P2 12 , what is the integer value of n ? (B) 6
(C) 7
(D) 8
(E) 10
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183
2
2 MATHEMATICS LEVEL 2 TEST - Continued USE THIS SPACE FOR SCRATCHWORK.
48.
The middle term of the expansion of x 2 y is 4
2 2 (A) 12x y 2 2 (B) 12x y
(C) 24x 2 y 2 2 2 (D) 24x y 2 2 (E) 32x y
49.
If f ( x)
9 x 2 and x 0 , what is the inverse of f ( x) ?
(A) f 1 ( x) x 2 3 (B) f 1 ( x)
9 x 2 and x 0
(C) f 1 ( x) 9 x 2 and x 0 (D) f 1 ( x) 9 x 2 and x 0 (E) f 1 ( x) x 2 9 and x 0
50.
y
0, 2
In Figure 4, what is the equation of line that is tangent to 2
2
the circle x y 1 and passes through the point (0, 2) ? (A) y x 2 (B) y 2 x 2
x
O
(C) y 3x 2 (D) y 2 x 2 (E) y 3 x 2
Figure 4
STOP IF YOU FINISH BEFORE TIME IS CALLED, YOU MAY CHECK YOUR WORK ON THIS TEST ONLY. DO NOT TURN TO ANY OTHER TEST IN THIS BOOK.
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184
No Test Material on This Page
Dr. John Chung's SAT II Math Level 2 Test 4
185
2
2
TEST 4 # 1 2 3 4 5 6 7 8 9 10
answer B D C C B D D C B D
ANSWERS # 11 12 13 14 15 16 17 18 19 20
answer E A C C C C D C D D
# 21 22 23 24 25 26 27 28 29 30
answer D A C D E C C C D A
# 31 32 33 34 35 36 37 38 39 40
answer B E B C E B E B A C
Explanations: Test 4 2
1. (B)
1 1 Since a a 2 2 2 and a a 2
2
1 1 2 a a 2 2 , a a
2
1 1 then a a 4 . a a 2. (D)
In Quadrant I:
3 x 1
2 2 1 sin 2 x 2sin x cos x 2 3 3 0.628539... 0.63
2 2
3. (C)
Only choice (C) passed a horizontal line test. Only one-to-one functions have an inverse function.
4. (C)
Since
5. (B)
x2 2 x y 2 4 y 9
x 2 x , x 5 . Therefore, x 5 .
x 12 y 2 2 14
r 2 14 r 3.74165... 3.74
2 1 1 , 2 1 3
2
2 6 1 1 1 f 9 3 3 3 3 3
6. (D)
g (2)
7. (D)
f ( x) 5 is a constant function for any real x . Therefore, 5 5 10.
186
# 41 42 43 44 45 46 47 48 49 50
answer D C D D C C B C C C
2 8. (C) 9. (B)
2 Since 10! 10 9! , x2 1 1
10. (D)
x 0
10 9! 10 9! 100 . 9! 9! 1 x2 1 1
2x 2
x
2
0 x2 2 0
x2 ( x k )2 4
Substitute y x k .
x 2 x 2 2kx k 2 4 2 x 2 2kx k 2 4 0 Since the line is tangent to the ellipse, its discriminant should be 0. D 4k 2 4(2)(k 2 4) 0 k2 8 k 2 2 y Or, since OM 2 , OM MP , OPR is an isosceles triangle, and MP 2 ,
OP 22 22 2 2 . Therefore, the values of k are 2 2 or 2 2 .
O
R 2 M
P
x
2
2
2
11. (E)
g f ( x) e2ln( x 1) eln( x 1) ( x 1)2 x 2 2 x 1
12. (A)
f 1 f ( x) x and f f 1 ( x) x.
13. (C)
Since log 2 3 log 24 34 log16 81 , then log16 81 log16 81 0 .
14. (C)
Let three numbers be a d , a, a d , where d 0 .
a d a a d 3a 36 12 d 12 12 d 1680
a 12
144 d 2 140 d 2 4 d 2
Therefore, the largest number is 12 2 14. 15. (C)
3 i 3 i 1 i 4 2i 2i 1 i 1 i 1 i 2 Since a bi 2 i , a 2 and b 1.
16. (C)
Since sin( A B) 0.2 , A B 165.5224... 165.5 in Quadrant II. sin A
3 2
A 60 or 120
Therefore, 165.5 60 105.5 or 165.5 120 45.5 .
Dr. John Chung's SAT II Math Level 2 Test 4
187
2 17. (D)
2 The standard form of the parabola is x 2
1 1 y 4 y . 2 8
1 1 Therefore, the focus is at 0, and the directrix is y . 8 8 18. (C)
Odd functions are symmetric with respect to the origin. (C) is odd function.
19. (D)
Let y 0 .
20. (D)
sin sin cos cos sin cos 2 2 2
21. (D)
Method 1) Graphic Solution: multiply by ( x 2) 2 0 .
x 12
x 12 20
1 1
10 x 1 2 5 x 5.47 or 3.47
( x 2) 2
( x 2)( x 1) 2 0 ( x 2)2 ( x 2)
x
2 1
Therefore, y 0 in the interval 2 x 2 . Method 2) Test value: Test value
3
0 2
1
1.5
()() 0 F () ()() 0 (T) At x 0 () ()() 0 (T) At x 1.5 () ()() 0 (F) At x 3 ( )
2
3
At x 2
Method 3) Use graphic utility directly.
188
20
( x 2)( x 2)( x 1)2 0 and x 2
y ( x 2)( x 2)( x 1) 2
y
2
x 1
2 x 2
2
2 4 4 C3 3
22. (A)
4 5 4 10 40 3 3
23. (C)
10 1 9 0.1937... 0.19 2 10 10
2
24. (D)
25. (E)
8
P( x) ( x 1)Q1 ( x) 3 -------(1) P( x) ( x 2)Q2 ( x) 5 ------(2) P( x) ( x 1)( x 2)Q( x) ax b -------(3) From equations (1) and (3) P(1) 3 a b --------(4) From equations (2) and (3) P(2) 5 2a b ------(5) From (4) and (5) a 2 and b 1 . Therefore, the remainder is 2 x 1.
10 10 , AB . tan AB 1 10 The area of ABC 10 2 tan Since tan
50 . tan
26. (C)
The length of the diagonal a 2 b 2 c 2
27. (C)
z a b 3 (2), 4 3 (5, 7) ,
28. (C)
z 52 (7) 2 8.602325 8.60
Since 3 4i is a root of the equation, then its conjugate 3 4i is also the root of the equation. b The product of the roots is . 2 b (3 4i )(3 4i ) 25 b 50 2
29. (D) 22
3
2 2 1 4 2 sin 2 2sin cos 2 9 3 3
II
1 III
30. (A)
D
IV
3(0) 4(0) 5(0) 10 32 4 5 2
Dr. John Chung's SAT II Math Level 2 Test 4
2
10 5 2
2 2
2
189
2
2
31. (B) n 1,
x2 x1 2
x2 x1 2 1 2
n 2,
x3 x2 4
x3 x2 4 1 2 4
n 3,
x4 x3 6
x4 x3 6 1 2 4 6
...... n 9,
x10 x9 18 1 2 4 6 18 (2 18) 9 Therefore, x10 1 2 4 6 18 1 91 2
32. (E)
33. (B)
34. (C)
35. (E)
x10 x9 18
4 , a 1 4 The area of the triangle A 2 a Since tan( A B)
4 b
4 8 b ab : area cannot be negative.
tan A tan B tan 70 tan 20 . , tan(70 20 ) 1 tan A tan B 1 tan 70 tan 20
15 20 sin B 3 3 90 sin B B sin 1 0.6435 2 5 5 Since B is obtuse, B 0.6435 2.50 . Area of ABC
Since the general term is the constant term is
36. (B)
y-intercept: by 4 y
x-intercept: ax 4 x
10 Cr
10 C5 x
x r
10 r
1 x
when r 5.
The lateral area: 2 rh 80 r
10 Cr x r x 10 r 10 Cr x 2 r 10 , That is
10 C5
252 .
80 80 4 2 h 20r
2
160 4 50.9295 50.9 V r 2 h 10 37. (E)
From the graph, (1) Concave down ----- a 0 b 0 b0 2a (3) y-intercept ----- f (0) c 0 (2) Axis of symmetry -----
(4) Two unequal roots ---- D b 2 4ac 0 b 2 4ac (E) is not true because y f ( x) has two real roots. 38. (B)
Since y x and y sin x are odd functions, then f ( x) x sin x is an odd function.
39. (A)
60 1 1 Since arccos 60 , sin sin 30 . 2 2 2
190
2 40. (C)
2 Graphic utility: The graph of f ( x) ( x 1)( x 2 x 1) is as follows. y
i) The function is increasing for x 1. ii) The domain of the function is all real x .
1
x
Algebraically: x 2 x 1 is positive for all real x , because its discriminant D 0 . Therefore, for the interval x 1 , f ( x) is always positive. 41. (D)
Since cos A
A 63.4349 acute angle 2
3 and A is obtuse, A 126.8698976 . 5
3 1 A 1 cos A 5 1 5 . Therefore, cos 2 2 2 5 5 Or, cos (63.4349) 0.447214 42. (C)
5 . 5
The standard expression of the ellipse: 4( x 2 4 x 4) ( y 2 4 y 4) 16 16 4 ( x 2)2 ( y 2) 2 1 a2 4 a 2 1 4 Therefore, the length of the major axis 2a 4 4( x 2)2 ( y 2) 2 4
43. (D) 44. (D)
f ( x) 2 x 2 3 , because f ( x) f ( x) . Since r
2 3 a and 1 , the sum of the series 3 4 1 r
2
45. (C)
lim
x
n2 n n
n2 n n
n2 n n
lim
x
2
n2 n n 2
n nn
2
1 3 . 2 5 1 3
lim 2 x
n2 n n
n
1 lim 2 1 1 4 x n
Dr. John Chung's SAT II Math Level 2 Test 4
191
2 46. (C)
2 The graph of f ( x) x 2 6 x 14 has a minimum of 23 at x 3 (axis of symmetry). Therefore, the minimum of y log 5 5 1. Or using graphic utility: Trace the minimum. y
x
O
Minimum x 2.999999982 y 1
47. (B)
n 1 P2
n P2 12 (n 1)n n(n 1) 1 n 2 n n 2 n 12
n6 48. (C)
49. (C)
4 2 2 The middle term is the third term: x 2 y 24 x 2 y 2 2 4 4 0 x 2 y is the first term. 0 The domain of function
f :x0
The range of f 1 ( x) : y 0
The range of function f : y 0 The domain of f 1 ( x) : x 0 Therefore, the inverse can be obtained as follows,
y 9 x2
switch x and y
x 9 f 1
2
f
1 2
9 x2
f 1 9 x 2
Since the range of the inverse is y 0 and its domain is x 0 , f 1 ( x) 9 x 2 and x 0 . y
y 9 x2
O
x
f 1 9 x 2
192
2 50. (C)
2 Let the equation of line be y mx 2 . Since the line is tangent to the circle, the discriminant of the equation x 2 (mx 2)2 1 must be 0.
x 2 m 2 x 2 4mx 4 1 0 (1 m 2 ) x 2 4mx 3 0 D 4m 4(1 m 2 )(3) 0 16m 2 12 12m 2 0 4m 2 12 2
m 2 3 m 3 , Since the line has a negative slope, the equation of line is y 3x 2 . END
Dr. John Chung's SAT II Math Level 2 Test 4
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No Test Material on This Page
194
Test 5 Dr. John Chung’s SAT II Mathematics Level 2
Dr. John Chung's SAT II Math Level 2 Test 5
195
2
2
MATHEMATICS LEVEL 2 TEST REFERENCE INFORMATION
THE FOLLOWING INFORMATION IS FOR YOUR REFERENCE IN ANSWERING SOME OF THE QUESTIONS IN THIS TEST 1 Volume of a right circular cone with radius r and height h : V r 2 h 3 1 Lateral Area of a right circular cone with circumference of the base c and slant height : S c 2
Volume of a sphere with radius r : V 4 r 3 3 Surface Area of a sphere with radius r : S 4 r 2 1 Volume of a pyramid with base area B and height h : V 3 Bh
196
2
2 Dr. John Chung’s SAT II Math Level 2
Answer Sheet 01 02
A
B
C
D
E
A
B
C
D
E
03
A
B
C
D
E
B
C
D
04 05
A A
B
C
06
A
B
07
A
08
26 27
A
B
C
D
E
A
B
C
D
E
A
B
C
D
E
E
28 29
A
B
C
D
E
D
E
30
A
B
C
D
E
C
D
E
A
B
C
D
E
B
C
D
E
31 32
A
B
C
D
E
A
B
C
D
E
33
A
B
C
D
E
09
A
B
C
D
E
34
A
B
C
D
E
10 11
A
B
C
D
E
A
B
C
D
E
A
B
C
D
E
35 36
A
B
C
D
E
12
A
B
C
D
E
37
A
B
C
D
E
13 14
A
B
C
D
E
B
C
D
E
A
B
C
D
E
38 39
A
A
B
C
D
E
15
A
B
C
D
E
40
A
B
C
D
E
16 17
A
B
C
D
E
A
B
C
D
E
A
B
C
D
E
41 42
A
B
C
D
E
18
A
B
C
D
E
B
C
D
E
19 20
B
C
D
E
43 44
A
A
A
B
C
D
E
A
B
C
D
E
A
B
C
D
E
A
B
C
D
E
A
B
C
D
E
21 22
45 46
A
B
C
D
E
47
A
B
C
D
E
23
A
B
C
D
E
48
A
B
C
D
E
24
A
B
C
D
E
49
A
B
C
D
E
25
A
B
C
D
E
50
A
B
C
D
E
The number of right answers:
The number of wrong answers:
# of correct
Dr. John Chung's SAT II Math Level 2 Test 5
1 4 # of wrong
Raw score
197
2
2 Score Conversion Table 6
Scaled Score 480
630
5
470
26
620
4
470
800
25
620
3
460
46
800
24
610
2
460
45
800
23
610
1
450
44
800
22
600
0
450
43
800
21
600
42
800
20
590
41
800
19
590
40
780
18
580
39
760
17
570
38
750
16
560
37
740
15
550
36
720
14
540
35
710
13
530
34
700
12
520
33
690
11
510
32
680
10
500
31
670
9
490
30
660
8
490
29
650
7
480
28
Scaled Score 640
800
27
48
800
47
50
Scaled Score 800
49
Raw Score
198
Raw Score
Raw Score
2
2
MATHEMATICS LEVEL 2 TEST For each of the following problems, decide which is the BEST of the choices given. If the exact numerical value is not one of the choices, select the choice that best approximates this value. Then fill in the corresponding circle on the answer sheet Note: (1) A scientific or graphing calculator will be necessary for answering some (but not all) of the questions in this test. For each question you will have to decide whether or not you should use a calculator. (2) For some questions in this test you may have to decide whether your calculator should be in the radian mode or the degree mode. (3) Figures that accompany problems in this test are intended to provide information useful in solving the problems. They are drawn as accurately as possible EXCEPT when it is stated in a specific problem that its figure is not drawn to scale. All figures lie in a plane unless otherwise indicated. (4) Unless otherwise specified, the domain of any function f is assumed to be the set of all real numbers
x for which f ( x) is a real number. The range of f is assumed to be the set of all real numbers f ( x) , where x is in the domain of f . (5) Reference information that may be useful in answering the questions in this test can be found on the page preceding Question 1. USE THIS SPACE FOR SCRATCHWORK 1.
Which is the negation of the statement “Some numbers are even”? (A) (B) (C) (D) (E)
2.
All numbers are even. Some numbers are not even. All numbers are not even. All numbers are odd. Some numbers are not odd.
In how many ways can 2 juniors and 2 seniors be selected from a group of 8 juniors and 6 seniors? (A) (B) (C) (D) (E)
4 48 420 480 840
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Dr. John Chung's SAT II Math level 2 Test 5
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2
2 MATHEMATICS LEVEL 2 TEST - Continued USE THIS SPACE FOR SCRATCHWORK.
3.
Find the largest integral value of k such that the roots of
x 2 5 x k 0 are real? (A) 4
4.
(E) 10
B
10
8 A
D
C
Figure1
If a and b are positive numbers, and a 2 b 2 29 and ab 10 , then a b
7
(B) 7
(C) 8
(D) 5 2
(E) 7 2
If x 2i is a solution to the equation x 3 kx 0 , what is the value of k ? (A) (B) (C) (D) (E)
7.
(D) 8
66.67 112.45 125.36 133.33 150
(A)
6.
(C) 7
In Figure 1, BD is the altitude to the hypotenuse AC . If BD 8 and BC 10 , which is the area of ABC ? (A) (B) (C) (D) (E)
5.
(B) 6
8 6 4 2 1
1 and g ( x) x 5 , what is the domain of x f g ( x) ?
If f ( x)
(A) (B) (C) (D) (E)
All All All All All
x such that x such that x such that x such that x such that
x 5 x0 x0 x 0 and x 5 x 0 and x 5
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200
2
2 MATHEMATICS LEVEL 2 TEST - Continued USE THIS SPACE FOR SCRATCHWORK.
8.
Which of the following graphs best describes the set of points a a , b for which b 2 in the xy -plane ? 2 (A) (B) y y
x
O
(C)
x
O
(E)
y
O
x
y
x
O
9.
(D)
y
x
O
Which of the following is symmetric with respect to the y -axis ? (A) y x 1
2
(B) x y 2 (C) x 2 4 y 2 4 (D) x 2 2 x y 2 3 (E) y x 2
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2
2 MATHEMATICS LEVEL 2 TEST - Continued USE THIS SPACE FOR SCRATCHWORK.
10.
If x 0.4 10 , then what is the value of x 0.6 ? (A) 10 10 (B) 15 5 (C)
100
(D) 15 10 (E) 11.
10000
1 If e x 3 , what is the value of 3 e
(A) 729
(B) 64
(C) 27
(D)
2 x
? 1 64
(E)
1 729
1 x 5 , what is the value of x ? If 1 1 2 x x
12.
(A) 5
13.
(C) 1.25
(D) 1.25
(E) 5
Figure 2 shows a hemisphere with a radius of 4. Find the surface area of that figure. (A) (B) (C) (D) (E)
14.
(B) 3
20 32 36 42 48
4
Figure 2
If two forces of 10 pounds and 15 pounds act on a body with an angle of 60 between them, what is the magnitude of the resultant? (A) 16.80 (B) 18.21 (C) 20.42 (D) 21.80 (E) 24.92
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2
2 MATHEMATICS LEVEL 2 TEST - Continued USE THIS SPACE FOR SCRATCHWORK.
15.
2 3 x 2 If the matrix equation , what is the 5 y 3 2 value of y ? (A) (B) (C) (D) (E)
16.
0.625 0.505 0.125 3.500 4.254
In the arithmetic progression an , a2 50 and a4 44 . Which is the first term that is a negative number? (A) (B) (C) (D) (E)
17.
If log 2 x log 2 ( x 1) 1 , which of the following is the solution set of the inequality? (A) (B) (C) (D) (E)
18.
17th 18th 19th 20th 21th
x x 1 x 1 x 2 x 0 x 2 x 1 x 2 x x 2
In a game, the probability of winning is
1 and the 4
3 . If 3 games are played, what is 4 the probability of winning at least 2 games?
probability of losing is
(A)
3 64
(B)
5 64
(C)
5 32
(D)
10 27
(E)
5 16
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2
2 MATHEMATICS LEVEL 2 TEST - Continued USE THIS SPACE FOR SCRATCHWORK.
19.
The graph of a polynomial function is shown in Figure 3. Which of the following could be the equation of the polynomial function?
y
(A) P( x) x( x 1)( x 3)( x2 1) (B) P( x) x( x 1)( x2 9)
x
O
(C) P( x) x( x 1)( x 3)( x2 5) (D) P( x) x( x 1)( x 3)( x2 5x 10) Figure 3
(E) P( x) x( x 1)( x 3)2 ( x2 1)
20.
If sin A 60 cos 40 , the measure of A is (A) 110
21.
(C) 80
(D) 45
(E) 20
If tan A 4 and tan B 3 , what is the value of tan( A B) ? (A) (B) (C) (D) (E)
22.
(B) 90
0.065 0.077 0.126 0.245 0.333
x If
2
3x 4
0 , which of the following is the solution x2 of the inequality?
(A) (B) (C) (D) (E)
x 1 x 1 1 x 4 1 x 0 or 0 x 4 All real x
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2
2 MATHEMATICS LEVEL 2 TEST - Continued USE THIS SPACE FOR SCRATCHWORK.
23. In Figure 4, line is the perpendicular bisector of AB at point E . What is the area of ADE ? (A) (B) (C) (D) (E)
y
A(0, 2)
2 3 4 5 10
E
x B (4, 0)
D
Note: Figure not drawn to scale. 24.
In Figure 5, if the radius of the semicircle is 5, what is the area of the inscribed square? (A) 9
25.
(C) 20
(D) 25
(E) 36
1 What is the value of sec arctan ? 3 (A)
26.
(B) 16
3 2
(B)
2 3 3
(C)
2 2 3
Figure 4
(D)
3 4 (E) 5 7
5
Figure 5
If two lines y 2 x 4 and y mx 5 are parallel, where m is a constant, then the distance between the two lines is
(A) (B) (C) (D) (E)
27.
4.02 5 5.4 6.25 8
If the parametric equations are x 4sin 2 and y 2 cos 2 , which of the following represents the graph of point x, y ? (A) (B) (C) (D) (E)
Line Parabola Hyperbola Ellipse Circle
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2
2 MATHEMATICS LEVEL 2 TEST - Continued USE THIS SPACE FOR SCRATCHWORK.
28.
If the equation of a circle is x2 2 x y 2 4 y 1 , then the area of the circle is (A) (B) (C) (D) (E)
29.
5 6 25 36 42
z
In Figure 6, the graph of plane 2 x 3 y 4 z 12 in three
B
dimensions forms a triangular pyramid with base AOC . What is the volume of the pyramid? (A) (B) (C) (D) (E)
12 18 36 48 72
2 x 3 y 4 z 12
C
O
x
A y
Note: Figure not drawn to scale. Figure 6
30.
sin sin tan 2
2
2
(A) sin 2 (B) cos 2 (C) tan 2 (D) cot 2 (E) sec 2
31.
If the value of f ( x) x2 3x k is always positive for any x , which of the following could be the value of k ? (A) 3 (B) 2 (C) 0 (D) 2 (E) 3
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2
2 MATHEMATICS LEVEL 2 TEST - Continued USE THIS SPACE FOR SCRATCHWORK.
32.
In Figure 7, AB 5 and BC 10 . What is the area of the quadrilateral? (A) (B) (C) (D) (E)
33.
25.6 28.4 32.5 42.6 62.9
A
5 130
B 10
If a( x 1) b( x 1) ( x 2) 0 for all real x , where
C D Note: Figure not drawn to scale. Figure 7
a and b are constants, what is the value of a ? (A) 1.5
34.
(B) 2.0
(C) 2.5
(D) 3.5
(E) 4.5
2 When polynomial f ( x) 2 x 5x k is divided by 2 x 3 ,
the remainder is 5. What is the value of constant k ? (A) (B) (C) (D) (E)
35.
1 3 5 7 8
What is the value of sin x cos x sin 2 x ? 2
(A) 1
36.
(B) 0
(C) 1
(D) 2
(E) 3
What is the value of in the interval 0
2
that satisfies
the equation 6 cos 1 5sec ? (A) (B) (C) (D) (E)
0.45 0.59 0.62 0.78 0.82
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2
2 MATHEMATICS LEVEL 2 TEST - Continued USE THIS SPACE FOR SCRATCHWORK.
37.
If the ratio of the two roots of the equation x 2 kx 18 0 is 1: 2 , which of the following is all the values of constant k? (A) 3,6 (B) (C) (D) (E)
38.
9, 9 9 10,12 6, 9
If f ( x)
2x 1 and f 1 ( x) is the inverse of f ( x) , then x 1
f 1 (3) (A) 3
39.
(C) 4
(D) 6
(E) 10
Figure 8 shows a triangle in a circle with center O . If the radius of circle O is 2, what is the area of the triangle in terms of ? (A) (B) (C) (D) (E)
40.
(B) 2
sin 2 2sin 2sin 2 2 cos 2 cos 2
O
2
Figure 8
In five years, the population of Spring Lake decreased steadily from 50,000 to 45,000. Find the rate of decrease per year? (A) (B) (C) (D) (E)
1.2% 2.1% 2.4% 2.5% 3.0%
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2
2 MATHEMATICS LEVEL 2 TEST - Continued USE THIS SPACE FOR SCRATCHWORK.
41.
Find the measure of the angle between two forces of 10 pounds and 20 pounds if the magnitude of their resultant is 25 pounds. (A) 45.3 (B) 71.8 (C) 108.2 (D) 123.5 (E) 135.7
42.
Find the asymptotes of
x2 y 2 1. 8 18
3 (A) y x 2 2 (B) y x 3 2 (C) y x 9 9 (D) y x 2 9 (E) y x 4
43.
If x log 27 3 (A) 3
44.
log3 27
(B) 3
, then log 3 x (C)
1 3
(D)
1 9
(E)
1 27
In Figure 9, the volume of the right circular cone is 12 and the radius of the base is 3. What is the lateral area of the cone? (A) (B) (C) (D) (E)
4 6 15 18 36
3
Figure 9
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2
2 MATHEMATICS LEVEL 2 TEST - Continued USE THIS SPACE FOR SCRATCHWORK.
45.
The area of a triangle whose sides are of lengths 7, 20, and 23 is (A) 20 5 (B) 25 (C) 25 5 (D) 30 (E) 30 5
46.
What is the value of 1 i ? 10
(A) 8i (B) 16i (C) 32i (D) 32 (E) 64 47.
In how many ways can 10 people be divided into three groups, one group with 4 people and the other two groups with 3 people each? (A) (B) (C) (D) (E)
48.
210 420 2100 4200 326000
Which of the following equations could be the graph shown in Figure 10? y (A)
x2 y 2 1 16 6
(B)
( x 8) 2 ( y 3) 2 1 8 3
(0,3)
( x 8) 2 ( y 3) 2 1 (C) 16 9
O
(D)
( x 8) 2 ( y 3) 2 1 64 9
(E)
( x 8) 2 ( y 3) 2 1 64 9
(8,0)
x
Figure 10
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2
2 MATHEMATICS LEVEL 2 TEST - Continued USE THIS SPACE FOR SCRATCHWORK.
49.
Day 1
Day 2
Day 3
16GB
10
15
20
64GB
5
9
13
128GB
11
17
18
The table above shows the number of smart phones that were sold during a three-day sale. The prices of models 16GB, 64GB, and 128GB were $300, $400, and $500, respectively. Which of the following matrix representations gives the total daily income, in dollars, received from the sale of the smart phones for each of the three days?
10 15 20 (A) 5 9 13 300 400 500 11 17 18 10 15 20 300 (B) 5 9 13 400 11 17 18 500 10 5 11 300 (C) 15 9 17 400 20 13 18 500 300 10 5 11 (D) 400 15 9 17 500 20 13 18
(E) 300 10 5 11 400 15 9 17 500 20 13 18
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2
2 MATHEMATICS LEVEL 2 TEST - Continued USE THIS SPACE FOR SCRATCHWORK.
50.
From the binomial expansion of 2 x 3 , what is the 6
coefficient of x 4 ? (A) (B) (C) (D) (E)
60 68 720 2160 4320
STOP IF YOU FINISH BEFORE TIME IS CALLED, YOU MAY CHECK YOUR WORK ON THIS TEST ONLY. DO NOT TURN TO ANY OTHER TEST IN THIS BOOK.
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212
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Dr. John Chung's SAT II Math Level 2 Test 5
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2
2
TEST 5 # 1 2 3 4 5 6 7 8 9 10
ANSWERS
answer C C B A B C A E C A
# 11 12 13 14 15 16 17 18 19 20
answer A E E D C C D C D A
# 21 22 23 24 25 26 27 28 29 30
answer B D D C B A D B A C
# 31 32 33 34 35 36 37 38 39 40
answer A E A E C B B C C B
# 41 42 43 44 45 46 47 48 49 50
answer B A B C E C C D C D
Explanations: Test 5 1. (C)
2. (C)
Remember the negation of the word “some” is “all.” Therefore, the negation is “All numbers are not even.” 8 6 28 15 420 2 2
3. (B)
Discriminant: D 25 4k 0 4k 25 k 6.25 The largest integer value of k is 6.
4. (A)
DC 6
B
10
8 8
A
D
6
C
BD 2 AD DC 64 AD 6 AD
32 3
1 32 The area of ABC 6 8 66.6666 66.7. 2 3 5. (B)
(a b)2 a 2 b 2 2ab 29 2(10) 49 a b 7 a and b are positive.
6. (C)
Substitution: 2i k (2i) 0 8i 2ki 0 2k 8 k 4
7. (A)
Since f g ( x)
3
Dr. John Chung's SAT II Math Level 2 Test 5
1 x5
, the domain is x 5 0 x 5.
215
2 8. (E)
2 If y 0 , then y
a a 2 , and if y 0 , then y 2 . The graph is as follows. 2 2
y
y
a 2, where y 0 2 x
O
a 2, where y 0 2
y
9. (C)
(A) is symmetric with respect to x 1. (B) is symmetric with respect to the x-axis. (C) is symmetric with respect to the x-axis or y -axis.
x 12 y 2 4
(D) is symmetric with respect to x 1 or the y -axis. (E) is symmetric with respect to x 2.
10. (A)
x 0.6 x 0.4
11. (A)
1 3 e
2 x
3 2
3
10 2 10 10
e3
2 x
e6 x e x
6
36 729 6
Or, since e x 3 x ln 3 , then e6 x e6ln 3 eln 3 36. 12. (E)
Compound fraction: multiply common denominator by x 2 . 1 2 2 x x x3 x x x 1 x x 1 2 x2 1 x2 1 x 1 x2 Therefore, x 5.
13. (E)
Since the surface area of a sphere is 4 r 2 4 (4) 2 64 , the surface area of a hemisphere is 32 . The area of the circular base is 16 . Therefore, the entire surface area is 48 .
14. (D)
In a parallelogram, two consecutive angles are supplementary. R
10 60
o
10 120
o
15
Law of cosine: R 102 152 2(10)(15) cos120 21.79441 21.8
216
2 15. (C)
2 From the matrix equation: 2x 3y 2 2x 5 y 3
y 0.125
,
8 y 1 16. (C)
Since a2 a1 d 50 and a4 a1 3d 44 , then d 3 and a1 53 . an a1 (n 1)d an 53 (n 1)(3) 0 3n 56 Therefore, n 18.666 and the first negative term is the 19th term.
17. (D)
log 2 x log 2 ( x 1) 1 log 2 x 2 x 1 log 2 x 2 x log 2 2 Since base 2 is greater than 1, x 2 x 2 x 2 x 2 0 ( x 2)( x 1) 0 . The solution of the inequality is 1 x 2 , but x 1 from the logarithmic equation. Therefore, 1 x 2 . 2
18. (C) 19. (D)
1
3
0
3 1 3 3 1 3 9 1 5 64 64 32 2 4 4 3 4 4
The function has one zero at x 0 , one negative zero, and one positive zero. Choice (D) has one zero at x 0 , one negative zero, and one positive zero. x 2 5 x 10 has imaginary roots.
20. (A)
Cofunction: A 60 40 90 A 110
21. (B)
tan( A B)
22. (D)
Test value: Graphing utility: x 2 3x 4 0 x 2 3x 4 0 ( x 4)( x 1) 0 1 x 4 and x 0. 2 x Therefore, the solution set is 1 x 0 0 x 4
23. (D)
The slope of AB
tan A tan B 43 1 0.07692 0.077 1 tan A tan B 1 4 3 13
2 1 04 20 and the midpoint E is , 2, 1 . 4 2 2 2 The slope of the perpendicular line is 2 and passes through (2,1). The equation of line is y 2 x 3 . y
h
(2,1)
D(0, 3)
Dr. John Chung's SAT II Math Level 2 Test 5
Since AD 5 and the height h 2 , 52 5. the area of ADE is 2
A(0, 2)
x B (4, 0)
217
2 24. (C)
2 From the figure below: 52 x 2 (2 x)2 . 25 5 x 2 x 5 The length of an edge is 2 5 . 5
x
25. (B)
1 1 cos tan Or , algebraically 1 Let X tan 1 . tan X 3
Use calculator:
28. (B)
2
20.
1 3 1 3
1.1547
2 3 3
and 90 X 90 .
Therefore, sec X 1
1 2 2 3 . cos X 3 3
Choose one point on y 2 x 4 : that is (0, 4) . Two lines are parallel: m 2. The distance between a point (0, 4) and the line 2 x y 5 0 is D
27. (D)
5
2 X 3
26. (A)
The area of the square is 2 5
2x
2(0) (4) 5 2
2 (1)
2
9 5
4.02492 4.02
x y x2 y 2 and cos 2 , sin 2 2 cos 2 2 1 1. 4 2 16 4 The graph of the parametric equations is an ellipse.
Since sin 2
x 2 2 x y 2 4 y 1 ( x 1) 2 ( y 2)2 6 r 6
Therefore, the area of the circle are, r 2 6 . 29. (A)
The coordinates of each intercept is: C ( x,0, 0), When y 0 and z 0 , 2 x 3(0) 4(0) 12 x 0 and z 0 , 2(0) 3 y 4(0) 12 x 0 and y 0 , 2(0) 3(0) 4 z 12
A(0, y ,0), and B (0, 0, z ) x6. y4 z 3
6 4 3 Bh 2 12 The volume of the cone: 3 3
30. (C)
218
sin 2 sin 2 tan 2 sin 2 1 tan 2 sin 2 sec 2 sin 2
1 tan 2 cos 2
2
2
31. (A)
Since f ( x) is always positive, f ( x) 0 must have imaginary roots. 9 2 D b 2 4ac 3 4(1)(k ) 0 k 4 9 Choice (A) : 3 4
32. (E)
EC 10sin 40 , BE 10cos 40 , and DC 5 EC sin 40 . A
5
B 130o 40
EC 10sin 40
10
BE 10 cos 40 is the height of the trapezoid. D
E
C
The area of the trapezoid: h(b1 b2 ) 10cos 40 (5 5 10sin 40) 62.922415 62.9 A 2 2 33. (A)
a ( x 1) b( x 1) ( x 2) 0 (a b 1) x (a b 2) 0 To be identical, a b 1 0 and a b 2 0 . 3 Therefore, 2a 3 a . 2
34. (E)
Remainder theorem: (or, long division)
f ( x) 2 x 2 5 x k 2 x 3 Q ( x ) 5 35. (C)
sin x cos x 2 sin 2 x
36. (B)
Use a graphic utility. Graph y
3 9 15 f k 5 k 8 2 2 2
sin 2 x 2sin x cos x cos 2 x 2sin x cos x 1
5 6cos x 1 and find the zero. cos x
Or algebraically, 6cos 1 5sec 6cos 1
6 cos 5 cos 1 0
5 cos
6cos 2 cos 5 0
5 and cos 1 . Since 0 , (calculator must be in radian mode) 2 6 5 5 cos 1 0.58568 0.58 cos 6 6
Therefore, cos
37. (B)
Define the two roots as n and 2n. The product of the two roots: 2n 2 18 n 3 The sum of the roots: n 2n 3n 9
Dr. John Chung's SAT II Math Level 2 Test 5
219
2
2 2y 1 y 1
x 1 4 f 1 (3) 4 x2 1 2y 1 y4 Or, 3 y 1
f 1 : y
38. (C)
From the inverse: x
39. (C)
Since AB 2 cos and OB 2sin , 4cos 2sin 4sin cos 2sin 2 the area of OAC 2
2
OB OB 2sin 2 AB cos AB 2 cos 2
sin
O
C
B
A
40. (B)
45000 50000(1 r )
r 41. (B)
1 1 0.9 5
1 r
1 0.9 5
The Law of Cosines: ABC and are supplementary. 102 202 252 A cos 0.3125 2(10)(20) B
43. (B)
0.9 (1 r )
5
0.0208516376 2.1%
25
10
42. (A)
5
10
20
C
cos 1 0.3125 108.2099569o mABC 180 108.2099569 71.79004 71.8
x2 y 2 1 a 8 and b 18 8 18 b 18 3 Therefore, the asymptotes are : y x x x a 2 8
Since log 27 3
log 3 1 log 27 and log3 27 3 , then x log 27 3 3 31 3log 3 3
3
33 .
Therefore, log3 33 3 . 44. (C)
A rs , where r is a radius and s is a slant height. r 2h 9 h V 3 h 3 h 12 h 4 3 3 Therefore, s 5 . s h A 3 5 15 3
220
2 45. (E)
2 Heron’s Formula: s
abc 2
7 20 23 50 25 2 2 The area is s ( s a)( s b)( s c) 25(25 7)(25 20)(25 23) 30 5 s
46. (C) 47. (C) 48. (D)
Since 1 i 1 2i i 2 2i , (1 i )10 2i 25 i 5 32i . 2
10 C4
5
6 C3 3 C3 2100 2!
From the graph: center 8,3 , a 8 , and b 3. y
The equation of the ellipse is
x 8 2 y 32 (3, 0) O
49. (C)
50. (D)
82 (8, 0)
32
1.
x
Total income each day: Day 1: 10 (300) 5 (400) 11 (500) Day 2: 15 (300) 9 (400) 17 (500) Day 3: 20 (300) 13 (400) 18 (500) Its matrix form is choice (C). 6 6 6 r nr r r 6r 2 x 3 2 3 x r r Since x6 r x 4 , r 2 . 6 2 The coefficient of x 4 is 24 3 2160. 2
END
Dr. John Chung's SAT II Math Level 2 Test 5
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222
Test 6 Dr. John Chung’s SAT II Mathematics Level 2
Dr. John Chung's SAT II Math Level 2 Test 6
223
2
2
MATHEMATICS LEVEL 2 TEST REFERENCE INFORMATION
THE FOLLOWING INFORMATION IS FOR YOUR REFERENCE IN ANSWERING SOME OF THE QUESTIONS IN THIS TEST 1 Volume of a right circular cone with radius r and height h : V r 2 h 3 1 Lateral Area of a right circular cone with circumference of the base c and slant height : S c 2
Volume of a sphere with radius r : V 4 r 3 3 Surface Area of a sphere with radius r : S 4 r 2 1 Volume of a pyramid with base area B and height h : V 3 Bh
224
2
2 Dr. John Chung’s SAT II Math Level 2
Answer Sheet 01 02
A
B
C
D
E
A
B
C
D
E
03
A
B
C
D
E
B
C
D
04 05
A A
B
C
06
A
B
07
A
08
26 27
A
B
C
D
E
A
B
C
D
E
A
B
C
D
E
E
28 29
A
B
C
D
E
D
E
30
A
B
C
D
E
C
D
E
A
B
C
D
E
B
C
D
E
31 32
A
B
C
D
E
A
B
C
D
E
33
A
B
C
D
E
09
A
B
C
D
E
34
A
B
C
D
E
10 11
A
B
C
D
E
A
B
C
D
E
A
B
C
D
E
35 36
A
B
C
D
E
12
A
B
C
D
E
37
A
B
C
D
E
13 14
A
B
C
D
E
B
C
D
E
A
B
C
D
E
38 39
A
A
B
C
D
E
15
A
B
C
D
E
40
A
B
C
D
E
16 17
A
B
C
D
E
A
B
C
D
E
A
B
C
D
E
41 42
A
B
C
D
E
18
A
B
C
D
E
B
C
D
E
19 20
B
C
D
E
43 44
A
A
A
B
C
D
E
A
B
C
D
E
A
B
C
D
E
A
B
C
D
E
A
B
C
D
E
21 22
45 46
A
B
C
D
E
47
A
B
C
D
E
23
A
B
C
D
E
48
A
B
C
D
E
24
A
B
C
D
E
49
A
B
C
D
E
25
A
B
C
D
E
50
A
B
C
D
E
The number of right answers:
The number of wrong answers:
# of correct
Dr. John Chung's SAT II Math Level 2 Test 6
1 4 # of wrong
Raw score
225
2
2 Score Conversion Table 6
Scaled Score 480
630
5
470
26
620
4
470
800
25
620
3
460
46
800
24
610
2
460
45
800
23
610
1
450
44
800
22
600
0
450
43
800
21
600
42
800
20
590
41
800
19
590
40
780
18
580
39
760
17
570
38
750
16
560
37
740
15
550
36
720
14
540
35
710
13
530
34
700
12
520
33
690
11
510
32
680
10
500
31
670
9
490
30
660
8
490
29
650
7
480
28
Scaled Score 640
800
27
48
800
47
50
Scaled Score 800
49
Raw Score
226
Raw Score
Raw Score
2
2
MATHEMATICS LEVEL 2 TEST For each of the following problems, decide which is the BEST of the choices given. If the exact numerical value is not one of the choices, select the choice that best approximates this value. Then fill in the corresponding circle on the answer sheet Note: (1) A scientific or graphing calculator will be necessary for answering some (but not all) of the questions in this test. For each question you will have to decide whether or not you should use a calculator. (2) For some questions in this test you may have to decide whether your calculator should be in the radian mode or the degree mode. (3) Figures that accompany problems in this test are intended to provide information useful in solving the problems. They are drawn as accurately as possible EXCEPT when it is stated in a specific problem that its figure is not drawn to scale. All figures lie in a plane unless otherwise indicated. (4) Unless otherwise specified, the domain of any function f is assumed to be the set of all real numbers
x for which f ( x) is a real number. The range of f is assumed to be the set of all real numbers f ( x) , where x is in the domain of f . (5) Reference information that may be useful in answering the questions in this test can be found on the page preceding Question 1. USE THIS SPACE FOR SCRATCHWORK 1.
If
1 1 x x , then x x x
(A) 1
2.
If
1 1 1 x
(A) 2
3.
(B) 0
3 5
(D) 2
(E) undefined
2 , what is the value of x ?
(C)
(B) 2
If 3x 5 y , then
(A)
(C) 1
(B)
5 3
1 2
(D)
1 2
(E)
1 4
x y
(C) log3 5
(D) log5 3
(E) 35
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2
2 MATHEMATICS LEVEL 2 TEST - Continued USE THIS SPACE FOR SCRATCHWORK.
4.
What is the number of arrangements of letters that can be formed from the letters of the word “abscissa”? (A) (B) (C) (D) (E)
5.
40320 20160 6720 3360 1680
If log x log x 2 3 , then which of the following could 2
be the value of x ? (A) 10 6.
8.
(C) 5
(D) 8
(E) 10
What is the minimum value of y sin x 3? (A) 0
7.
(B) 3
(B) 1
(C) 2
(D) 3
(E) 4
1 A B for all real x , what is the value of x 1 x 1 x 1 constant B ?
If
2
(A) 1
(B)
If x
1 3
1 2 2
(C)
1 2
(D) 1
(E) 2
3 , then x
(A) 1 3
(B) 4 3 (C) 2 3
9.
(D)
32
(E)
34
y
In Figure 1, if ABC is equilateral, what is the slope of BC ? (A) 3
(B) 2
(C) 1
(D) 1
(E)
3
O
C
A (2, 0)
B (6, 0)
x
Figure 1
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228
2
2 MATHEMATICS LEVEL 2 TEST - Continued USE THIS SPACE FOR SCRATCHWORK.
10. If tan
(A) 1.8
1 2 , then sin cos 2 (B) 2.0
(C) 2.2 (D) 2.4
(E) 2.6
11. The graph of y f ( x) is shown in Figure 2. Which of the following is the equation of the graph? y (3,3)
(A) y x 3
y f ( x)
(B) y x 3 3 (C) y x 3 3
O
(D) y x 3 3
x
Figure 2
(E) y x 3 3
(6, 0)
12. If f ( x) log 2 x 2 7 and f g (1) 4, which of the
following could be g ( x)? (A) g ( x) x 2 x 2 (B) g ( x) 2 x 2 x 1 (C) g ( x) cos x 4 (D) g ( x) sin x 2 (E) g ( x) 3x 1 13. If the roots of 2 x 2 kx 14 0 are integers, then which of the following could be the value of constant k ?
(A) 6
(B) 8
(C) 10
(D) 12
(E) 16
14. Which of the following is an equation with roots 0 and
2 3? (A) 0 x3 3x 2 x (B) 0 x3 4 x 2 x (C) 0 x3 4 x 2 x (D) 0 x 2 2 x 2 (E) 0 x3 2 x 2 2 x
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2
2 MATHEMATICS LEVEL 2 TEST - Continued USE THIS SPACE FOR SCRATCHWORK.
15. How far is the point 2,1 from the line 3x y 4? (A) (B) (C) (D) (E)
0.316 0.542 1.358 2.855 3.282
2 , which of 3 the following are the rectangular coordinates of point A ?
16. If the polar coordinates of point A are 10,
(A) (B) (C) (D) (E)
5 3, 5 5,5 3 5, 5 3 5, 5 3 5, 5 3 B
17. In Figure 3, AD 20 and BD is perpendicular to AC . What is the length of CD ? (A) (B) (C) (D) (E)
10.35 12.07 13.06 14.85 15.50
A
32
46 20
D
C
Note: Figure not drawn to scale. Figure 3
18. If
3
(A)
1 3
46 x
27 x , then x
(B)
1 2
(C)
2 3
(D)
3 4
(E)
3 2
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230
2
2 MATHEMATICS LEVEL 2 TEST - Continued USE THIS SPACE FOR SCRATCHWORK.
19. Which of the following is the solution of
x 0? ( x 1)( x 2)
(A) x 1 (B) 5 x 2 (C) x 2 (D) x 0 or 1 x 2 (E) x 1 or x 2
20. What are the asymptotes of f ( x)
x ? x x 3
(A) x 0 (B) x 0 and x 1 (C) x 1 and y 0 (D) y 0 (E) x 0 and y 0
21. If f ( x) log3
x 3 and g ( x) is the inverse of
f ( x),
what is the value of g (2)? (A)
1 27
(B)
1 9
(C)
1 3
(D) 3
1 22. If sin 2 , what is the value of 4 (A) (B) (C) (D) (E)
(E) 9
cos sin 2 ?
1.25 0.75 0.50 0.25 0.15
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2
2 MATHEMATICS LEVEL 2 TEST - Continued USE THIS SPACE FOR SCRATCHWORK.
23. Which of the following is a horizontal tangent to the ellipse
x 32 y 2 2 49
25
1?
(A) y 2 (B) y 3 (C) y 5 (D) y 7 (E) y 10
24. Which of the following is true of the graph of the function
xy x 2 1 ?
(A) (B) (C) (D) (E)
Even function Odd function Symmetric with respect to x-axis Symmetric with respect to y -axis Symmetric with respect to y x
25. What is the value of
(A) 1
(B)
2
2i ? i2 (C)
3
(D) 2
(E) 2 3
B
26. In ABC , if A 30 , a 5 and b 10, then ABC in Figure 5 is
(A) (B) (C) (D) (E)
A
An acute triangle A right triangle An obtuse triangle An acute or an obtuse triangle An isosceles triangle
a
30 b
C
Note: Figure not drawn to scale. Figure 5
27. If cos 2 cos 1 , then sin 4 sin 2
(A) 1
(B)
2
(C)
3
(D)
1 2
(E)
3 2 GO ON TO THE NEXT PAGE
232
2
2 MATHEMATICS LEVEL 2 TEST - Continued USE THIS SPACE FOR SCRATCHWORK.
28.
x What is the period of the function y 2 tan 1 4? 3 (A)
29.
1 3
(B) 2
(C) 3
(D) 6
(E) 8 y
In Figure 6, point P is on the x-axis. What is the
(A) (B) (C) (D) (E) 30.
7.48 8.60 9.25 9.75 13.75
O
P ( x, 0)
x
Note:Figure not drawn to scale. Figure 6
The roots of x 2 kx 1 0 are p and q, where k is a constant. If (A) (B) (C) (D) (E)
31.
B (3, 4)
A(2, 3)
minimum length of AP PB ?
1 1 10, what is the value of k ? p q
10 5 5 10 15
If x 1 is a factor of x 6 5 x 4 4 x3 x k , then what is the value of k ? (A) 1
32.
(B) 2
(C) 3
(D) 4
(E) 5
If the quadratic equation x 2 2ax 2a 2 2a 3 0 has real roots, then which of the following could NOT be the value of a ? (A) 2 (B) 1 (C) 0 (D) 1 (E) 2 GO ON TO THE NEXT PAGE
Dr. John Chung's SAT II Math Level 2 Test 6
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2
2 MATHEMATICS LEVEL 2 TEST - Continued USE THIS SPACE FOR SCRATCHWORK.
33.
What is the distance from the plane x 2 y 3z 5 to the point 2, 2, 0 ? (A) (B) (C) (D) (E)
34.
1.871 2.225 2.786 3.125 4.750
If the line through (5, 4) and (2, k ) is perpendicular to the line with equation 3x 4 y 4, what is the value of k ? (A) (B) (C) (D) (E)
35.
If the radius of a right circular cone is 6 and the height of the cone is 8, what is the lateral surface area of the cone? (A) (B) (C) (D) (E)
36.
20 40 60 96 120
If y 3log ( 10 x x 2 ), what is the maximum value of y ? (A) (B) (C) (D) (E)
37.
2 4 6 8 10
3.56 4.19 5.25 6.32 7.41
What is the length of the major axis of the ellipse whose equation is 5 x 2 18 y 2 90 0 ? (A) 6.25
(B) 7.25
(C) 8.49
(D) 9.34
(E) 10.25
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234
2
2 MATHEMATICS LEVEL 2 TEST - Continued USE THIS SPACE FOR SCRATCHWORK.
38.
In Figure 7, A, B, and C , the vertices of the squares, are
C
collinear. What is the value of k ? (A) (B) (C) (D) (E)
39.
k 7 Figure 7
4
5 6
(B)
6
(C)
6
(D)
2
(E)
5 6
In a box there are 4 red marbles and 5 white marbles. If marbles are drawn one at a time and replaced after each drawing, what is the probability of drawing exactly 2 red marbles when 3 marbles are drawn? (A) (B) (C) (D) (E)
41.
A
3 If Arc cos , then 2
(A)
40.
B
8.45 10.38 12.25 13.12 13.74
0.329 0.235 0.198 0.110 0.102
In Figure 8, P is a point in the square of side-length 10 such that it is equally distant from two consecutive vertices
B
and from the opposite side AD. What is the length of BP ? (A) (B) (C) (D) (E)
5 5.25 5.78 6.25 7.07
C P
A
D Figure 8
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2
2 MATHEMATICS LEVEL 2 TEST - Continued USE THIS SPACE FOR SCRATCHWORK.
42.
If log 2 log 3 log 2 x 1 , what is the value of x ?
(A) (B) (C) (D) (E)
43.
1 a 2 1 If matrix A , B , C , and AB C , 2 1 b 3 where a and b are constants, what is the value of a ? (A) (B) (C) (D) (E)
44.
126 256 512 1024 2048
1 2 3 4 5
In the arithmetic progression, the first term is 5 and the common difference is 3. What is the sum of the first 20 terms? (A) (B) (C) (D) (E)
300 475 670 850 925
45.
The sum of the infinite series (A) (B) (C) (D) (E)
k
k
1 1 2 3 is k 1 k 1
1 1.5 2 3 4.5
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236
2
2 MATHEMATICS LEVEL 2 TEST - Continued USE THIS SPACE FOR SCRATCHWORK.
46.
1 2 3 n n n2 lim
(A) (B) (C) (D) (E)
47.
A committee of 5 is to be chosen from 8 men and 5 women. What is the probability that the committee consists of 2 men and 3 women? (A) (B) (C) (D) (E)
48.
0.5 1 2 3 4
0.185 0.218 0.302 0.387 0.425
From the expansion of the binomial ax y , where a is a 6
positive constant, the coefficient of x 2 y 4 is 60. What is the value of a ? (A) 2
49.
(B) 3
(C) 4
(D) 6
(E) 8
If f ( x) e 2 x and g ( x) ln( x 2 1) , then f g x (A) x 2 1 (B) x 3 x (C) 2 x( x 2 1) (D) x 4 2 x 2 1
(E) 2 x ln x 2 1
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2
2 MATHEMATICS LEVEL 2 TEST - Continued USE THIS SPACE FOR SCRATCHWORK.
50.
If
ab 0 , then which of the following could be true? ab
I. 0 b a II. b a 0 III. a b 0 (A) (B) (C) (D) (E)
I only I and II only II and III only I and III only I, II, and III
STOP IF YOU FINISH BEFORE TIME IS CALLED, YOU MAY CHECK YOUR WORK ON THIS TEST ONLY. DO NOT TURN TO ANY OTHER TEST IN THIS BOOK.
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238
No Test Material on This Page
Dr. John Chung's SAT II Math Level 2 Test 6
239
2
2 S ANSWERS
TEST 6 # 1 2 3 4 5 6 7 8 9 10
answer E B C D E C C B A A
# 11 12 13 14 15 16 17 18 19 20
answer D C E C A D B A D D
# 21 22 23 24 25 26 27 28 29 30
answer B B D B A B A C B A
# 31 32 33 34 35 36 37 38 39 40
answer A E A D C B C C E A
# 41 42 43 44 45 46 47 48 49 50
answer D C C C B A B A D B
Explanations: Test 6 1. (E)
2. (B)
1 1 x x x0 x x But x 0 is extraneous. The solution is undefined. 1( x) x x 2 x 2x 2 x 2 1 x 1 x 1 1 ( x ) x x
y
1 3x y
1 5y y
3. (C)
3 5
4. (D)
Permutation with repetition:
5. (E)
log x 2 log x 2 3
x y
3 5
x log3 5 y
8! 3360 2!3!
log x 2 2 log x 3 0
log x 3 log x 1 0
3
Since log x 3 and log x 1 , x 10 or 10. 6. (C)
The graph is symmetric with respect to y-axis. Therefore, the minimum of the function is 1 3 2.
2
240
2 7. (C)
8. (B)
2 1 A B 1 A( x 1) B( x 1) 1 ( A B) x B A 2 2 2 2 x 1 x 1 x 1 x 1 x 1 x 1 x 1 x2 1 1 To be identical, A B 0 and A B 1 . Therefore, B . 2 2
x
1 3
2
1 3
3
2
3 x 1 3 3 x
3 1 3 x
Therefore, x 4 3 . 9. (A)
CM 2 3 3. Since BM 2 , A 60 , and CM 2 3 , then the slope of BC MB 2 y
O
10. (A)
C
A (2, 0) M
Since tan
B (6, 0)
x
1 , then 2 1 2 (1) sin 5 and cos 5 or 1 2 (2) sin and cos 5 5
1
2 2
1
2
(1)
2 9 1 sin cos 2 1.8 5 5 5
(2)
sin cos 2
1
5
2
2 9 1.8 5 5
11. (D)
Translations: y x
T3,3 y x 3 3
12. (C)
Let g (1) be k , then f k log 2 k 2 7 4 .
2
4
2
k 7 2 16 k 9 k 3 (B) g (1) 2 (C) g (1) 3 (A) g (1) 4 (E) g (1) 2 (D) g (1) 2
Dr. John Chung's SAT II Math Level 2 Test 6
241
2 13. (E)
2 k 14 , Product of the roots: r1 r2 7 2 2 Therefore, the roots are 7 and 1, or 7 and 1 . k 16 and 16. Sum of the roots: r1 r2
(Check) Choice (E): 2 x 2 16 x 14 0 x 2 8 x 7 0 ( x 1)( x 7) 0 The roots are 1 and 7.
14. (C)
The polynomial equation is x x 2 3 x 2 3 0 . x( x 2 3)( x 2 3) 0 x ( x 2) 2 ( 3)2 0
Therefore, x( x 2 4 x 1) 0 0 x3 4 x 2 x Or, use sum and product of the roots: b c Let the quadratic equation be x 2 x 0 . a a b c SUM 2 3 2 3 4 4 , PRODUCT 2 3 (2 3) 1 1 a a The equation is x 2 4 x 1 0 . Because of zero at x 0 , the equation is x( x 2 4 x 1) 0 .
15. (A)
16. (D)
17. (B)
Distance from a point (2, 1) to a line 3x y 4 0 : D
BD 20 tan 32 and CD
A
32 (1) 2
BD 20 tan 32o 12.06858 12.07 tan 46 tan 46
3
32o
46o 20
46 x
27
x
D
1 32
C
46 x
33 x 32 3 x 33 x
Exponents: 2 3x 3x 6 x 2 x
242
3(2) 1 4
2 x r cos x 10cos 5 3 2 y r sin y 10sin 5 3 3
B
18. (A)
1 3
0.316227766 0.316
2 19. (D)
2 Test value:
x 0 ( x 1)( x 2)
0
1
2
( ) ( ) ( ) 0 (Ok). At x 0.5, 0 . At x 1.5, 0 ()() ()() ()() ( ) 0 . Therefore, the solution set is x 0 1 x 2 . At x 3, ()() At x 1,
Or, multiply by ( x 1)2 ( x 1) 2
1
0
x( x 1)( x 2) 0
2
The solution set is x 0 or 1 x 2. 20. (D)
f ( x)
x 1 2 x x x 1 3
x 2 1 0 No vertical asymptote 1 y lim 2 0 y 0 : Horizontal asymptote x x 1 21. (B)
Since g ( x) f 1 ( x) 32( x 3) , g (2)
22. (B)
cos sin
23. (D)
The graph of
2
1 . 9
cos 2 sin 2 2 cos sin 1 sin 2 1
x 32 y 2 2
a 7 and b 5.
49
25
1 3 4 4
1 is as follows.
y
y7 5 (3, 2) 5
x y 3
There are two horizontal tangent lines: y 7 and y 3
Dr. John Chung's SAT II Math Level 2 Test 6
243
2 24. (B)
25. (A)
2 x2 1 x Choice (A): f ( x) f ( x) : not even function Choice (B): f ( x) f ( x) : odd function Choice (C): f ( x) f ( x) : not symmetric with respect to x-axis Choice (D): f ( x) f ( x) : not symmetric with respect to y-axis Choice (E): not symmetric with respect to y x xy x 2 1 y
2i i2
2i 2 i
22 12 2
2
(2) 1
5 5
1
sin 30 sin B 10sin 30 sin B 1 B 90 5 10 5
26. (B)
The law of sines:
27. (A)
cos 2 cos 1 cos 1 cos 2 cos sin 2 Since sin 4 cos 2 , then sin 4 sin 2 cos 2 sin 2 1 . 1 , the period is 3 . 1 3 3
28. (C)
Since the frequency is
29. (B)
To have a minimum length point P( x,0) should be on the segment AB . Since AP AP , the minimum of APB is equal to the length of AB . y
AB (3 2)2 (4 3)2 74 8.602325 8.60 A( 2, 3)
O
B (3, 4)
P ( x, 0)
x
A( 2, 3)
1 1 p q k . p q pq 1
30. (A)
Since p q k and pq 1 ,
31. (A)
Factor Theorem: f (1) 1 5 4 1 k 0 k 1
32. (E)
Discriminant: D 2a 4(1)(2a 2 2a 3) 0 (a 3)(a 1) 0 2
The solution is: 3 a 1
244
k 10 k 10 .
2
2
33. (A)
Distance from a point to a line: D
34. (D)
Since the slope of 3x 4 y 4 is
2 2(2) 3(0) 5 2
2
2
1 (2) 3
7
14
1.870828 1.871
3 4 , the slope perpendicular to the line is . 4 3
k 4 4 k 4 4 k 8 25 3 3 3 62 82 10 . The lateral surface area is rs (6)(10) 60 .
35. (C)
The slant height is
36. (B)
y 3log(10 x x 2 ) has a maximum at x 5 [Axis of symmetric of ( f ( x) 10 x x 2 ) ] Therefore, the minimum of y is 3log(10 5 52 ) 3log 25 4.193820026 4.19
37. (C)
38. (C)
x2 y 2 1 18 5 The length of the major axis is 2 18 8.485281374 8.49 5 x 2 18 y 2 90 0
The slopes of AB and BC are equal. C B A
3
7
4
4
39. (E)
3 k 7 k 12.25 4 7
k 7
7
k
1) Using calculator: 2) Algebraic solution: 3 3 Let Arccos , where 0 . cos 2 2
5 6
2
40. (A)
4 5 0.329218107 0.329 9 9 Or, the ways to draw two red marbles: 3 C2
For WRR, the probability is
Dr. John Chung's SAT II Math Level 2 Test 6
WRR, RWR, RRW
5 4 4 80 80 . Therefore, 3 0.329. 9 9 9 729 729
245
2 41. (D)
2 If PM x , then BN 10 x . 10
B 10 x N
P
5
x
A
42. (C)
43. (C)
44. (C)
Pythagorean Theorem: x 2 (10 x) 2 52 x 6.25
C
x
D
M
log 2 log 3 log 2 x 1 log 3 log 2 x 2 log 2 x 9 x 29 512
1 a 2 1 2 ab 1 and 4 b 3 b 1 2 1 b 3 Therefore, 2 a (1) 1 a 3 . a20 5 (20 1)3 62 , S 20
n(a1 a20 ) 20(5 62) 670 2 2
1 1 k k 1 1 2 2 1 1 and 3 3 1 0.5 k 1 k 1 1 1 2 3 Therefore, the sum is 1 0.5 1.5 .
45. (B)
46. (A)
1 2 3 n lim lim n n n2
47. (B)
P
48. (A)
Since 6 Cr ax
n(1 n) n2 n 1 2 lim n 2n 2 2 n2
5 C3 0.2175602176 0.218 13 C5
8 C2
y
6 r
r
60 x 2 y 4 , 6 Cr a 6 r x 6 r 1 y r 6 Cr a 6 r 1 x 6 r y r . r
r must be 4. Therefore, 6 C4 a 2 1 60 15a 2 60 a 2 4
49. (D)
246
f
g x e
eln x2 1
2ln x 2 1
2
x2 1
2
x4 2 x2 1
r
2 50. (B)
2 ab 0 ab Method 1: ab 0 a b and a 0, b 0 a b 0 a b a b and a 0, b 0 0 a b
ab 0 a b and a 0, b 0 a b 0 (Not working) a b a b and a 0, b 0 b 0 b Method 2: Plug-in test number 2 1 I. a 2, b 1 2 0 (OK) 2 1 (1)(2) 1 0 (OK) II. b 2, a 1 (1) (2) (2)(1) 2 0 (NO) III. a 2, b 1 (2) (1) END
Dr. John Chung's SAT II Math Level 2 Test 6
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248
Test 7 Dr. John Chung’s SAT II Mathematics Level 2
Dr. John Chung's SAT II Math Level 2 Test 7
249
2
2
MATHEMATICS LEVEL 2 TEST REFERENCE INFORMATION
THE FOLLOWING INFORMATION IS FOR YOUR REFERENCE IN ANSWERING SOME OF THE QUESTIONS IN THIS TEST 1 Volume of a right circular cone with radius r and height h : V r 2 h 3 1 Lateral Area of a right circular cone with circumference of the base c and slant height : S c 2
Volume of a sphere with radius r : V 4 r 3 3 Surface Area of a sphere with radius r : S 4 r 2 1 Volume of a pyramid with base area B and height h : V 3 Bh
250
2
2 Dr. John Chung’s SAT II Math Level 2
Answer Sheet 01 02
A
B
C
D
E
A
B
C
D
E
03
A
B
C
D
E
B
C
D
04 05
A A
B
C
06
A
B
07
A
08
26 27
A
B
C
D
E
A
B
C
D
E
A
B
C
D
E
E
28 29
A
B
C
D
E
D
E
30
A
B
C
D
E
C
D
E
A
B
C
D
E
B
C
D
E
31 32
A
B
C
D
E
A
B
C
D
E
33
A
B
C
D
E
09
A
B
C
D
E
34
A
B
C
D
E
10 11
A
B
C
D
E
A
B
C
D
E
A
B
C
D
E
35 36
A
B
C
D
E
12
A
B
C
D
E
37
A
B
C
D
E
13 14
A
B
C
D
E
B
C
D
E
A
B
C
D
E
38 39
A
A
B
C
D
E
15
A
B
C
D
E
40
A
B
C
D
E
16 17
A
B
C
D
E
A
B
C
D
E
A
B
C
D
E
41 42
A
B
C
D
E
18
A
B
C
D
E
B
C
D
E
19 20
B
C
D
E
43 44
A
A
A
B
C
D
E
A
B
C
D
E
A
B
C
D
E
A
B
C
D
E
A
B
C
D
E
21 22
45 46
A
B
C
D
E
47
A
B
C
D
E
23
A
B
C
D
E
48
A
B
C
D
E
24
A
B
C
D
E
49
A
B
C
D
E
25
A
B
C
D
E
50
A
B
C
D
E
The number of right answers:
The number of wrong answers:
# of correct
Dr. John Chung's SAT II Math Level 2 Test 7
1 4 # of wrong
Raw score
251
2
2 Score Conversion Table 6
Scaled Score 480
630
5
470
26
620
4
470
800
25
620
3
460
46
800
24
610
2
460
45
800
23
610
1
450
44
800
22
600
0
450
43
800
21
600
42
800
20
590
41
800
19
590
40
780
18
580
39
760
17
570
38
750
16
560
37
740
15
550
36
720
14
540
35
710
13
530
34
700
12
520
33
690
11
510
32
680
10
500
31
670
9
490
30
660
8
490
29
650
7
480
28
Scaled Score 640
800
27
48
800
47
50
Scaled Score 800
49
Raw Score
252
Raw Score
Raw Score
2
2
MATHEMATICS LEVEL 2 TEST For each of the following problems, decide which is the BEST of the choices given. If the exact numerical value is not one of the choices, select the choice that best approximates this value. Then fill in the corresponding circle on the answer sheet Note: (1) A scientific or graphing calculator will be necessary for answering some (but not all) of the questions in this test. For each question you will have to decide whether or not you should use a calculator. (2) For some questions in this test you may have to decide whether your calculator should be in the radian mode or the degree mode. (3) Figures that accompany problems in this test are intended to provide information useful in solving the problems. They are drawn as accurately as possible EXCEPT when it is stated in a specific problem that its figure is not drawn to scale. All figures lie in a plane unless otherwise indicated. (4) Unless otherwise specified, the domain of any function f is assumed to be the set of all real numbers
x for which f ( x) is a real number. The range of f is assumed to be the set of all real numbers f ( x) , where x is in the domain of f . (5) Reference information that may be useful in answering the questions in this test can be found on the page preceding Question 1. USE THIS SPACE FOR SCRATCHWORK 1.
If x 10 , then
(A) 5
2.
(B) 9
x x 1 x x xx
(C) 10
(D) 100
(E) 1000
If x and y are positive integers and x 2 y 2 21 , then which of the following could be the value of x ? (A) (B) (C) (D) (E)
13 11 10 9 4
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Dr. John Chung's SAT II Math Level 2 Test 7
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2
2 MATHEMATICS LEVEL 2 TEST - Continued USE THIS SPACE FOR SCRATCHWORK.
3.
If f ( x) 4 x 2 , then the inverse function f 1 ( x) is x8 4 2x 1 2 x2 4 2x 1 2 x2 4
(A) (B) (C) (D) (E)
4.
In how many points do the graphs of x 2 y 2 1 and y 2 x 2 1 intersect? (A) (B) (C) (D) (E)
5.
If f ( x) 3x and g ( x) 2 x , then f g (5.4) (A) (B) (C) (D) (E)
6.
0 1 2 4 8
33 30 25 20 3
A
In Figure 1, AD CD and AB 5 . Which of the following is the value of tan ACD ? (A) (B) (C) (D) (E)
0.201 0.309 0.407 0.414 0.500
C
45
D
B
Note: Figure not drawn to scale. Figure 1
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254
2
2 MATHEMATICS LEVEL 2 TEST - Continued USE THIS SPACE FOR SCRATCHWORK.
7.
If sin cos
1 , then which of the following is the value(s) of 4
sin cos ?
1 (A) 2 1 (B) 2
2 2
(C)
(D)
2 2
2 2 , 2 2
(E)
8.
If x increases from (A) (B) (C) (D) (E)
9.
2
to
2
, then the value of sec x
decreases, then increases. increases, then decreases. increases. decreases. none of these
3x 2 If f x 1 , what is the value of f (6) ? 1 x (A) (B) (C) (D) (E)
3 5 18 37 42
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2
2 MATHEMATICS LEVEL 2 TEST - Continued USE THIS SPACE FOR SCRATCHWORK.
10.
x 1 1 and f g ( x) , then which of the x x 1 following could be g ( x ) ?
If f ( x)
(A) x 1 1 x 1
(B)
(C) x 1 1 x 1 x 1 (E) 1 x
(D)
11.
What is the range of the function defined by
x2 , x 0 f ( x) 1 , x0 x (A) y 0 (B) y 0 (C) y 0 (D) y 0 (E) All real numbers
12.
If 2log x log 2 x 3 , then which of the following is the solution set of x ? (A) (B) (C) (D)
1 1, 3 3 3, 4
(E) All real numbers
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256
2
2 MATHEMATICS LEVEL 2 TEST - Continued USE THIS SPACE FOR SCRATCHWORK.
13.
Which of the following could be the graph of y (A)
(B)
y
O
x x
?
y
x
x
O
x
(C)
(D)
y
y
x
O
(E)
x
y
O
14.
O
x
Which of the following is the equation of the polynomial with roots 1 2 and i ? (A) x3 x 1 (B) x 3 2 (C) x 4 2 (D) x 4 3x 2 2 (E) x 4 2 x 3 2 x 1
15.
If a log 5 9 , then 252 a is (A) 81
(B) 729
(C) 2187
(D) 6561 (E) 13122
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2
2 MATHEMATICS LEVEL 2 TEST - Continued USE THIS SPACE FOR SCRATCHWORK.
16.
If the volume of a cube is 63a , then the surface area of the cube is (A) 62 a (B) 6(6a ) (C) 6(32 a ) (D) 62 a1 (E) 62 a 3
17.
If cos 2 x
1 , then sin x 2
3 2 1 1 (B) or 2 2 (A)
(C)
3 2
(D)
3
(E)
18.
3 If Arc sin k , then tan k 2 (A) 1
19.
2 2 or 2 2
(B) 3
(C)
3
(D)
1 2
(E) 2
Which of the following is the domain of f ( x) (A) (B) (C) (D) (E)
x 1 ? x2
All real numbers All real numbers except 2 All real numbers greater than or equal to 1 All real numbers greater than or equal to 2 All real numbers greater than or equal to 1 except 2
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258
2
2 MATHEMATICS LEVEL 2 TEST - Continued USE THIS SPACE FOR SCRATCHWORK.
20.
If x C x 2 21 , then x (A) (B) (C) (D) (E)
21.
On a multiple- choice test, there are 5 choices for each question. What is the probability that a student who guesses every answer will have exactly 10 correct answers on a test that consists of 20 questions? (A) (B) (C) (D) (E)
22.
4 5 6 7 8
0.002 0.02 0.2 0.25 0.5
What is the sum of the numerical coefficients of x 2 y ? 4
(A) (B) (C) (D) (E)
0 1 16 32 48
100
23.
What is the value of
in ? n 1
(A) 0 (B) 1 (C) i (D) i (E) 1
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2
2 MATHEMATICS LEVEL 2 TEST - Continued USE THIS SPACE FOR SCRATCHWORK.
24.
If 3 i 2 is the root of the quadratic equation ax 2 12 x c 0 , what is the value of c ?
(A) (B) (C) (D) (E) 25.
11 22 25 33 44
In Figure 2, which of the following is the equation of the graph?
y
(A) y 3sin 2
3
(B) y 3sin 4
O
(C) y 3sin 8 (D) y 3sin (E) y 3sin 26.
2
4
3
2
Figure 2
4
What is the length of the major axis of an ellipse whose equation is x 2 4 y 2 4 x 8 y 8 ? (A) 4
27.
(B) 8
(C) 12
(D) 16
(E) 20
What is the slope of the line tangent to the circle
x 12 y 12 25
at the point 4,5 ?
4 3 3 4 5 4 4 5 4 5
(A) (B) (C) (D) (E)
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260
2
2 MATHEMATICS LEVEL 2 TEST - Continued USE THIS SPACE FOR SCRATCHWORK.
28.
If f ( x) f x , then which of the following could be 2 f ( x) ? (A) f ( x) sin x (B) f ( x ) 2sin 2 x (C) f ( x) cos x (D) f ( x) 2cos 2 x (E) f ( x) 2 tan 2 x
29.
In Figure 3, ABCD is a rectangle and tan CBE tan EAD
(A) (B) (C) (D) (E) 30.
1 and 7
1 . What is the value of tan BDA ? 3
C E
0.488 0.476 0.434 0.421 0.306
If f ( x)
B
A
D
Note:Figure not drawn to scale. Figure 3 1 2 , what is the range of the function? x3
(A) y 2 (B) y 2 (C) y 3 (D) y 2 (E) All real 31.
y
y f ( x)
Figure 4 shows the graph of f ( x) . Which of the following could be the function f ( x) ? (A) y x x 1
1
O
x 1
(B) y x 1 x 1 (C) y x 1 x 1 (D) y x 1 x 1
Note: Figure not drawn to scale. Figure 4
(E) y x 1 x
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2
2 MATHEMATICS LEVEL 2 TEST - Continued USE THIS SPACE FOR SCRATCHWORK.
32.
The right circular cone is sliced horizontally forming two pieces, each of which has the same height. What is the ratio of the volume of the smaller piece to the volume of the larger piece? 1 (A) 2 1 (B) 4 1 (C) 5 1 (D) 7 1 (E) 8
33.
How many possible rational zeros does
f ( x) 2 x3 3x2 8x 4 have? (A) (B) (C) (D) (E) 34.
6 8 10 12 14
What is the polar form of the rectangular equation
x2 y 2 4 x 0 ? (A) r sin
35.
(B) (C) (D) (E)
r 2 4sin r 4 cos r 4sin r 2 cos
lim
x 1 x x2 x 1
(A)
1 4
x 1
3
(B)
1 2
(C)
2 3
(D) 2
(E) 5
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262
2
2 MATHEMATICS LEVEL 2 TEST - Continued USE THIS SPACE FOR SCRATCHWORK.
36.
If the demand equation for a graphic utility is given by
D x 200 0.4 e0.005 x , where D is in dollars, which of the following is the demand x for a price D of $150? (A) 966
37.
(D) 1450
(E) 2048
Parabola Circle Ellipse Hyperbola Two perpendicular lines
If g ( x) 3 (A) (B) (C) (D) (E)
39.
(C) 1368
What is the graph of the parametrically defined equations x 4 2 cos and y sin 1 ? (A) (B) (C) (D) (E)
38.
(B) 1024
x 1 , then g 1 (1.5) 2
4.25 5.75 6.52 7.12 8.45
On a math exam, the scores of ten students were 66, 81, 85, 97, 86, 58, 76, 73, 88, and 80. What is the standard deviation of the scores? (A) (B) (C) (D) (E)
10.72 12.29 12.88 13.16 13.58
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2
2 MATHEMATICS LEVEL 2 TEST - Continued USE THIS SPACE FOR SCRATCHWORK.
40.
Which of the following is the function of the graph in Figure 5? y
f ( x) x3 ax 2 bx c
I. II.
f ( x) x5 ax 4 bx3 cx 2 dx e
III.
f ( x) x 7 ax 6 bx5 cx 4 dx3 ex 2 fx g
(A) (B) (C) (D) (E)
41.
I only II only II and III only I, II, and III None of those
y f ( x)
O
x
Figure 5
What is the smallest positive value of , in radians, which satisfies the equation 2sin 2 2 cos 1 0 ? (A) (B) (C) (D) (E)
42.
0.56 0.86 1.24 1.56 1.95
Which of the following is the solution set of the equation log 3 ( x 5) log 9 (2 x 5) ? (A) 2 (B) 5 (C) 10 (D) 2,10 (E) 2, 5, 10
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264
2
2 MATHEMATICS LEVEL 2 TEST - Continued USE THIS SPACE FOR SCRATCHWORK.
43.
What is the distance between two points of 1, 2, 3 and
0, 4, 2 ? (A) (B) (C) (D) (E)
44.
7.87 8.24 8.48 10.25 11.24
Which of the following could be the solution to
n 2 2 n 3 , where n (A) (B) (C) (D) (E)
is the greatest integer function?
2n3 0 n 1 1 n 0 1 n 0 1 n 0
y C
45. In Figure 6, ABCDE is a regular pentagon with side of length 6. What is the x -coordinate of D ?
(A) (B) (C) (D) (E)
D
B
10.3 10.5 10.7 10.9 11.9 O
A (4, 0)
E
x
FIgure 6 46.
If the line y 3x k is tangent to the hyperbola whose equation is 4 x 2 y 2 16 , which of the following could be the value of k ? (A) 2 (B) 1 (C)
5
(D) 2 5 (E) 3 5
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2
2 MATHEMATICS LEVEL 2 TEST - Continued USE THIS SPACE FOR SCRATCHWORK.
47.
What is the area of the polygon formed by the points ( x, y ) which satisfy the inequality x (A) (B) (C) (D) (E)
48.
49.
2
1?
2 3 4 8 10
If lim
x 1
(A) (B) (C) (D) (E)
y
x2 2 x k 4 , then what is the value of k ? x 1
4 3 2 3 4
What is the radius of the sphere whose equation is
x2 y2 z 2 2x 4 y 6z 0 ? (A) (B) (C) (D) (E)
50.
3 3.74 8.56 12.45 14
If a (2, 1) and b (1, 2) , then 3a 2b (A) (B) (C) (D) (E)
7.44 8.06 8.45 9.12 10.14
STOP IF YOU FINISH BEFORE TIME IS CALLED, YOU MAY CHECK YOUR WORK ON THIS TEST ONLY. DO NOT TURN TO ANY OTHER TEST IN THIS BOOK.
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266
No Test Material on This Page
Dr. John Chung's SAT II Math Level 2 Test 7
267
2
2
TEST 7 # 1 2 3 4 5 6 7 8 9 10
ANSWERS
answer B B C A A D E A B E
# 11 12 13 14 15 16 17 18 19 20
answer E C E E D D B B E D
# 21 22 23 24 25 26 27 28 29 30
answer A B A B E B B E B D
Explanations: Test 7 1. (B)
2. (B)
x x 1 x x x x ( x 1) x 1 xx xx Since x 10, x 1 10 1 9 .
x y x y 21 x and y are positive integers x y 21 x y 1
or
x y 7 x y 3
2 x 22
2 x 10
x 11 x5 Or, substitute choices and check.
x 4y 2
f 1 : y
3. (C)
y 4x 2
4. (A)
The asymptotes of both graphs are y x . The graphs are as follows.
x2 4
y
x
5. (A)
268
g (5.4) 10.8 11
f (11) 33 33
# 31 32 33 34 35 36 37 38 39 40
answer D D B C B A C B A C
# 41 42 43 44 45 46 47 48 49 50
answer E C A D E D C B B B
2 6. (D)
2 tan C
5
0.414
55 2
A
5
5 2 45o
C
7. (E)
D
5 2
B
5
1 1 (sin cos ) 2 1 2sin cos 1 2 4 2 2 2 1 1 1 sin 2 2 30 ,150 ,390 ,510... Or, sin cos sin 2 2 4 2 15 , 75 ,195 , 255... Therefore, 2 sin15 cos15 0.707 2 2 sin 75 cos 75 0.707 2 sin 75 cos 75 0.707 ….. The answer is (E). sin cos
8. (A)
sec x
1 cos x
y sec x y cos x
9. (B)
10. (E) 11. (E)
Since
2
2
3x 6 x 2 , then f (6) 22 1 5. x 1
f (g)
g 1 1 1 x xg x g 1 g ( x) g 1 x 1 x
The graph is as follows. 1 lim 0 , f (0) 0 x x 1 Since lim f (0) , the range is all real numbers x 0 x
Dr. John Chung's SAT II Math Level 2 Test 7
y
x
269
2
2
12. (C)
2log x log(2 x 3) x 2 2 x 3 x 2 2 x 3 0 ( x 3)( x 1) 0 x 3 or x 1 (rejected)
13. (E)
Piecewise-defined function: x x 0, 1 y x x 0, 1
14. (E)
Reconstruct equation: conjugate roots [ x (1 2)][ x (1 2)][ x i ][ x i] 0 [( x 1) 2
2 ][ x 2
2
1] 0 ( x 2 2 x 1)( x 2 1) 0
x 4 2 x3 2 x 1 0 15. (D)
Since a log5 9 9 5a ,
252 a 54 a 5a 16. (D)
4
94 6561 .
x : the length of an edge
Volume x3 63a x 63a
Surface area 6 x 2 6 6a 17. (B)
18. (B)
2
1 1 1 sin 2 x sin x 2 4 2 Or, 2 x 60 , 300 x 30 , 150 1 1 Therefore, sin 30 or sin 300 2 2 cos 2 x 1 2sin 2 x
Use a calculator. 3 3 , where 90 k 90 . Or, Arcsin k sin k 2 2
2
N ( x) x 1 x 1 D ( x) x 2 x 2 Domain: x x 1 but x 2
270
6a
62 a 1
tan k 3
1
19. (E)
1 3
3
2
2 x! x( x 1) x( x 1) 21 ( x 7)( x 6) 0 2!( x 2)! 2 2 x 7 ( x must be a positive integer and greater than 2)
20. (D)
x Cx 2
21. (A)
P
22. (B)
Let x 1 and y 1 .
20 C10
0.2 10 (0.8)10 0.0020314137 0.002
x 2 y 4 4 C0 x 4 2 y 0 4 C1 x 3 2 y 1 4 C2 x 2 2 y 2 4 C3 x 1 2 y 3 0 4 4 C4 x 2 y When you substitute x 1 and y 1 , you can get the sum of the coefficients. Both sides are equal for any x . Therefore, the sum of all coefficients is 1 2 1 1 . 4
100
23. (A)
i n i i 2 i3 i 4 ... i100 : Geometric sequence n 1
a (1 r n ) i (1 i100 ) i (1 1) 0 1 r 1 i 1 i Or, i i 2 i 3 i 4 0 : The sum of every four terms is 0. There are 25 of the terms. 0 25 0 . S100
24. (B)
Since 3 i 2 is the root, the other root is 3 i 2 . 12 Sum of the roots: S 6 a 2 a c Product of the roots: P 3 i 2 3 i 2 11 c 22 2
25. (E)
From the graph: Period is 8 and frequency is b Amplitude : 3
26. (B)
2 . 8 4
y 3sin 4
x 2 4 y 2 4 x 8 y 8 ( x 2)2 4( y 1)2 16 a 4 Major axis 2a 8
( x 2) 2 ( y 1) 2 1 16 4
m
27. (B)
(4,5) (1,1)
Dr. John Chung's SAT II Math Level 2 Test 7
5 1 4 4 1 3 3 The slope of line 4
The slope of line m
271
2 28. (E)
2 f ( x) is periodic with period of The period of f ( x) 2 tan 2 x is
29. (B)
30. (D)
2
2
. .
CE a ED b and tan EAD . BC 7a AD 3b AB a b 7a tan BDA B AD 3b 3b Since 7 a 3b a , ab 7 3b b ab 7 10b 10 A 0.476 . 3b 3b 3b 21b 21 tan CBE
C a E b
D
1 : shift to the right by 3 and up by 2 x The range: y 2
The graph of f : Transformation of y y
2 O
3
x
31. (D)
From the graph: y2 If x 1, If 1 x 1, y 2 x If x 1, y 2 Choice (D): If x 1, then y ( x 1) ( x 1) 2 If 1 x 1, then y ( x 1) ( x 1) 2 x If x 1, then y ( x 1) ( x 1) 2
32. (D)
Since the ratio of the lengths is 1:2, the ratio of their volumes is 13 : 23 1: 8 . Therefore, the ratio of the volumes is 1:7.
h
2h
33. (B)
272
The rational zero test: 1, 2, 4 1 1, 2, 4, 1, 2 2
: 8 possible rational zeros
2 34. (C)
2 Since r 2 x 2 y 2 and r cos x , x 2 y 2 4 x 0 r 2 4r cos 0 r 4 cos ( x 1) 1 x 1 ( x 1) lim 2 lim 2 2 1 1 x x x x x 1 x ( x 1) ( x 1) ( x 1) ( x 1) 2
35. (B)
lim
36. (A)
150 200 0.4 e0.005 x
37. (C)
38. (B)
x 1
3
x ln125 0.005 966
125 e0.005 x
x4 , sin y 1 2 ( x 4)2 Since cos 2 sin 2 1, ( y 1) 2 1 : Ellipse 4 cos
g 1 (1.5) k (1.5, k ) Since g 1 ( x) is the inverse of g ( x) , point (k , 1.5) must be on g ( x). Therefore,
3
k 1 k 1 1.5 1.53 k 2 1.53 1 5.75 . 2 2
Or, you can find g 1 ( x) . 39. (A)
Use your calculator : Statistics. x 10.72
40. (C)
Check the end of the graph. f ( x) have three real roots. Therefore, II and III could be f ( x). I. lim f ( x) ( False) x
II. lim f ( x) and lim f ( x) . Three real and two imaginary roots. x
x
III. lim f ( x) and lim f ( x) . Three real and four imaginary roots. x
41. (E)
x
Use a graphic utility:
Zero x 1.9455308
42. (C)
y0
Theorem: log a b log a 2 b 2 log3 ( x 5) log9 (2 x 5) log9 ( x 5)2 log9 (2 x 5) ( x 5) 2 2 x 5 ( x 2)( x 10) 0 x 2 or x 10 But x 5 0 , x 2 rejected.
Dr. John Chung's SAT II Math Level 2 Test 7
273
2
2
43. (A)
Distance (1 0) 2 (2 4) 2 (3 2)2 62 7.87
44. (D)
Let n x . x 2 2 x 3 0 ( x 3)( x 1) 0 x 3 or x 1
If n 3, then 3 n 4 .
If n 1, then 1 n 0 .
45. (E)
y
( x, y ) 6
x 10 6 cos 72 11.854 11.9 72
x
10
46. (D)
Discriminant: Substitute y 3x k into the equation of the hyperbola. 4 x 2 (3x k ) 2 16 5 x 2 6kx k 2 16 0 D (6k )2 4(5)(k 2 16) 0 k 2 20
y 3x 2 5
Therefore, k 2 5 . y 2
2
x y 1 4 16 Since a 2 and b 4, Asymptotes: y 2 x . 4 x 2 y 2 16
x
y 3x 2 5
274
2 47. (C)
2 Remember: The graph of x y 1 is as follows. y 1
1
1
x
1
Therefore, the graph of x
y
2
1 is
y 2
1
1
x
2
The area
48. (B)
2 1 4 4 2
x2 2 x k 4 : Numerator must have a factor of ( x 1) . x 1 x 1 N (1) 1 2 k 0 k 3
lim
( x 3) ( x 1) x2 2 x k x2 2 x 3 4 4 lim 4 lim x 1 x 1 x 1 x 1 x 1 ( x 1)
lim
49. (B)
Since ( x 1)2 ( y 2)2 ( z 3)2 14 , r 14 3.74 .
50. (B)
3a 2b 3(2,1) 2(1, 2) (4, 7) 3a 2b 42 7 2 65 8.06
END
Dr. John Chung's SAT II Math Level 2 Test 7
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276
Test 8 Dr. John Chung’s SAT II Mathematics Level 2
Dr. John Chung's SAT II Math Level 2 Test 8
277
2
2
MATHEMATICS LEVEL 2 TEST REFERENCE INFORMATION
THE FOLLOWING INFORMATION IS FOR YOUR REFERENCE IN ANSWERING SOME OF THE QUESTIONS IN THIS TEST 1 Volume of a right circular cone with radius r and height h : V r 2 h 3 1 Lateral Area of a right circular cone with circumference of the base c and slant height : S c 2
Volume of a sphere with radius r : V 4 r 3 3 Surface Area of a sphere with radius r : S 4 r 2 1 Volume of a pyramid with base area B and height h : V 3 Bh
278
2
2 Dr. John Chung’s SAT II Math Level 2
Answer Sheet 01 02
A
B
C
D
E
A
B
C
D
E
03
A
B
C
D
E
B
C
D
04 05
A A
B
C
06
A
B
07
A
08
26 27
A
B
C
D
E
A
B
C
D
E
A
B
C
D
E
E
28 29
A
B
C
D
E
D
E
30
A
B
C
D
E
C
D
E
A
B
C
D
E
B
C
D
E
31 32
A
B
C
D
E
A
B
C
D
E
33
A
B
C
D
E
09
A
B
C
D
E
34
A
B
C
D
E
10 11
A
B
C
D
E
A
B
C
D
E
A
B
C
D
E
35 36
A
B
C
D
E
12
A
B
C
D
E
37
A
B
C
D
E
13 14
A
B
C
D
E
B
C
D
E
A
B
C
D
E
38 39
A
A
B
C
D
E
15
A
B
C
D
E
40
A
B
C
D
E
16 17
A
B
C
D
E
A
B
C
D
E
A
B
C
D
E
41 42
A
B
C
D
E
18
A
B
C
D
E
B
C
D
E
19 20
B
C
D
E
43 44
A
A
A
B
C
D
E
A
B
C
D
E
A
B
C
D
E
A
B
C
D
E
A
B
C
D
E
21 22
45 46
A
B
C
D
E
47
A
B
C
D
E
23
A
B
C
D
E
48
A
B
C
D
E
24
A
B
C
D
E
49
A
B
C
D
E
25
A
B
C
D
E
50
A
B
C
D
E
The number of right answers:
The number of wrong answers:
# of correct
Dr. John Chung's SAT II Math Level 2 Test 8
1 4 # of wrong
Raw score
279
2
2 Score Conversion Table 6
Scaled Score 480
630
5
470
26
620
4
470
800
25
620
3
460
46
800
24
610
2
460
45
800
23
610
1
450
44
800
22
600
0
450
43
800
21
600
42
800
20
590
41
800
19
590
40
780
18
580
39
760
17
570
38
750
16
560
37
740
15
550
36
720
14
540
35
710
13
530
34
700
12
520
33
690
11
510
32
680
10
500
31
670
9
490
30
660
8
490
29
650
7
480
28
Scaled Score 640
800
27
48
800
47
50
Scaled Score 800
49
Raw Score
280
Raw Score
Raw Score
2
2
MATHEMATICS LEVEL 2 TEST For each of the following problems, decide which is the BEST of the choices given. If the exact numerical value is not one of the choices, select the choice that best approximates this value. Then fill in the corresponding circle on the answer sheet Note: (1) A scientific or graphing calculator will be necessary for answering some (but not all) of the questions in this test. For each question you will have to decide whether or not you should use a calculator. (2) For some questions in this test you may have to decide whether your calculator should be in the radian mode or the degree mode. (3) Figures that accompany problems in this test are intended to provide information useful in solving the problems. They are drawn as accurately as possible EXCEPT when it is stated in a specific problem that its figure is not drawn to scale. All figures lie in a plane unless otherwise indicated. (4) Unless otherwise specified, the domain of any function f is assumed to be the set of all real numbers x for which f ( x) is a real number. The range of f is assumed to be the set of all real numbers f ( x) , where x
is in the domain of f . (5) Reference information that may be useful in answering the questions in this test can be found on the page preceding Question 1. USE THIS SPACE FOR SCRATCHWORK 1.
If e x 32 , then x (A) 1.4 (B) 1.7 (C) 1.9 (D) 2.2 (E) 2.4
2.
If f
(A) 21 (B) 46 (C) 84 (D) 117 (E) 256
x x
2
4 x , then f (3)
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Dr. John Chung's SAT II Math Level 2 Test 8
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2
2 MATHEMATICS LEVEL 2 TEST - Continued USE THIS SPACE FOR SCRATCHWORK.
3.
If 3
1 3 1 9 , then 3 x x x
(B) 1
(A) 3
4.
ab 18
(B)
(E) 9
2a b
(C)
2b a
b 2a
(D)
(E)
a 2b
If 32 y1 5 , then y (A) 0.23
6.
(D) 3
3 ax 6 bx (A)
5.
(C) 0
(B) 0.32
(C) 2.75
(D) 3.12
(E) 3.44
Which of the following could be the graph of the polar equation r 2cos ? y
(A)
y
(B)
x
x
O
y
y
(C)
(D) x
O
x
y
(E) x
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2
2 MATHEMATICS LEVEL 2 TEST - Continued USE THIS SPACE FOR SCRATCHWORK.
7.
Which of the following ordered pairs is not an element of x the greatest- integer function y ? 2
(D)
8, 4 2.78, 1 4.8, 3 5.64, 2
(E)
(A) (B) (C)
8.
10, 1
What is the range of the function f ( x) x 2 for the domain 3 x 3 ? (A) 3 y 5 (B) 1 y 5 (C) 0 y 5 (D) 1 y 5 (E) 3 y 5
9.
If 2 5 3 5 a b 5 , where a and b are rational numbers, then a b (A) 1
10.
(B) 2
(C) 3
(D) 4
(E) 5
a a 4a 3
(A) a 16 (B)
1 a8 3
(C) a 4 (D) a (E) a 2
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2
2 MATHEMATICS LEVEL 2 TEST - Continued USE THIS SPACE FOR SCRATCHWORK.
11.
Figure 1 shows the prism with dimensions 3, 5, and 7. What is the perimeter of the triangle ABC ? (A) (B) (C) (D) (E)
A
B
15 17.45 20.71 22.38 23.12
7
5 3
C
Figure 1 12.
cos 4 sin 4
(A) 1 (B) sin 2 (C) 2 cos (D) cos 2 (E) sin cos
13.
If 2 3 is a root of the polynomial P( x) , then a factor of P( x) is (A) x 2 3 (B) x 2 4 (C) x 2 4 x 1 (D) x 2 4 x 1 (E) x 2 4 x 1
B 14.
The angle of elevation from point A to point B is 40 and the angle of elevation from D to B is 60 . If the length of AD is 100, then what is the length of BC ? (A) (B) (C) (D) (E)
80.50 93.97 98.56 120.45 162.76
A
40
60
D
C
Note: Figure not drawn to scale. Figure 2
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2
2 MATHEMATICS LEVEL 2 TEST - Continued USE THIS SPACE FOR SCRATCHWORK.
15.
1 and f g ( x) x , then which of the x 1 following is g ( x) ?
If f ( x)
(A) x 1 x x 1 x 1 (C) x x 1 (D) x x (E) 1 x
(B)
16.
In ABC , if A 30 , b 1 , and c 3 , then B (A) 30 (B) 45 (C) 52 (D) 60 (E) 75
17.
If sin 0.4 , then which of the following could be cos(90 ) ? (A) (B) (C) (D) (E)
0.2 0.4 0.5 0.6 0.8
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2
2 MATHEMATICS LEVEL 2 TEST - Continued USE THIS SPACE FOR SCRATCHWORK.
18.
The set of points ( x, y, z ) such that ( x 1) 2 ( y 1) 2 z 2 0 is (A) (B) (C) (D) (E)
19.
A point A circle A plane A sphere Empty
If the graph of the rational function R( x)
x 2 ax b x x 1
does not have vertical asymptotes, then what is the value of a? (A) (B) (C) (D) (E)
20.
0 1 2 3 4
If the four numbers log 3, x, log81, and y form a geometric sequence in that order, which of the following could be the value of y ? (A) log126 (B) log162 (C) log 324 (D) log 1296 (E) log 6561
21.
If 52 x 3 7 x 1 , then x (A) (B) (C) (D) (E)
4.35 2.26 1.32 2.26 4.35
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2
2 MATHEMATICS LEVEL 2 TEST - Continued USE THIS SPACE FOR SCRATCHWORK.
22.
If lim
x
(A) (B) (C) (D) (E)
ax 2 bx 1 4 , then what is the value of a b ? 2x 5
4 6 8 10 14
n
23.
n lim
i 2
(A) (B) (C) (D) (E)
24.
1 3i
1 6 1 3 2 3 3 2 4 3
If a and b are values in the domain of f ( x) and f (a) f (b) , where b a , then which of the following must be true? (A) f ( x) is an odd function. (B) f ( x) is an even function. (C) f ( x) increases as x increases. (D) f ( x) decreases as x increases. (E) f ( x) is a linear function.
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2
2 MATHEMATICS LEVEL 2 TEST - Continued USE THIS SPACE FOR SCRATCHWORK.
25.
What is the measure of the largest angle in a triangle with sides of lengths 3, 4, and 6? (A) 62 (B) 84 (C) 98 (D) 117 (E) 128
x 3
26.
1 If f ( x) 2 and g ( x) is the inverse of f , then 2 what is the value of a which satisfies g (a ) 5 ?
(A) (B) (C) (D) (E)
27.
2 2.25 2.5 4.45 5.25
If f ( x ) 3 x , then what is the range of the function? (A) y 0 (B) y 0 (C) y 1 (D) y 1 (E) 0 y 1 y
y 3 x
28.
Figure 3 shows the graph of y 3 x and three inscribed rectangles. What is the sum of the areas of the rectangles? (A) 0.25 (B) 0.50 (C) 2.45 (D) 6.26 (E) 12.68
O
2
4
8
x
Note: Figure not drawn to scale. Figure 3
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2
2 MATHEMATICS LEVEL 2 TEST - Continued USE THIS SPACE FOR SCRATCHWORK. 2
29.
What is the domain of f ( x) ( x 2 4) 3 ? (A) (B) (C) (D) (E)
30.
3 5 , A is acute, sec B , and B is obtuse, 5 3 then what is the value of tan( A B ) ?
If sin A
(A) (B) (C) (D) (E) 31.
0.58 0.29 0.29 0.58 0.86
If f (3x 4) 6 x 5 for all real numbers x , then f ( x ) (A) (B) (C) (D) (E)
32.
x0 x 2 or x 2 2 x 2 2 x 2 All real numbers
3x 4 2 x 15 2 x 13 2 x 10 3 x 13
Which of the following is true for the graph of the equation x 2 y 2 kx , where k is a positive constant? (A) A circle with center on the y -axis is tangent to the x-axis. (B) A circle with center on the x-axis is not tangent to the y -axis. (C) A circle with center on the x-axis is tangent to the y -axis. (D) An ellipse with center on the x-axis is tangent to the y -axis. (E) None of these
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2
2 MATHEMATICS LEVEL 2 TEST - Continued USE THIS SPACE FOR SCRATCHWORK.
33.
What is the area of an equilateral triangle that is inscribed in a circle of radius 2? (A) 3 3 (B) 2 3 (C) 3 2 (D) 2 2 (E) 4 2
34.
What is the remainder when the polynomial x 4 3 x 2 10 is divided by x 2 1 ? (A) (B) (C) (D) (E)
35.
In Figure 4, if the radius of the base is 6 and the height of the cone is 8, what is the surface area of the cone? (A) (B) (C) (D) (E)
36.
10 14 x 10 x 14 2x 5
36 64 96 136 148
6
8
FIgure 4
A cylinder with height 10 is inscribed in a sphere with radius 8. What is the volume of the cylinder? (A) (B) (C) (D) (E)
201 652 844 1225 1412
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2
2 MATHEMATICS LEVEL 2 TEST - Continued USE THIS SPACE FOR SCRATCHWORK.
37.
If one zero of f ( x) x 2 kx k 7 is 3, where k is a constant, then what is the other zero of f ? (A) (B) (C) (D) (E)
38.
If $5,000 is invested in a bank at a rate of 5% annual interest compounded monthly for 3 years, what is the amount of the balance after 3 years? (A) (B) (C) (D) (E)
39.
1 2 3 4 5
$5300.45 $5800.12 $5807.36 $6000.50 $6900.56
2x 3 If f 4 x 8 , then f (2) 3
(A) 8 (B) 10 (C) 12 (D) 14 (E) 16
40.
In Figure 5, if the volume of the inscribed circular cone is 100, then what is the volume of the square pyramid? (A) (B) (C) (D) (E)
112.43 127.32 135.25 151.38 167.12
h r
2r
2r Figure 5
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2
2 MATHEMATICS LEVEL 2 TEST - Continued USE THIS SPACE FOR SCRATCHWORK.
41.
Which of the following is the point(s) of intersection of x 2 y 2 16 and y 2 x 4 ? (A) 4, 0 (B) 5, 0 (C) 4, 0 and 5, 3 (D) 4, 0 and 4, 0 (E) 5, 0 and 5, 0
42.
What is the coefficient of x3 in the expansion of 5 x 1 ? 9
(A) (B) (C) (D) (E)
43.
The standard deviation of a data set is 8.5. If a new data set is created by subtracting 5.5 from each data value, what is the standard deviation of the new data set? (A) (B) (C) (D) (E)
44.
5 25 125 9500 10500
3.0 5.5 7.0 8.5 8.7
Which of the following is the unit vector of a (3, 4, 12) ?
(A) (1, 0, 0) (B) (1, 1, 1) (C) (0.31, 0.23, 0.92) (D) (0.23, 0.31, 0.92) (E) (0.23, 0.92, 0.31)
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2
2 MATHEMATICS LEVEL 2 TEST - Continued USE THIS SPACE FOR SCRATCHWORK.
45.
The polynomial equation xy x 2 0 can be expressed by a set of parametric equations as a function of t . If y (t ) 2t 3 , then x(t ) (A) 1 3t 1 2t 1 (C) t2 1 3t (D) 2
(B)
(E) 3t 1
46.
If i 1 , what is the third term in the binomial expansion of (2 3i )6 ? (A) 1120 (B) 1120i (C) 2160 (D) 2160 (E) 2160i
47.
If there are 10 points on a circle, how many line segments can be made by connecting any two given points? (A) (B) (C) (D) (E)
48.
100 90 80 72 45
In ABC , if A 45 , a 7 , and b 10 , how many distinct triangles can be formed? (A) (B) (C) (D) (E)
0 1 2 3 4 GO ON TO THE NEXT PAGE
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2
2 MATHEMATICS LEVEL 2 TEST - Continued USE THIS SPACE FOR SCRATCHWORK.
49.
Which of the following could be the curve represented by 1 2t 1 and y ? the parametric equations x t t y
(A)
x
O
y
(C)
(E)
x
O
y
(D)
x
O
y
(B)
O
x
y
O
x
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2
2 MATHEMATICS LEVEL 2 TEST - Continued USE THIS SPACE FOR SCRATCHWORK.
50.
Figure 6 shows the rectangular solid with dimensions 3, 4, and 5. If PQ is the diagonal of the solid, what is the value of formed by the diagonals and the side QR ?
4
3
(A) 30
P
(B) 32.07
5
(C) 34.45 (D) 46.28
Q
(E) 55.55
R Note: Figure not drawn to scale. Figure 6
STOP IF YOU FINISH BEFORE TIME IS CALLED, YOU MAY CHECK YOUR WORK ON THIS TEST ONLY. DO NOT TURN TO ANY OTHER TEST IN THIS BOOK.
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2
2
TEST 8 # 1 2 3 4 5 6 7 8 9 10
ANSWERS
answer D D C D A D D C B D
# 11 12 13 14 15 16 17 18 19 20
answer C D E E C A B A B E
# 21 22 23 24 25 26 27 28 29 30
answer D C A D D B E A E C
# 31 32 33 34 35 36 37 38 39 40
answer C C A B C D E C B B
# 41 42 43 44 45 46 47 48 49 50
answer A E D D B D E A C E
Explanations: Test 8 1. (D)
x ln 9 2.2
2. (D)
Since
3. (C)
3x 1 9 x 3 6 x 2 x
4.
3 abx 3b b ax 6a 2a 6 abx bx
(D)
x 3
x 9.
log 5 log 3
f ( 9) 92 4(9) 117
1 log 5 y 1 0.23 2 log 3
5. (A)
2 y 1 log3 5
6. (D)
Since r x 2 y 2 and x r cos , r 2 cos
1 33 0 3
x2 y 2 2
x 2
x y
2
7. (D)
5.64 2 2.82 3
8. (C)
From the graph of f ( x) x 2 , the range is 0 y5
296
x 12 y 2 1 .
x2 y 2 2 x 0
y
(3,5)
3
O
3
x
2 9. (B)
10. (D)
11. (C)
2 The equation can be simplified as follows. 1 5 a b 5 Since a and b are rational numbers, a 1 and b 1. a b 2 . 1 a a2
3 a2
1
3 3 2 a2 a4 .
AC 32 52 7 2 83
3
1
4
Therefore, a 4 a 4 a 4 a.
AB 3
BC 5 7 74 2
2
Perimeter 83 74 3 20.71
12. (D)
cos 4 sin 2 cos 2 sin 2 cos 2 sin 2 cos 2 sin 2 cos 2
13. (E)
x 2 3 x 2 3 0 , x 2 3 x 2 3 x 2
14. (E)
15. (C)
sin 20 sin 40 100sin 40 y 100 y sin 20 100sin 40 sin 60 z y sin 60 162.76 sin 20
Law of sines:
2
3 x2 4 x 1 0 B
20o
z
y 40o 100
A
If f g ( x) x , then g ( x) f 1 .
60o
C
D
1 1 1 y 1 y 1 y 1 x x 1 1 1 x 1 x g 1 g 1 Or, f ( g ) g 1 x x x For f 1 , x
B
16. (A)
Law of cosines: a2
3
2
12 2
3 1 cos30
4 3 1
a 1. Since AC BC , A B . B 30
a
3 A
30o
C
1
17. (B)
cos(90 ) cos 90 cos sin 90sin sin 0.4
18. (A)
It is not a sphere because the radius is 0. It indicates the point (1, 1, 0).
19. (B)
In order not to have any vertical asymptotes, x 2 ax b must have factors of x( x 1) . Therefore, N (0) b 0 and N (1) 1 a 0 . a 1.
20. (E)
r
x log81 x 2 log 3 . log 3 x y log 81 2 log 6561
Dr. John Chung's SAT II Math Level 2 Test 8
Therefore, r 2 or 2 .
297
2
2
21. (D)
52 x 3 7 x 1 (2 x 3) log 5 ( x 1) log 7 x(2log 5 log 7) log 7 3log 5 log 7 3log 5 x 2.26 2log 5 log 7
22. (C)
To have a finite limit value, degree of numerator and degree of denominator should be same. Therefore, a must be 0. To have limit value of 4, b must be 8. Therefore, a b 8 .
23. (A)
Sum of geometric infinite series: S
a , where a is the first term and r 1 . 1 r
1 1 1 1 1 1 9 lim t 1 n 9 27 81 6 t 2 3 1 3 n
24. (D)
Since f (b) f (a ) where b a, f ( x) is decreasing for all real number x .
25. (D)
The longest side is opposite the greatest angle. a 2 b 2 c 2 32 42 62 11 cos C 2ab 2(3)(4) 24
11 C cos 1 117 24 26. (B)
1 If g (a ) 5 , then f (5) a . Therefore, 2
27. (E)
f ( x) has a maximum at x 0 and lim 3
53
x
x
2a a
9 . 4
0.
Therefore, 0 y 1 . Or use graphic utility. y
1 O
28. (A) 29. (E)
x
1 1 1 The areas 2 f (2) 2 f (4) 4 f (8) 2 2 4 0.25 9 81 6561 y
Using a graphic utility: Domain: All real numbers. O
298
x
2
2 3 4 and tan B , 4 3 3 4 tan A tan B 4 3 tan( A B ) 0.29 . 1 tan A tan B 3 4 1 4 3
30. (C)
Since tan A
31. (C)
Let f (3k 4) 6k 5 . Then x 3k 4 or k
x4 . 3
x4 Therefore, f ( x) 6 5 2 x 13. 3 2
32. (C)
k k x 2 y 2 kx x y 2 2 2 k , 0 2
y
O
33. (A)
2
The area of triangle is A
x
(2)(2)sin120 3 3 3 . 2
2 120o 2
34. (B)
Use long division. x2 4
x 2 1 x 4 3x 2 10 x4 x2 4 x 2 10 4x2 4 14 35. (C)
Lateral area r where is a slant height. 10. Therefore, the lateral area is (6)(10) 60 and the base area is 36 . The surface area is 96 .
Dr. John Chung's SAT II Math Level 2 Test 8
299
2 36. (D)
2 The radius of the base is
V r 2h
156 39 . Therefore, the volume of the cylinder is 2
2
39 10 390 1225 .
16
10
d 162 102 156
d
37. (E)
Because 3 is a zero of f ( x) , f (3) 0 . f (3) 9 3k k 7 0 k 8.
f ( x) x 2 8 x 15 ( x 3)( x 5) 0 Therefore, the other root is 5. 12(3)
38. (C)
0.05 A 5, 000 1 12
39. (B)
2x 3 2 3
40. (B)
Since
41. (A)
Substitute y 2 x 4 in the other equation. That is,
r 2h 3
x
100
5807.36 9 2
9 Therefore, f (2) 4 8 10. 2
r 2h r 2 h 100 100 . The volume of a pyramid : V 4 4 127.32 . 3 3
x 2 x 4 16 x 2 x 20 0
x 5 x 4 0 ,
x 4, 5
2
But when x 5, y 5 4 9 (rejected). When x 4, y 2 4 4 0 y 0 . Therefore, the intersection is (4, 0) . y
O
42. (E)
43. (D)
300
y2 x 4 x
9 9 r r For the coefficient of each term 5 x 1 , r 6 to have x3 . r 9 3 6 Therefore, the coefficient is 5 1 10500 . 6 Because all data values have been decreased by 5, the standard deviation has not been changed.
2 44. (D)
45. (B)
2
a u , a
1 a 32 42 122 13 . Therefore u 3, 4,12 0.23, 0.31, 0.92 . 13
Since y 2t 3 , the polynomial equation will be as follows. 2 1 x(2t 3) x 2 0 x(2t 4) 2 x x 2t 4 2t
46. (D)
6 4 2 The third term is 2 3i 2160 . 2
47. (E)
10 45 2
48. (A)
Law of sines: 7 10 10sin 45 sin B 1.01 , sin B 1.01 (No solution) sin 45 sin B 7
49. (C)
x
50. (E)
PQ 32 42 52 50 and QR 4 .
1 t
cos
0 and t
1 2t 1 1 . y 2 y 2 x 2 (Domain : x 0) . 2 t t x
4 o cos 1 55.55 50 50
4
END
Dr. John Chung's SAT II Math Level 2 Test 8
301
No Test Material on This Page
302
Test 9 Dr. John Chung’s SAT II Mathematics Level 2
Dr. John Chung's SAT II Math Level 2 Test 9
303
2
2
MATHEMATICS LEVEL 2 TEST REFERENCE INFORMATION
THE FOLLOWING INFORMATION IS FOR YOUR REFERENCE IN ANSWERING SOME OF THE QUESTIONS IN THIS TEST 1 Volume of a right circular cone with radius r and height h : V r 2 h 3 1 Lateral Area of a right circular cone with circumference of the base c and slant height : S c 2
Volume of a sphere with radius r : V 4 r 3 3 Surface Area of a sphere with radius r : S 4 r 2 1 Volume of a pyramid with base area B and height h : V 3 Bh
304
2
2 Dr. John Chung’s SAT II Math Level 2
Answer Sheet 01 02
A
B
C
D
E
A
B
C
D
E
03
A
B
C
D
E
B
C
D
04 05
A A
B
C
06
A
B
07
A
08
26 27
A
B
C
D
E
A
B
C
D
E
A
B
C
D
E
E
28 29
A
B
C
D
E
D
E
30
A
B
C
D
E
C
D
E
A
B
C
D
E
B
C
D
E
31 32
A
B
C
D
E
A
B
C
D
E
33
A
B
C
D
E
09
A
B
C
D
E
34
A
B
C
D
E
10 11
A
B
C
D
E
A
B
C
D
E
A
B
C
D
E
35 36
A
B
C
D
E
12
A
B
C
D
E
37
A
B
C
D
E
13 14
A
B
C
D
E
B
C
D
E
A
B
C
D
E
38 39
A
A
B
C
D
E
15
A
B
C
D
E
40
A
B
C
D
E
16 17
A
B
C
D
E
A
B
C
D
E
A
B
C
D
E
41 42
A
B
C
D
E
18
A
B
C
D
E
B
C
D
E
19 20
B
C
D
E
43 44
A
A
A
B
C
D
E
A
B
C
D
E
A
B
C
D
E
A
B
C
D
E
A
B
C
D
E
21 22
45 46
A
B
C
D
E
47
A
B
C
D
E
23
A
B
C
D
E
48
A
B
C
D
E
24
A
B
C
D
E
49
A
B
C
D
E
25
A
B
C
D
E
50
A
B
C
D
E
The number of right answers:
The number of wrong answers:
# of correct
Dr. John Chung's SAT II Math Level 2 Test 9
1 4 # of wrong
Raw score
305
2
2 Score Conversion Table 6
Scaled Score 480
630
5
470
26
620
4
470
800
25
620
3
460
46
800
24
610
2
460
45
800
23
610
1
450
44
800
22
600
0
450
43
800
21
600
42
800
20
590
41
800
19
590
40
780
18
580
39
760
17
570
38
750
16
560
37
740
15
550
36
720
14
540
35
710
13
530
34
700
12
520
33
690
11
510
32
680
10
500
31
670
9
490
30
660
8
490
29
650
7
480
28
Scaled Score 640
800
27
48
800
47
50
Scaled Score 800
49
Raw Score
306
Raw Score
Raw Score
2
2
MATHEMATICS LEVEL 2 TEST For each of the following problems, decide which is the BEST of the choices given. If the exact numerical value is not one of the choices, select the choice that best approximates this value. Then fill in the corresponding circle on the answer sheet Note: (1) A scientific or graphing calculator will be necessary for answering some (but not all) of the questions in this test. For each question you will have to decide whether or not you should use a calculator. (2) For some questions in this test you may have to decide whether your calculator should be in the radian mode or the degree mode. (3) Figures that accompany problems in this test are intended to provide information useful in solving the problems. They are drawn as accurately as possible EXCEPT when it is stated in a specific problem that its figure is not drawn to scale. All figures lie in a plane unless otherwise indicated. (4) Unless otherwise specified, the domain of any function f is assumed to be the set of all real numbers x for which f ( x) is a real number. The range of f is assumed to be the set of all real numbers f ( x) , where x
is in the domain of f . (5) Reference information that may be useful in answering the questions in this test can be found on the page preceding Question 1. USE THIS SPACE FOR SCRATCHWORK 1.
If 2(a b) 5(a b) , then which of the following must be true? (A) (B) (C) (D) (E)
2.
a0 b0 ab a b ab0
If f ( x) f ( x) for all real x and a point (3, 5) is on the line, then which of the following points is also on the line? (A) (3, 5) (B) (3, 5) (C) (5, 3) (D) (3, 5) (E) (5, 3)
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Dr. John Chung's SAT II Math Level 2 Test 9
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2
2 MATHEMATICS LEVEL 2 TEST - Continued USE THIS SPACE FOR SCRATCHWORK.
3.
If
10! n ! , then n 90 56
(A) 3
4.
(B) 4
(C) 5
(D) 6
(E) 7
Figure 1 shows the graph of the linear function whose 3 equation is defined by f ( x) x 3 . What is the value of 4 x -axis ? formed by the line and (A) (B) (C) (D) (E)
36.9 36.9 45.5 45.5 55.8
y
y f ( x)
O
x
Note: Figure not drawn to scale. FIgure 1
5.
If f ( x) 3 x 1 and g ( x) 5 , then g f (7.82) (A) (B) (C) (D) (E)
6.
10.3 5 5 10.3 15.6
Two circles are symmetric with respect to y x . If the equation of a circle is x2 y 2 2 x 4 y 1 0 , then which of the following is the equation of the other circle ? 2 2 (A) ( x 2) ( y 1) 4
(B) ( x 1)2 ( y 2)2 2 2 2 (C) ( x 1) ( y 2) 4
(D) ( x 2)2 ( y 2)2 4 (E) ( x 1)2 ( y 2)2 4
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308
2
2 MATHEMATICS LEVEL 2 TEST - Continued USE THIS SPACE FOR SCRATCHWORK.
7.
If 6 (A) (B) (C) (D) (E)
8.
cos ?
0.116 0.201 0.217 0.328 0.466
If x 3, then (A) (B) (C) (D) (E)
10.
1 n 13 2 n 13 2 n 26 n 1 or n 26 n 2 or n 13
If tan 4.5 , what is the value of (A) (B) (C) (D) (E)
9.
n 3 4 , which of the following is the solution set? 2
x 10 2
10 x 10 x x 10 x 10
( x 10)
Which of the following is true? (A) sin sin (B) cos cos (C) tan tan (D) sec sec (E) csc csc
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2
2 MATHEMATICS LEVEL 2 TEST - Continued USE THIS SPACE FOR SCRATCHWORK.
11.
When a polynomial P( x) x2 ax b is divided by ( x 1) , the remainder is 3, and when the polynomial is divided by ( x 2) , the remainder is 3 . What are the values of a and b? (A) a 9, b 11 (B) a 9, b 11 (C) a 5, b 3 (D) a 5, b 3 (E) a 3, b 5
12.
If sin a cos(2a 30) , then what is the value of tan a ? (A) (B) (C) (D) (E)
13.
0.21 0.36 0.42 0.60 0.75
What is the range of the function f ( x) 3x 9 4 ? (A) y 3 (B) y 3 (C) y 4 (D) y 4 (E) y 4 y
14.
If line is perpendicular to the line y 3x , then what is the
y 3x
area of ABO ?
A (A) (B) (C) (D) (E)
1 1.5 2 2.5 3
O
B (3, 0)
x
Note: Figure not drawn to scale.
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310
2
2 MATHEMATICS LEVEL 2 TEST - Continued USE THIS SPACE FOR SCRATCHWORK.
15.
If xy 1 , then (A) 1
16.
(B) 2
If log
3
(E) 5
x 10 , then log 3 x3 (B) 15
(C) 30
If tan 3 and (A) (B) (C) (D) (E)
19.
(D) 4
2 2 5 5 10
(A) 10
18.
(C) 3
If one of the roots of 2 x 2 ax b 0 is 1 2i , what is the value of b ? (A) (B) (C) (D) (E)
17.
x y x 1 y 1
(D) 45
(E) 60
3 , what is the value of cos 2 ? 2
0.2 0.4 0.8 0.8 0.4
If the surface area of a cylinder, whose height is twice the radius, is 50, then what is the value of the radius? (A) (B) (C) (D) (E)
r
1.63 1.84 2.45 3.87 4.56
h
FIgure 3
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311
2
2 MATHEMATICS LEVEL 2 TEST - Continued USE THIS SPACE FOR SCRATCHWORK.
20.
If f ( x) log( x 1) log( x 1) , then f 1 ( x) 2
1
(A)
10 x
(B)
x 2 10
10 x 1
(C)
(D) 10 x 1 (E) 10 x 1 y
21.
In Figure 2, f ( x) x b is tangent to the graph of a circle
f ( x) x b
whose equation is x 2 y 2 4 . What is the value of b ? (A) (B) (C) (D) (E)
22.
5.45 3.48 2.14 2.21 2.83
O
x
Figure 4
What is the distance between the two points of intersection of the circles whose equations are x 2 y 2 16 and
x 4 2 y 4 2 16 ? (A) (B) (C) (D) (E)
23.
1.12 2.73 3.35 4.87 5.66
If a 3, 2 , 3 and b 1, 5, 2 , which of the following is the value of a b ?
(A) 2.52 (B) 3.74 (C) 4.25 (D) 7.58 (E) 8.02
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312
2
2 MATHEMATICS LEVEL 2 TEST - Continued USE THIS SPACE FOR SCRATCHWORK.
24.
A pencil holder contains only five black pencils and three white pencils. If three pencils are drawn at random, what is the probability to have two black pencils and one white pencil? (A)
25.
3 5
(B)
3 8
(C)
15 28
(D)
5 7
(E)
2 3
( x 1) 2 0 , then which of the following is the complete x solution set of the inequality?
If
(A) x 0 (B) 0 x 1 (C) x 0 (D) x 0 or x 1 (E) x 0 or x 1 26.
Which of the following includes all asymptotes of the x3 rational function f ( x) 2 ? x 1 (A) x 1, x 1 (B) x 1, y 0 (C) x 1, x 1, and y 0 (D) x 1, x 1, and y x (E) x 1, x 1, and y 1
27.
Which of the following is the distance from the origin to the plane x y z 3 0 ? (A)
2
(B) 3 (C) 2 (D)
5
(E)
6
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2
2 MATHEMATICS LEVEL 2 TEST - Continued USE THIS SPACE FOR SCRATCHWORK.
28.
If cos 2 3cos 1 0 , then what is the smallest positive value of ? (A) (B) (C) (D) (E)
29.
What is the interquartile range of the following set of data 10, 13, 15, 18, 25, 30, 40, 60, 75, 80, 80 ? (A) (B) (C) (D) (E)
30.
20 40 50 60 70
3 If angle A is obtuse and tan A , which of the 2 following is the value of cos 2A ?
(A) (B) (C) (D) (E)
31.
8.16 4.40 1.88 0.92 0.46
0.38 0.30 0.15 1.5 3.6
3 3 What is the value of arccos arcsin ? 2 2 (A) 0 (B) 30 (C) 45 (D) 30 (E) 45
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314
2
2 MATHEMATICS LEVEL 2 TEST - Continued USE THIS SPACE FOR SCRATCHWORK.
32.
If f ( x) 3 2 x 3 , then f 1 3 (A) (B) (C) (D) (E)
33.
The function f ( x) x 2 4 x 9 is a shift of f ( x) x 2 (A) (B) (C) (D) (E)
34.
3.6 5 8 10 12
4 units to the right and 9 units up 2 units to the right and 5 units down 2 unit to the left and 5 units up 4 units to the left and 9 units up 2 units to the right and 5 units up
If cos i sin (cos i sin ) a 1 bi , where a and b are real numbers, which of the following is true? (A) a 1, b 1 (B) a 1, b 0 (C) a 2, b 0 (D) a 2, b 2 (E) a 2, b 2
35.
If the difference of the roots of x 2 2mx 7 is 8, then what is the positive integer value of m ? (A) 0
(B) 1 (C) 2
(D) 3 (E) 4
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2
2 MATHEMATICS LEVEL 2 TEST - Continued USE THIS SPACE FOR SCRATCHWORK.
36.
What is the measure of one of the larger angles of a parallelogram in the xy -plane that has vertices with coordinates 3, 2 , 6, 2 , 4, 6 and 7, 6 ? (A) 76.0 (B) 98.2 (C) 104.0
(D) 103.5 (E) 108.6
37.
A used car was purchased for $20,000 and the car loses k % of its value each year. If the car is worth $10,000 after 5 years, what is the value of k ? (A) (B) (C) (D) (E)
38.
Which of the following is the polar form of the rectangular equation y 4 ? (A) (B) (C) (D) (E)
39.
10.5 11.6 12.9 13.6 14.8
r 3 r4 r 4 sin r 4 cos r 4 csc
If function f ( x )
1 x 3 , and f 1 ( x) is the inverse 4
function of f ( x) , then f 1 f 1 (3) (A) 0 (B) 3 (C) 6 (D) 12 (E) 18
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316
2
2 MATHEMATICS LEVEL 2 TEST - Continued USE THIS SPACE FOR SCRATCHWORK.
40.
(A) (B) (C) (D) (E) 41.
42.
If 2 32 x5 5 10 , what is the value of x ? 1.28 2.46 3.42 3.68 4.12
What is the domain of f x (A) (B) (C) (D) (E)
x 3 x 1 1 x 3 x3 1 x 5
lim
x2 x x2 x 2
x 1
x 1 x2 x 6
?
1 3 1 (B) 2
(A)
(C) 1 (D) 2 (E) Undefined 43.
How many integer values of x satisfy the inequality x( x 6)( x 8)( x 2) 0 ? (A) 4
(B) 6
(C) 8
(D) 10
(E) 11
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2
2 MATHEMATICS LEVEL 2 TEST - Continued USE THIS SPACE FOR SCRATCHWORK.
44.
What is the range of the following function?
1 x, f ( x) x 1 1, (A) (B) (C) (D) (E) 45.
, 0 0, 1 0, 1 0, 0,
10 100 1024 2048 4096
What is the coefficient of x in the binomial expansion of
4
x 5 ?
(A) (B) (C) (D) (E)
47.
if x 1
In how many ways can a 10 question true-false math exam be answered? (Assume that no questions are omitted.) (A) (B) (C) (D) (E)
46.
if x 1
20 150 500 625 875
If 14 n P3 value of n ? (A) (B) (C) (D) (E)
n 2 P4
, then which of the following could be the
6 7 8 9 15
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2
2 MATHEMATICS LEVEL 2 TEST - Continued USE THIS SPACE FOR SCRATCHWORK.
48.
Assume that the probability of having a boy is 40%. In a family with three children, what is the probability that there is at least one boy? (A) (B) (C) (D) (E)
49.
0.40 0.486 0.562 0.765 0.784
Figure 5 shows the graph of an ellipse whose equation is
x h
2
y k
2
1 . If the area A of an ellipse is given a2 b2 by A ab , then what is the area of the ellipse in Figure 5?
y
(A) 9.42 (B) 15.71 (C) 28.42
(5, 0)
O
x (10, 3)
(D) 47.12 (E) 52.25
Figure 5
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319
2
2 MATHEMATICS LEVEL 2 TEST - Continued USE THIS SPACE FOR SCRATCHWORK.
50.
Which of the following could be the graph of the parametric equations x t 1 and y t 2 ? (A)
(B)
y
x
O
(C)
x
x
O
(D)
y
O
y
y
O
x
(E) y
O
x
STOP IF YOU FINISH BEFORE TIME IS CALLED, YOU MAY CHECK YOUR WORK ON THIS TEST ONLY. DO NOT TURN TO ANY OTHER TEST IN THIS BOOK.
GO ON TO THE NEXT PAGE
320
No Test Material on This Page
Dr. John Chung's SAT II Math Level 2 Test 9
321
2
2
TEST 9 # 1 2 3 4 5 6 7 8 9 10
ANSWERS
answer C D D A C A C E A D
# 11 12 13 14 15 16 17 18 19 20
answer B B D B A E B D A C
# 21 22 23 24 25 26 27 28 29 30
answer E E B C C D B C D A
# 31 32 33 34 35 36 37 38 39 40
answer B E E C D C C E D C
# 41 42 43 44 45 46 47 48 49 50
Explanations: Test 9 1. (C)
ab 0 a b
2. (D)
Since f ( x) is an odd function, (3, 5) must be on the line.
3. (D)
10! 10! 6! , n 6 90 56 10 9 8 7
4. (A)
Since tan
5. (C)
g ( x) 5 is a constant function for any value of x .
6. (A)
Switch x and y. y 2 x 2 2 y 4 x 1 0
7. (C) 8. (E)
6
( x 2)2 ( y 1)2 4
Since tan 4.5 and cos 0 , lies in Quadrant I. y 1 21.25 Therefore, cos 0.466 4.5 21.25 1
x
( x 10) 2 x 10 If x 3 , then x 10 x 10 .
322
n n n n 6 7 7 6 7 1 13 2 n 26 2 2 2 2
O
9. (A)
3 3 , tan 1 36.9 . 4 4
answer D A D E C B A E D D
2
2
10. (D)
sec is an even function.
11. (B)
P(1) 1 a b 3 and P (2) 4 2a b 3 . ab2 2a b 7 a
a 9 and b 11
9
12. (B)
Cofunction: a 2a 30 90 a 20 tan 20 0.36
13. (D)
Check with a graphing calculator.
f ( x) 3x 9 4 3( x 3) 4
y
(3, 4)
x
O
14. (B)
Range: y 4
1 The equation of line is y x b . The line passes through a point (3,0). Therefore, the 3 1 3 1 1.5. equation of line is y x 1 . AO 1 and OB 3 . The area is 3 2 x y x( y 1) y ( x 1) x y2 2 xy x y 1. x 1 y 1 xy x y 1 xy x y 1 x y 2 1 1 x 1 x 1 1 y Or, since y , . Therefore, 1. x 1 x 1 x 1 x 1 x y 1 1 x 1 x
15. (A)
Since xy 1,
16. (E)
Product of the roots:
17. (B)
log
3
1 2i 1 2i
b b 5 b 10 2 2
x log 3 x 2 10 2 log 3 x 10 log3 x 5
log3 x3 3log3 x 3(5) 15
Dr. John Chung's SAT II Math Level 2 Test 9
323
2 18. (D)
2 Since lies in Quadrant III, cos
2
4 1 . cos 2 2cos 2 1 2 1 5 10 10
1
1 3
10
19. (A)
Surface area = 2 r 2 2 r (2r ) 50 6 r 2 50 r
20. (C)
f ( x) log( x 1) log( x 1) log( x 2 1) (Domain: x 1 )
50 1.63 6
For the inverse f 1 :
x log y 2 1
21. (E)
y 2 1 10 x y 10 x 1 (Range: y 1 )
y x b and x 2 y 2 4 intersect at one point. Substitute y x b into a circle equation. x 2 ( x b) 2 4 2 x 2 2bx b2 4 0 To have two equal roots, its discriminant should be zero. D 2b 4(2)(b 2 4) 4b 2 32 0 b 2 8 b 2 2 or 2 2 2
In Figure 2, the y-intercept b is negative. Therefore, b 2 2 .
Or, the distance from the origin to x y b 0 is d
00b 12 (1) 2
2 b 2 2 b 2 2 : the y -intercept is negative.
y
22. (E)
The length is 4 2 5.66.
( x 4)2 ( y 4)2 16
(0, 4)
(4, 4) (4,0)
x 2 y 2 16
23. (B)
324
x
The graphs intersect at points 0, 4 and 4,0 .
a b (3, 2, 3) (1, 5, 2) (2, 3, 1) a b (2, 3, 1) 22 (3) 2 12 14 3.74
2 24. (C)
25. (C)
2 3 C1 15 28 8 C3 Or, find the probability to have BBW , BWB, and WRR. 5 4 3 5 P( BBW ) . P( BBW ) P( BWB) P(WBB) . 8 7 6 28 5 15 Therefore, 3 . 28 28 P
5 C2
( x 1)2 ( x 1) 2 0 multiply by x 2 x 2 0 x 2 x( x 1) 2 0 x 0 x x Using graphic solution: The complete solution set is x 0 .
0
Or, use test value. 26. (D)
1
x3 x x 2 2 x 1 x 1 Vertical asymptotes : D ( x) x 1 x 1 0 x 1, x 1 f ( x)
Slant asymptote: y x as x . 0003
D
28. (C)
Use a graphing calculator: Radian mode.
2
2
2
1 1 1
3
27. (B)
3
3
x 1.8784
y0
Algebraic Solution: Quadratic formula. 3 13 cos cos 3.30277 (rejected) or 0.3027756377 2 cos 1 (0.3027756377) 1.8783999 1.88 29. (D)
Interquartile range upper quattile lower quartile = 75 15 = 60 10, 13, (15), 18, 25, (30), 40, 60, (75), 80, 80 10
15
Dr. John Chung's SAT II Math Level 2 Test 9
30
75
80
Box and Whisker Plot
325
2 30. (A)
2 Method 1: Find the angle using a calculator. 3 3 tan A A tan 1 56.30993247 180 123.6900675 A is obtuse 2 2 cos 2 A cos(2 123.6900675) 0.3846153846 0.38 Method 2: Using diagram A
2
2 cos 2 A 2cos 2 A 1 2 1 13 0.3846153846 0.38
13
3 2
31. (B)
Method 1: Using a calculator (Degree mode) 3 3 arccos arcsin 30 2 2 Method 2: Algebraic solution 3 3 , where 0 180 . cos 30 Let arccos 2 2 3 3 60 Let arcsin , where 90 90 . sin 2 2 Therefore, 30 60 30
32. (E)
33. (E)
f 1 : y
x3 3 2
f 1 3 12
Method1:
x 3 2y 3
Method 2:
3 3 2 y 3 27 2 y 3 y 12
f ( x) x 2 5 : Two units to the right and 5 units up 2
34. (C)
cos 2 sin 2 a 1 bi 1 a 1 bi a 1 1 and b 0 Therefore, a 2 and b 0.
35. (D)
If r1 r2 , then r1 r2 8 , r1 r2 2m, and r1 r2 7 .
r1 r2 2 r1 r2 2 4r1r2
4m 2 64 28 m 2 9 m 3
Or, substitute the choices into the equation. For m 3, x 2 6 x 7 ( x 1)( x 7) 0 x 1, 7
326
(Difference is 8.)
2
2
36. (C)
(4,6)
(7,6)
x (6, 2) 1
(3, 2)
tan x 4 37. (C)
4
x tan 1 4 75.96
Therefore, 180 75.96 104.0 .
Compound decay yearly: A P(1 r )t 10,000 20,000 1 0.01k
5
1 (1 0.01k )5 2
1
1 5 1 0.01k 2
1
k
1 0.5 5 0.01
12.94404367 12.9
38. (E)
y 4 r sin 4 r
39. (D)
f 1 ( x) 4 x 12
40. (C)
41. (D)
4 4csc sin
f 1 (3) 0
f 1 (0) 12
15 7.5 2 x 5 log3 7.5 2 1 log 7.5 Therefore, x 5 3.417021884 3.42 2 log 3 2(32 x 5 ) 15 32 x 5
N ( x) x 1 x 1 D( x) x 2 x 6 x 2 x 6 0 ( x 3)( x 2) 0 x 3 or x 2
x 1 x 3 x 3
or x 1 x 2
Therefore, the solution set is x 3 . x( x 1) x 1 lim x 1 ( x 2)( x 1) x 1 x 2 3
42. (A)
lim
43. (D)
The graph of y x( x 6)( x 8)( x 2) 0 is as follows.
8
0
2
6
x
Therefore, the intervals for y 0 is 8 x 0 or 2 x 6 . The integers x are 7, 6, 5, 4, 3, 2, 1 and 3, 4, 5 10 integers
Dr. John Chung's SAT II Math Level 2 Test 9
327
2 44. (E)
2 The graph of the piecewise function is as follows. y
y x 1 1
y 1 x
O
x
1
Therefore, the range is 0, . 45. (C)
210 1024
46. (B)
4 r
x
4r
5 r
4 Therefore, 2 47. (A)
14 n P3
4 4r x 2 (5)r r
x
n 2 P4
2
r must be 2
52 150 x
14 n(n 1) (n 2) (n 2)(n 1) n(n 1)
14n 28 n 2 3n 2 n 2 11n 30 0 (n 6)(n 5) 0 Therefore, n 5 or 6. 48. (E)
Since P( B)
2 3 and P(G ) , 5 5
1 2 2 1 3 0 3 2 3 3 2 3 3 2 3 98 0.784 The probability is 1 5 5 2 5 5 3 5 5 125 3 2 Or, 1 P(all three girs) 1 0.6 (0.4)0 0.784. 3
49. (D)
Since a 5 and b 3 , A ab 15 47.12 .
50. (D)
Since x t 1 0 and x 2 t 1 t x 2 1 , Therefore, y x 2 1 and x 0.
END
328
y t 2 x2 1 2 y x2 1 .
Test 10 Dr. John Chung’s SAT II Mathematics Level 2
Dr. John Chung's SAT II Math Level 2 Test 10
329
2
2
MATHEMATICS LEVEL 2 TEST REFERENCE INFORMATION
THE FOLLOWING INFORMATION IS FOR YOUR REFERENCE IN ANSWERING SOME OF THE QUESTIONS IN THIS TEST 1 Volume of a right circular cone with radius r and height h : V r 2 h 3 1 Lateral Area of a right circular cone with circumference of the base c and slant height : S c 2
Volume of a sphere with radius r : V 4 r 3 3 Surface Area of a sphere with radius r : S 4 r 2 1 Volume of a pyramid with base area B and height h : V 3 Bh
330
2
2 Dr. John Chung’s SAT II Math Level 2
Answer Sheet 01 02
A
B
C
D
E
A
B
C
D
E
03
A
B
C
D
E
B
C
D
04 05
A A
B
C
06
A
B
07
A
08
26 27
A
B
C
D
E
A
B
C
D
E
A
B
C
D
E
E
28 29
A
B
C
D
E
D
E
30
A
B
C
D
E
C
D
E
A
B
C
D
E
B
C
D
E
31 32
A
B
C
D
E
A
B
C
D
E
33
A
B
C
D
E
09
A
B
C
D
E
34
A
B
C
D
E
10 11
A
B
C
D
E
A
B
C
D
E
A
B
C
D
E
35 36
A
B
C
D
E
12
A
B
C
D
E
37
A
B
C
D
E
13 14
A
B
C
D
E
B
C
D
E
A
B
C
D
E
38 39
A
A
B
C
D
E
15
A
B
C
D
E
40
A
B
C
D
E
16 17
A
B
C
D
E
A
B
C
D
E
A
B
C
D
E
41 42
A
B
C
D
E
18
A
B
C
D
E
B
C
D
E
19 20
B
C
D
E
43 44
A
A
A
B
C
D
E
A
B
C
D
E
A
B
C
D
E
A
B
C
D
E
A
B
C
D
E
21 22
45 46
A
B
C
D
E
47
A
B
C
D
E
23
A
B
C
D
E
48
A
B
C
D
E
24
A
B
C
D
E
49
A
B
C
D
E
25
A
B
C
D
E
50
A
B
C
D
E
The number of right answers:
The number of wrong answers:
# of correct
Dr. John Chung's SAT II Math Level 2 Test 10
1 4 # of wrong
Raw score
331
2
2 Score Conversion Table 6
Scaled Score 480
630
5
470
26
620
4
470
800
25
620
3
460
46
800
24
610
2
460
45
800
23
610
1
450
44
800
22
600
0
450
43
800
21
600
42
800
20
590
41
800
19
590
40
780
18
580
39
760
17
570
38
750
16
560
37
740
15
550
36
720
14
540
35
710
13
530
34
700
12
520
33
690
11
510
32
680
10
500
31
670
9
490
30
660
8
490
29
650
7
480
28
Scaled Score 640
800
27
48
800
47
50
Scaled Score 800
49
Raw Score
332
Raw Score
Raw Score
2
2
MATHEMATICS LEVEL 2 TEST For each of the following problems, decide which is the BEST of the choices given. If the exact numerical value is not one of the choices, select the choice that best approximates this value. Then fill in the corresponding circle on the answer sheet Note: (1) A scientific or graphing calculator will be necessary for answering some (but not all) of the questions in this test. For each question you will have to decide whether or not you should use a calculator. (2) For some questions in this test you may have to decide whether your calculator should be in the radian mode or the degree mode. (3) Figures that accompany problems in this test are intended to provide information useful in solving the problems. They are drawn as accurately as possible EXCEPT when it is stated in a specific problem that its figure is not drawn to scale. All figures lie in a plane unless otherwise indicated. (4) Unless otherwise specified, the domain of any function f is assumed to be the set of all real numbers
x for which f ( x) is a real number. The range of f is assumed to be the set of all real numbers f ( x) , where x is in the domain of f . (5) Reference information that may be useful in answering the questions in this test can be found on the page preceding Question 1. USE THIS SPACE FOR SCRATCHWORK 1.
If a 0 , then
2a 2
9a 2
(A) a (B) 2a (C) 5a (D) 2a (E) 5a
2.
If 28k is an integer, which of the following is the smallest integer value of k ? (A) (B) (C) (D) (E)
2 4 6 7 28
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Dr. John Chung's SAT II Math Level 2 Test 10
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2
2 MATHEMATICS LEVEL 2 TEST - Continued USE THIS SPACE FOR SCRATCHWORK.
3.
If a3 64 and
a2 b , then b
(A) 4 (B) 4 (C) 8 (D) 8 (E) 16 4.
How many positive values of a are there to satisfy that 40 a is an integer? (A) 3
5.
(C) 5
(D) 6
(E) 7
Which of the following sets of data has a standard deviation of 0 ? (A) (B) (C) (D) (E)
6.
(B) 4
3, 2, 1, 0, 1, 2, 3 3, 3, 3, 1, 3, 3, 3 2, 2, 2, 0, 2, 2, 2 0, 0, 0, 3, 5, 5, 5 5, 5, 5, 5, 5, 5, 5
How many integers satisfy with the inequality
5 x2 10 ? (A) 4 (B) 6 (C) 8 (D) 10 (E) Infinitely many
7.
If a 3 , then
(A) (B) (C) (D) (E)
a2
a 2 2
8 5 3 5 8
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2
2 MATHEMATICS LEVEL 2 TEST - Continued USE THIS SPACE FOR SCRATCHWORK.
8.
Which of the following is true? (A)
5
25 5
(C)
( 11) 2 11
(D)
( 6) 2 6
16
1 2
4
If x 2 5 , then ( x 3)( x 2) ( x 1)( x 6) (A) 4
10.
2
(B)
(E)
9.
5
(B) 2
(C) 0
(D) 5
(E) 10
If the polynomial x 2 kx 1 has a factor of x 2 ,
then k (A)
11.
2 3
(B)
3 2
(C)
2 3
(D)
3 2
(E)
5 2
If x 2 10 x a x b for all real x , where a and b are 2
constants, what is the value of a ? (A) 30
12.
(B) 20
(C) 0
(D) 25
(E) 30
If one root of the equation x 2 10 x a 2 0 is 5 10 , what is the value of the constant a ? (A) (B) (C) (D) (E)
7 13 15 18 23
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2
2 MATHEMATICS LEVEL 2 TEST - Continued USE THIS SPACE FOR SCRATCHWORK.
13.
Which of the following equations has no solution? (A) x3 8
x2 4
(B)
(C) x 2 2 x 1 0 (D) e x 0.5 (E) x 5 2 14.
If f ( x) 8sin x 2 , which of the following includes all values of x in the interval 0 x , where f ( x) 6 ? (A) (B)
3
6 5 (C) 6
3
5 6 6 2 and (E) 3 3
(D)
15.
and
and
2 If the coordinates of the vertex of f ( x) x 4 x k are
(2, 5) , what is the value of k ? (A) 5
16.
(B) 7
(C) 9
(D) 11
(E) 14
What is the range of the quadratic function
f ( x) x2 7 x 12 ? (A) y y 4.25 (B) y y 3.50
(C) y y 2.36 (D) y y 1.25 (E) y y 0.25
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336
2
2 MATHEMATICS LEVEL 2 TEST - Continued USE THIS SPACE FOR SCRATCHWORK.
17.
2 If the minimum value of y x 2kx k is 12 , then what
is all value of k ? (A) (B) (C) (D) (E)
3 3 3, 4 3, 4 4, 5 A
18.
In Figure 1, a circular cone is inscribed in the sphere at center O with radius 10. If AH 16 , what is the volume of the cone? (A) (B) (C) (D) (E)
4188.8 3217.0 2495.2 1072.3 1010.6
O
B
H
C
Note: Figure not drawn to scale. Figure 1
19.
3 If f ( x) 2 x 1 , then the inverse of f ( x) is
(A) 2 x 2 1 (B)
3
x 1
3
x 1 2
(C) (D)
3
x 1 2
(E)
3
x 1 2
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2
2 MATHEMATICS LEVEL 2 TEST - Continued USE THIS SPACE FOR SCRATCHWORK.
20.
If the period of the function y 2sec(2kx 0.3) 5 is 2, then what is the value of k ? (A) (B) (C) (D) (E)
21.
If a sphere has a volume of 64 , then what is its surface area? (A) (B) (C) (D) (E)
22.
6.28 4.77 3.14 1.57 0.79
166 175 184 225 289
Which of the following functions has an inverse function? 2 (A) f ( x) x
(B) f ( x) x 5 (C) f ( x) x3 x 1 (D) f ( x ) x 3 2 (E) f ( x) x 10
23.
In Figure 2, the surface area of the cylinder is 24 and the volume of the cylinder is 12 . Which of the following is the value of the radius? (A) (B) (C) (D) (E)
r
0.52 1.33 1.59 2.77 3.78
h
Figure 2
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338
2
2 MATHEMATICS LEVEL 2 TEST - Continued USE THIS SPACE FOR SCRATCHWORK.
24.
If two six-sided dice are tossed, what is the probability that a total of 7 is rolled? (A) (B) (C) (D) (E)
25.
1 18 1 12 3 6 2 6 1 6
If a vector P has the magnitude of 10 and the same
direction as the vector Q (3, 4) , which of the following
is the vector P ? (A) (6, 8) (B) (10, 1) (C) (3, 4) (D) (6, 8) (E) (9, 12)
26.
If sin t
(A) (B) (C) (D) (E)
4 , then sin( t ) 5
3 5 3 5 4 5 4 5 3 4
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2
2 MATHEMATICS LEVEL 2 TEST - Continued USE THIS SPACE FOR SCRATCHWORK.
27.
If f (t ) sin t , g (t ) cos t , and h(t ) f (t ) g (t ) , which of the following is not true? (A) f (t ) is an odd function. (B) g(t) is an even function (C) h(t ) is an odd function (D) h(t ) is an even function
f (t ) is an odd function g (t )
(E)
28.
y f ( x)
Which of the following is the equation of the graph in Figure 3 ? 1 (A) y 2sin x 2 2 (B) y 2cos2x 2
4 2 4
1 (C) y 2cos x 2 2 1 (D) y 2cos x 2 2 1 (E) y 2cos x 2 4
29.
4
O
2 4 Figure 3
In the xy-coordinate plane, which of the following is the set of points whose distance from the origin is two times the distance from the point (6, 0) ? (A) (B) (C) (D) (E)
a line a parabola a hyperbola an ellipse a circle
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340
2
2 MATHEMATICS LEVEL 2 TEST - Continued USE THIS SPACE FOR SCRATCHWORK.
30.
If f ( x) 2 x 1 and g ( x) x 2 2 x 1 , then ( f g )(4) (A)
19
(B) 10 (C) 3 (D) 9 (E) 27
31.
If g ( x) is the inverse function of f ( x) , which of the following is not true? I. The graphs of g ( x) and f ( x) are reflections of each other in the line y x . II. f g ( x) x III. g f ( x) x (A) (B) (C) (D) (E)
32.
I only II only III only I and II only I and III only
If a soft-drink manufacturer has daily production costs of
C (n) 80, 000 120n 0.05n 2 , where C is the total cost in dollars and n is the number of units produced, what is the minimum cost each day? (A) (B) (C) (D) (E)
8,000 7,500 7,000 6,700 6,000
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2
2 MATHEMATICS LEVEL 2 TEST - Continued USE THIS SPACE FOR SCRATCHWORK.
33.
In Figure 4, if a line is drawn around the box from vertex P to Q passing through AB , what is the length of the shortest distance of the line? (A) (B) (C) (D) (E)
34.
6 B A 7
Q
Note: Figure not drawn to scale. Figure 4
If a polynomial function y x7 100 x2 5x 4 , what is the right-end and left-end behavior of the graph of the function? (A) (B) (C) (D) (E)
35.
12.6 13.9 14.6 15.1 15.8
5 P
The graph falls to the left and rises to the right. The graph falls to the left and falls to the right. The graph rises to the left and rises to the right. The graph rises to the left and falls to the right. The graph approaches to the x axis.
( x 2)( x 5) 2 0 , which of the following is the ( x 3) solution set of the inequality?
If
(A) x x 3 (B) x 3 x 2 (C) x 3 x 2 (D) x 2 x 5 (E) x 3 x 2 x x 5
36.
a If log 3 (ab) 10 , log3 2 , and b 0, what is the b value of a ? (A) (B) (C) (D) (E)
9 81 243 729 2187 GO ON TO THE NEXT PAGE
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2
2 MATHEMATICS LEVEL 2 TEST - Continued USE THIS SPACE FOR SCRATCHWORK.
37.
If the ellipse x 2 2 y 2 2 x 4 y k has a major axis of 10, where k is a constant, then what is the value of k ? (A) (B) (C) (D) (E)
5 10 15 22 25
2
38.
If x 3 4 , which of the following is the complete solution set ? (A) x x 8 (B) x x 8 (C) x x 4 (D) x x 8 or 8 (E) x x 4 or 4
39.
What is the amplitude of y 3sin 4cos ? (A) (B) (C) (D) (E)
3.5 4 5 5.8 7
n
40.
lim 5(0.2)k
n
(A) (B) (C) (D) (E)
k 1
1 1.25 5 6.26 Infinite
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2
2 MATHEMATICS LEVEL 2 TEST - Continued USE THIS SPACE FOR SCRATCHWORK.
41.
What is the coefficient of the term x6 y 5 in the expansion of
2 x 5 y 11 ? (A) (B) (C) (D) (E)
42.
Matrices A, B, C, and D are of orders 2 3 , 2 3, 3 2, and 2 2 , respectively. Which of the following matrices are of proper order to perform the operation? (A) (B) (C) (D) (E)
43.
A 3C B 2C AB BC 3D CB 2D
If the sum of the first n terms of a series is S n n 2 4n , then what is the 10th term? (A) (B) (C) (D) (E)
44.
3,125 29,568 324,567 90,400,000 92, 400, 000
23 40 85 125 140
In how many ways can 5 different prizes be given to any 5 of 12 people, if no person receives more than one prize? (A) (B) (C) (D) (E)
124020 95040 7650 792 60
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2
2 MATHEMATICS LEVEL 2 TEST - Continued USE THIS SPACE FOR SCRATCHWORK.
45.
If n C2 (A) (B) (C) (D) (E)
46.
7 6 5 4 3
0.084 0.096 0.125 0.358 0.612
If three marbles are chosen at random from a bag containing 4 red marbles and 5 white marbles, what is the probability that exactly two marbles are red? (A) (B) (C) (D) (E)
48.
and n 3, then n
If three people are randomly chosen, what is the probability that all were born on different days of the week? (A) (B) (C) (D) (E)
47.
n 1 P2
0.36 0.42 0.52 0.63 0.81
1 1 x x 0 x
lim
(A) 1.5 (B) 0.5 (C) 1.5 (D) 2.0 (E) Undefined
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2
2 MATHEMATICS LEVEL 2 TEST - Continued USE THIS SPACE FOR SCRATCHWORK.
49.
The frequency table in Figure 5 shows the test score for students sampled in a statistics class. What is the standard deviation of the data?
(A) 5.9 (B) 6.1
Test score
70 78 82 92
Frequency
3
(C) 6.6
8
5
2
Figure 5
(D) 7.2 (E) 7.5
50.
What is the surface area of the cast iron solid in Figure 6?
(A) 100 (B) 120 (C) 132
15
(D) 165
12
(E) 190
8 Figure 6
STOP IF YOU FINISH BEFORE TIME IS CALLED, YOU MAY CHECK YOUR WORK ON THIS TEST ONLY. DO NOT TURN TO ANY OTHER TEST IN THIS BOOK.
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346
No Test Material on This Page
Dr. John Chung's SAT II Math Level 2 Test 10
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2
2
TEST 10 # 1 2 3 4 5 6 7 8 9 10
ANSWERS
answer A D A E E C E C E D
# 11 12 13 14 15 16 17 18 19 20
answer D B B D C E D D D D
# 21 22 23 24 25 26 27 28 29 30
answer A D D E D C D D E E
# 31 32 33 34 35 36 37 38 39 40
answer C A B D E D D D C B
# 41 42 43 44 45 46 47 48 49 50
answer E D A B D E A B B C
Explanations: Test 10 2a 2
1. (A)
Since
x2 x ,
2. (D)
Since
28k 2 2 7 k , k should be 7. (Prime factorization)
3. (A)
a3 64 a 4 and b a 2 (4)2 16 4 .
4. (E)
9a 2 4a 2 9a 2 2a 3a 2a (3a) a .
40 a 0,1, 2,3, 4,5, 6 40 a 0,1, 4,9,16, 25,36
5. (E)
If all data are equal, the standard deviation is 0.
6. (C)
Since
a 40,39,36,31, 24,15, 4
x 2 x , 5 x 10 . Therefore,
x 6,7,8,9 x 6, 7, 8, 9 (8 integers) .
7. (E)
a a 2 3 3 2 8
5
2
5 (B)
25 5 (C)
112
11 (D)
6 2
6
8. (C)
(A)
9. (E)
( x 3)( x 2) ( x 1)( x 6) x 2 5 x 6 x 2 5 x 6 2 x 2
2
(E)
2
Since x 5, 2 x 10. 10. (D)
Factor Theorem:
3 Let x 2 kx 1 ( x 2)Q( x) . When x 2, 4 2k 1 0 k . 2
348
16
1 2
2
2 11. (D)
2 x 2 10 x a x b
2
x 2 10 x a x 2 2bx b 2
Therefore, b 5 and a b 2 . a 25. 12. (B)
Substitution: 5 10
2
10 5 10 a 2 0 35 10 10 50 10 10 a 2 0
15 a 2 0 a 13 Or,
Product of the roots: 5 10 5 10 a 2 15 a 2 a 13 13. (B)
(A) x3 8 x 2
(B)
x 2 4
x 4
(C) x 2 2 x 1 0 Discriminant 8 0 :two real roots (D) e x 0.5 x ln 0.5
(E) x 5 2 x 7 or 3
1 5 x or 2 6 6
14. (D)
8sin x 2 6 sin x
15. (C)
f (2) 4 8 k 5 k 9
16. (E)
Axis of symmetry: x The range: y 0.25
7 3.5 , 2
f (3.5) (3.5)2 7(3.5) 12 0.25
Graphing utility:
Maximum x 3.4999
y 0.25
2k k f (k ) k 2 2k 2 k 12 k 2 k 12 0 2 (k 4)(k 3) 0 k 4 or 3 A
17. (D)
Axis of symmetry: x
18. (D)
The radius of the circular base is 8 and the height of the cone is 16. r 2 h 64 16 V 1072.330292 1072.3 3 3 B
19. (D)
f ( x ) 2 x3 1
Dr. John Chung's SAT II Math Level 2 Test 10
f 1 ( x) : x 2 y 3 1 y 3
O 10 6 H 8
C
x 1 2
349
2 20. (D)
2 Since the frequency is 2k , then the period is
k
2 k
2
2 . 2k k
1.570796327 1.57 1
21. (A)
4 V r 3 64 r 48 3 3 2
2 1 The surface area: 4 r 4 48 3 4 48 3 165.9729662 166 2
22. (D)
Since f ( x) x3 is a one- to- one function, its inverse is also a function. (C) y x3 x 1 is not one-to-one. Use a graphing utility to show the graph.
23. (D)
Since 2 r 2 2 rh 24 and r 2 h 12 , then r 2 rh 12 and r 2 h 12. h
12 12 12 r 2 r 2 12 r 2 12 r 3 12r 12 0 2 r r r x 1.1157494 x 2.7687343
Zero x 2.7687343
y0 y0
y0
Therefore, r 1.12 or 2.77 . 24. (E)
The total of 7: P
1, 6 2,5 3, 4 4,3 5, 2 6,1
6 36
25. (D)
Choice (D): Magnitude 62 8 10 6, 8 2 3, 4 : Same direction Because P 10 and Q 32 (4) 2 5 , P 2Q .
26. (C)
sin( t ) sin cos t cos sin t sin t
27. (D)
1 h(t ) sin t cos t sin 2t : odd function 2
28. (D)
period 4 , middle line 2 , frequency
2
1 Therefore, y 2cos x 2 2
350
sin t
2 1 4 2
4 5
2
2
29. (E)
y
2
(0, 0)
:
( x, y )
x2 y 2 2
(6 x) (0 y )
2
1
(6, 0)
x
2 x 2 y 2 4(36 12 x x 2 y 2 ) 3 x 2 3 y 2 48 x 144 0 1
x 2 y 2 16 x 48 0 ( x 8)2 y 2 42 : a circle 30. (E)
Multiplication : f g ( x)
2 x 1 x2 2x 1
f g (4) 3 9 27
cf. ( f g )( x) ; composition 31. (C) 32. (A)
f f 1 ( x) x , y f ( x) and y f 1 ( x) is symmetric with respect to y x. Axis of symmetry: x
120 1200 2 (0.05)
f (1, 200) 80,000 120(1, 200) 0.05(1, 200) 2 8, 000 33. (B)
Since PQ is the shortest distance, PQ 52 132 13.92838828 13.9 6
7
Q
5 P
34. (D)
lim f ( x) : rises to the left
x
lim f ( x) : falls to the right
x
Dr. John Chung's SAT II Math Level 2 Test 10
351
2 35. (E)
2 By multiplying ( x 3)2 to both sides: ( x 2)( x 5)2 0 ( x 3)( x 2)( x 5) 2 0 , where x 3. ( x 3) 3
2
5
x5
3 x 2
The solution set: 3 x 2 or x 5 . At x 5 , the inequality is also true. Or, you can use test value method. 36. (D)
Since ab 310 and
a a a 32 9 , then b and a 310. 9 b 9
a 2 312 a 36 729 37. (D)
x 2 2 y 2 2 x 4 y k ( x 1) 2 2( y 1) 2 k 3
( x 1)2 ( y 1) 2 1 k 3 k 3 2
a2 k 3 a k 3 Major axis : 2a 2 k 3 10 k 3 25 k 22
38. (D)
Algebraic solution:
2 x3
2
3
1 2 2 x 3 4 x 3 43 64 x 2 64 x 8
Graphing calculator:
Intersection x 8 x 8
39. (C)
2 y1 x 3 y2 4
y 4 y 4
If y a sin and y b cos , then the amplitude of y a sin b cos is
a 2 b2 .
The amplitude is 32 42 5 . 4 4 3 3 Because y 5 sin cos 5sin( 1 ) , where sin 1 and cos 1 . 5 5 5 5
352
2 40. (B)
2 Infinite series: n n 0.2 lim 5(0.2)k 5 lim 0.2k 5 1.25 n n 1 0.2 k 1 k 1
41. (E)
r 42. (D)
2 11 r 5r x 11r y r 6 5 must be 5. Therefore, 11C5 2 5 x 6 y 5 92, 400,000
11 Cr
2 x 11 r 5 y r
11 Cr
Dimension operation: (A) A 3C 2 3 3 2 : wrong
2 3 3 2 : wrong AB 2 3 2 2 :wrong BC 3D 2 3 3 2 2 2 2 2 2 2 2 2 : true
(B) B 2C (C) (D)
3 2 2 3 2 2 3 3 2 2 : wrong
(E) CB 2 D
43. (A)
a10 S10 S9 102 4(10) 92 4(9) 140 117 23
44. (B)
Choose 5 people and assign them to the 5 prizes:
45. (D)
n C2
n 1 P2
n (n 1) 2!
Therefore, n 4. 46. (E)
P
n 1 n 2
12 C5
5! 95040
n (n 2) 2
n 1
7 6 5 210 0.612244898 0.612 343 73
people #1 M T W TH F SA S
people #2 M T W TH F SA S
people #3 M T W TH F SA S
All possible outcomes: 7 7 7 343 47. (A)
Probability: P
5 C1 30 0.3571428571 0.36 84 9 C3
4 C2
Dr. John Chung's SAT II Math Level 2 Test 10
353
2
2 1
1 x 1 1 x
48. (B)
lim
49. (B)
Calculator: statistics S x 6.097251068 6.1
50. (C)
END
354
x 0
x 1 1 x
lim
x 0
x
x 1 1 x
Lateral area of the cone rs 4 5 20 Lateral area of the cylinder 2 rh 96 The area of the base 16 Therefore, 20 96 16 132
lim
x 0
1
1 1 x
1 2
5
3 4 12
8
15
Test 11 Dr. John Chung’s SAT II Mathematics Level 2
Dr. John Chung's SAT II Math Level 2 Test 11
355
2
2
MATHEMATICS LEVEL 2 TEST REFERENCE INFORMATION
THE FOLLOWING INFORMATION IS FOR YOUR REFERENCE IN ANSWERING SOME OF THE QUESTIONS IN THIS TEST 1 Volume of a right circular cone with radius r and height h : V r 2 h 3 1 Lateral Area of a right circular cone with circumference of the base c and slant height : S c 2
Volume of a sphere with radius r : V 4 r 3 3 Surface Area of a sphere with radius r : S 4 r 2 1 Volume of a pyramid with base area B and height h : V 3 Bh
356
2
2 Dr. John Chung’s SAT II Math Level 2
Answer Sheet 01 02
A
B
C
D
E
A
B
C
D
E
03
A
B
C
D
E
B
C
D
04 05
A A
B
C
06
A
B
07
A
08
26 27
A
B
C
D
E
A
B
C
D
E
A
B
C
D
E
E
28 29
A
B
C
D
E
D
E
30
A
B
C
D
E
C
D
E
A
B
C
D
E
B
C
D
E
31 32
A
B
C
D
E
A
B
C
D
E
33
A
B
C
D
E
09
A
B
C
D
E
34
A
B
C
D
E
10 11
A
B
C
D
E
A
B
C
D
E
A
B
C
D
E
35 36
A
B
C
D
E
12
A
B
C
D
E
37
A
B
C
D
E
13 14
A
B
C
D
E
B
C
D
E
A
B
C
D
E
38 39
A
A
B
C
D
E
15
A
B
C
D
E
40
A
B
C
D
E
16 17
A
B
C
D
E
A
B
C
D
E
A
B
C
D
E
41 42
A
B
C
D
E
18
A
B
C
D
E
B
C
D
E
19 20
B
C
D
E
43 44
A
A
A
B
C
D
E
A
B
C
D
E
A
B
C
D
E
A
B
C
D
E
A
B
C
D
E
21 22
45 46
A
B
C
D
E
47
A
B
C
D
E
23
A
B
C
D
E
48
A
B
C
D
E
24
A
B
C
D
E
49
A
B
C
D
E
25
A
B
C
D
E
50
A
B
C
D
E
The number of right answers:
The number of wrong answers:
# of correct
Dr. John Chung's SAT II Math Level 2 Test 11
1 4 # of wrong
Raw score
357
2
2 Score Conversion Table 6
Scaled Score 480
630
5
470
26
620
4
470
800
25
620
3
460
46
800
24
610
2
460
45
800
23
610
1
450
44
800
22
600
0
450
43
800
21
600
42
800
20
590
41
800
19
590
40
780
18
580
39
760
17
570
38
750
16
560
37
740
15
550
36
720
14
540
35
710
13
530
34
700
12
520
33
690
11
510
32
680
10
500
31
670
9
490
30
660
8
490
29
650
7
480
28
Scaled Score 640
800
27
48
800
47
50
Scaled Score 800
49
Raw Score
358
Raw Score
Raw Score
2
2
MATHEMATICS LEVEL 2 TEST For each of the following problems, decide which is the BEST of the choices given. If the exact numerical value is not one of the choices, select the choice that best approximates this value. Then fill in the corresponding circle on the answer sheet Note: (1) A scientific or graphing calculator will be necessary for answering some (but not all) of the questions in this test. For each question you will have to decide whether or not you should use a calculator. (2) For some questions in this test you may have to decide whether your calculator should be in the radian mode or the degree mode. (3) Figures that accompany problems in this test are intended to provide information useful in solving the problems. They are drawn as accurately as possible EXCEPT when it is stated in a specific problem that its figure is not drawn to scale. All figures lie in a plane unless otherwise indicated. (4) Unless otherwise specified, the domain of any function f is assumed to be the set of all real numbers x for which f ( x) is a real number. The range of f is assumed to be the set of all real numbers f ( x) , where x
is in the domain of f . (5) Reference information that may be useful in answering the questions in this test can be found on the page preceding Question 1. USE THIS SPACE FOR SCRATCHWORK
1.
3x 3 2 5y (A)
2.
2
3x 4 9 x6 25 x 6 9x4 25 y 4 (D) (B) (C) (E) 5 y5 25 y 4 9 y4 25 y 6 9 x6
What is the value of k for which x 2 2kx k 2 is a perfect square? (A) (B) (C) (D) (E)
1 4 9 16 Any real number
GO ON TO THE NEXT PAGE
Dr. John Chung's SAT II Math Level 2 Test 11
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2
2 MATHEMATICS LEVEL 2 TEST - Continued USE THIS SPACE FOR SCRATCHWORK.
3.
3 If x 2 x 3 is a factor of 2x px q , what are the values
of p and q ? (A) p 4 and q 6 (B) p 8 and q 4 (C) p 8 and q 6 (D) p 8 and q 6 (E) p 8 and q 6
4.
What is the sum of the roots of the equation x3 3x 52 0 ?
(A) (B) (C) (D) (E)
5.
3 0 2 3 52
If a polynomial P ( x) has a remainder of three when divided by ( x 1) and remainder of one when divided by ( x 1) , then the remainder when divided by ( x2 1) is (A) (B) (C) (D) (E)
6.
4 x2 x4 x2 x4
If x 3, then 1 x 2 (A) (B) (C) (D) (E)
x3 x3 x 3 x 3 x 1
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360
2
2 MATHEMATICS LEVEL 2 TEST - Continued USE THIS SPACE FOR SCRATCHWORK.
7.
Which of the following values of x is true for the inequality sin x cos x ?
(A)
8.
4
4
(C)
3 4
(B)
2
(C)
4
7 7 25 25 6
If
(D)
5 4
(E)
3 2
(D)
2
(E)
If
3 4
10
10
10
Figure 1
a 2 4 , then a a2 a2 2 2 4 6 No solution
(A) (B) (C) (D) (E)
11.
2
What is the product of 3 4i and its conjugate? (A) (B) (C) (D) (E)
10.
In Figure 1, a right cylinder is inscribed in a cube. If the cube has an edge of 10, what is the ratio of the volume of the cylinder to the volume of the cube? (A)
9.
(B)
3
27 x3 3 , then x
(A) 3 (B) 3 3 2 (C) 33 2 (D) 9 (E) 9
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2
2 MATHEMATICS LEVEL 2 TEST - Continued USE THIS SPACE FOR SCRATCHWORK.
12.
What is the domain of f ( x) (A) (B) (C) (D) (E)
13.
x ? 10 x
x0 0 x 10 0 x 10 0 x 10 0 x 10
Which of the following is the slant asymptote(s) of 2 x 2 3x 1 f ( x) ? x 1 (A) x 1 (B) y 2 (C) y 2 x (D) y 2 x 5 (E) y 2 x 10
14.
2 If f 3x 5 , then f ( x) x (A) (B) (C) (D) (E)
15.
3x 5 2 2 3x 5 2 2 3 1 x 5 6 5 x 6 1 x 5
If f ( x) x 1 , what value does the inverse function
f 1 ( x) have at the point x 9 ? (A) 8
(B) 10
(C) 100
(D) 120
(E) 125
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362
2
2 MATHEMATICS LEVEL 2 TEST - Continued USE THIS SPACE FOR SCRATCHWORK.
16.
The symbol x is the greatest integer which is less than or equal to x. If
x 2 6 x 9 , which of the following
could be the value of x ? (A) (B) (C) (D) (E) 17.
3.3 1.3 2.5 3.9 4.01
How many real numbers are equal to their multiplicative inverses? (A) 4
18.
(D) 1
(E) 0
24 120 360 2880 40320
Find all values of k which satisfy the determinant k 3 27 ? 2k k (A) (B) (C) (D) (E)
20.
(C) 2
How many ways can 8 books be arranged on a shelf if 5 of them are math books and must be kept together? (A) (B) (C) (D) (E)
19.
(B) 3
9 3 6 6 or 3 9 or 3
If y varies directly as x and inversely as the square of z , and y 15 when x 3 and z 5 , what is the value of y when x 5 and z 2.5 ? (A) 1
(B) 25
(C) 50
(D) 100
(E) 125
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2
2 MATHEMATICS LEVEL 2 TEST - Continued USE THIS SPACE FOR SCRATCHWORK.
21.
What is the number of points of intersection of the graphs of
x 2 y 2 1 and x 2 4 y 2 4 ? (A) 0
22.
(B) 1
(C) 2
(D) 3
(E) 4
If the equation of the graph in Figure 2 is y x 10 k ,
y
A
where k is a constant, what is the area of ABO ? (A) (B) (C) (D) (E) 23.
x
Figure 2
100 192.5 225 475.8 1000
If cos x x , then how many real solutions are there? (A) (B) (C) (D) (E)
25.
B O
If the length of a diagonal of a cube is 10, then what is the volume of the cube? (A) (B) (C) (D) (E)
24.
100 75 50 25 12.5
0 1 2 3 4
Which of the following is the graph of the polar equation 1 r ? cos sin (A) (B) (C) (D) (E)
A line A circle An ellipse A parabola An hyperbola
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364
2
2 MATHEMATICS LEVEL 2 TEST - Continued USE THIS SPACE FOR SCRATCHWORK.
26.
What is the distance between the line y 2 x 5 and the point (1, 4)? (A) (B) (C) (D) (E)
27.
3.25 1.75 1.34 1.19 0.75
If the area of the triangle bounded by the lines y 2 x , x k , and y 6 is 16, what is the positive value of k ?
(A) (B) (C) (D) (E) 28.
What is the sum of all factors of 210? (A) (B) (C) (D) (E)
29.
556 576 584 616 625
If log 2 ( x 2 3x 2) log 2 ( x 1) 5 , what is the value of x? (A) (B) (C) (D) (E)
30.
4 5 6 7 8
10 30 34 40 48
If 103 x 27 , then 102 x (A)
1 3
(B)
1 6
(C)
1 9
(D)
1 18
(E)
1 27
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365
2
2 MATHEMATICS LEVEL 2 TEST - Continued USE THIS SPACE FOR SCRATCHWORK.
31.
If log 2 0.3010 , what is the number of digits in 2100 ? (A) (B) (C) (D) (E)
32.
50 46 34 31 30
If 2i is a root of x3 2 x 2 4 x 8 0 , which of the following are the other roots? (A) 2i (B) 2i and 3 (C) 2i and 2 (D) 2i and 1 (E) 2i and 2
33.
2 3 4 n 1 lim 2 2 2 2 2 n n n n n
n
(A) 0
34.
1 2
(C) 1
(D)
3 2
(E) 2
Three numbers have a sum of 30 and a product of 640. If these three numbers form an arithmetic sequence, what is the smallest number? (A) (B) (C) (D) (E)
35.
(B)
1 2 4 10 16
If f ( x) 22 x and g ( x) log 4 x , then f g ( x) (A) 0.5 x (B) x (C) 1.5x (D) 2 x (E) 2.5x GO ON TO THE NEXT PAGE
366
2
2 MATHEMATICS LEVEL 2 TEST - Continued USE THIS SPACE FOR SCRATCHWORK.
36.
Which of the following is not a function? 1 (A) y 2
x
(B) y x5 (C) xy 9 (D) y 10 x 2 (E) y 2 4 x
37.
In Figure 3, what is the value of the angle between the lines y x 1 and y 3x 5 ? (A) (B) (C) (D) (E)
63.4 65.8 69.2 72.5 75.6
y
y x 1
y 3x 5 O
x
Note: Figure not drawn to scale. 38.
sin 2 1 cos 2
(A) (B) (C) (D) (E)
39.
Figure 3
sin cos tan cot sec
If cos 0.61, then cos( ) (A) (B) (C) (D) (E)
0.61 0.39 0.39 0.61 0.93
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2
2 MATHEMATICS LEVEL 2 TEST - Continued USE THIS SPACE FOR SCRATCHWORK.
40.
x2 y2 1 , then which of the following is the foci of the 9 16 hyperbola?
If
(A) (B) (C) (D) (E)
41.
0, 3 0, 4 3, 0 4, 0 5, 0
Which of the following is the graph of the curve represented by the parametric equations x t 2 and y t 5 ? (A)
(B)
y
x
O
(C)
x
(E)
x
O
(D)
y
O
y
y
O
x
y
O
x
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368
2
2 MATHEMATICS LEVEL 2 TEST - Continued USE THIS SPACE FOR SCRATCHWORK.
42.
In Figure 4, if AB 3 , BC 5 , and CD 7 , what is the degree measure of angle ADF ? (A) (B) (C) (D) (E)
50.2 48.4 46.8 45.6 43.1
A
F
B
E C
D
Figure 4 43.
If the rectangular equation is x 2 y 2 4 x 0 , which of the following is an equivalent equation in polar form? (A) (B) (C) (D) (E)
44.
2 0 1 2 If the matrices A and B , what is the 3 1 1 4 determinant of A+B ? (A) (B) (C) (D) (E)
45.
r2 r 0 r 2cos 0 r 4cos 0 r 2sin 0 r 4cos 0
12 12 23 23 28
If the equation of a sphere is given by x 2 y 2 z 2 4 x 6 z 0 , what is the surface area of the sphere? (A) (B) (C) (D) (E)
13 52 84 128 169
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2
2 MATHEMATICS LEVEL 2 TEST - Continued USE THIS SPACE FOR SCRATCHWORK.
46.
If the plane passes through the point 3, 2, 5 and is parallel to the xy -plane , then what is the equation of the plane? (A) x 3 (B) y 5 (C) x 5 (D) z 5 (E) y z
47.
x 2 px q 5 , where p and q are constants, what x 1 x 1 is the value of p ? If lim
(A) (B) (C) (D) (E)
48.
What is the minimum value of the function defined as x 2 x , f ( x) ? x 2 2, (A) (B) (C) (D) (E)
49.
2 1 1 3 4
4 2 0 1 2
How many positive integers are there in the solution set of ( x 5)( x 2) 0? x 1 (A) (B) (C) (D) (E)
2 3 4 5 6
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370
2
2 MATHEMATICS LEVEL 2 TEST - Continued USE THIS SPACE FOR SCRATCHWORK.
50.
What is the constant term in the binomial expansion of 12
2 1 x ? x (A) (B) (C) (D) (E)
924 792 495 220 66
STOP IF YOU FINISH BEFORE TIME IS CALLED, YOU MAY CHECK YOUR WORK ON THIS TEST ONLY. DO NOT TURN TO ANY OTHER TEST IN THIS BOOK.
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371
No Test Material on This Page
372
2
2
TEST 11 # 1 2 3 4 5 6 7 8 9 10
ANSWERS
answer D E A B B D E A D E
# 11 12 13 14 15 16 17 18 19 20
answer B E D D C D C D E D
# 21 22 23 24 25 26 27 28 29 30
answer C A B B A C D B C C
# 31 32 33 34 35 36 37 38 39 40
answer D C B C B E A D A E
# 41 42 43 44 45 46 47 48 49 50
answer D A C D B D D C C C
Explanations: Test 11 2
1. (D)
3x 3 5 y 2
2. (E)
2k Since k 2 , then k 2 k 2 . It is always true for all real number k . 2
3. (A)
Long division:
32 x 6 25 x 6 52 y 4 9 y 4 2
2x 2 2
x x 3 2 x3
px
q
2 x3 2 x 2 6 x 2 x 2 ( p 6) x q
remainder
2 x2 2 x 6 ( p 4) x q 6 Therefore, p 4 and q 6 . 4. (B)
Sum of the roots of polynomial equation an x n an 1 x n 1 an 2 x n 2 a1 x a0 a 0 is n 1 . Therefore, S 0. an 1
5. (B)
P ( x) can be expressed in three different ways as follows.
( x 1)Q1 ( x) 3 P( x) ( x 1)Q2 ( x) 1 ( x 1)( x 1)Q( x) ax b
Dr. John Chung's SAT II Math Level 2 Test 11
(1) (2) (3)
373
2
2 When x 1 3 P(1) a b
(1) (3)
When x 1 1 P(1) a b
(2) (3)
ab 3
a b 1
Therefore, a 1 and b 2 remainder x 2 6. (D)
Since x 3 , 1 x 2 1 ( x 2) x 3 ( x 3) x 3 .
7. (E)
Graphing calculator: Check each choice: If If sin x cos x If If If
8. (A)
x x
4
, then sin , then sin
cos . 4 4
cos
2 2 2 3 3 3 cos , then sin x 4 4 4 5 5 5 cos x , then sin 4 4 4 3 3 3 x , then sin cos (True) 2 2 2
Volume of the cylinder: r 2 h 250
Volume of the cube: 103 1000 250 Therefore, the ratio: . 1000 4 9. (D)
(3 4i )(3 4i ) 3 4i 25
10. (E)
Multiply by the common denominator (a 2) : a 4(a 2) 2 a 2. (Extraneous root)
11. (B) 12. (E)
2
3
2
27 x3 3 27 x3 27 x3 54 x 3 3 2
x x is 0 , where x 10 . 10 x 10 x (10 x) x 0 x( x 10) 0 0 x 10
Domain of f ( x)
Or, find the critical points; From
374
x 0, critical points are x 0 and x 10. 10 x
2
2 Therefore, for 0 x 10,
13. (D)
Since f ( x)
14. (D)
Method 1:
x is positive. 10 x
2 x 2 3x 1 6 6 2x 5 , y lim 2 x 5 2x 5 . x x 1 x 1 x 1
2 2 2 If x is replaced with , f is changed to f f ( x) x x 2 x 6 2 Therefore, f ( x) 3 5 5 x x Method 2: 2 2 Let f 3k 5 and x . k k 2 2 k . Replace k with . x x 6 2 2 f 3k 5 f ( x) 3 5 f ( x) 5 x k x 15. (C)
Method 1: The inverse function: f 1 ( x) x 1 Method 2: From f ( x) : f 1 (9) y is equivalent to 9 2
16. (D)
x 2 6 x 9 x 2 6 x 9 0 x 3 3 x 4
17. (C)
k
y 1.
y 100
x 3
2
0
1 k 2 1 k 1 k
18. (D)
4!
4! 5! 2880 19. (E)
f 1 (9) 100
5!
Determinant: k 3 k 2 6k k 2 6k 27 k 2 6k 27 0 2k k
Dr. John Chung's SAT II Math Level 2 Test 11
375
2
2 (k 9)(k 3) 0 k 9 or k 3
20. (D)
Compound proportion: y k 15 k
21. (C)
3 25
k 125
x z2 y 125
,
5 100 2.52
x2 y 2 1 4 1
x 2 y 2 1 and x 2 4 y 2 4 y
1 2
2
22. (A)
1
f (0) 0 10 k 0 k 10 y
10
A
B
O
10
23. (B)
x
Area A x
20 10 100 2
Let the length of a edge be x. The length of the diagonal:
x 2 x 2 x 2 x 3 10
3
10 Volume V x3 192.4500897 192.5 3 24. (B)
Graphic utility: One point of intersection.
25. (A)
376
1 cos sin x y 1 r
r cos r sin 1 r
x y r 1 r r
2 26. (C)
2 Distance between 1, 4 and 2 x y 5 0 . D
2(1) 4 5 22 1
2
3
27. (D)
5
1.341640786 1.34
y
y 2x
( k , 2k ) (3, 6)
2k 6 y6
x
O
k 3 xk
The area of because k 0 .
(k 3)(2k 6) 16. (k 3)2 16 k 3 4, 4 2
28. (B)
Prime factorization: 210 21 31 51 71 Therefore, the sum of all factors (1 2)(1 3)(1 5)(1 7) 576
29. (C)
log 2 ( x 2 3x 2) log 2 ( x 1) 5 x 2 25
30. (C)
31. (D)
( x 2)( x 1) log 2 ( x 2) 5 ( x 1)
x 34
103 x 27
32
log 2
k 7, 1
103 x
2 3
27
2 3
10 x 33
2 3
32
1 9
Using a calculator: 2100 1.2676506 1030 Algebraically: Let X 2100
The number of digits 30 1 31
log X log 2100 100log 2 30.10
X 1030.10 1030 X 1031 Therefore, the number of digits is 31. Ex. 210 1024 (4 digits number) log 210 10log 2 10 0.3010 3.010 . 3 log 210 4 . Therefore, it is 4digits number. 32. (C)
b 2 2. a 1 Since 2i is a root, then 2i is also a root. Let the third root be w . Sum of the roots: 2i (2i ) w 2 w 2 Therefore, the roots are 2i, 2i, and 2 . Sum of the roots of x3 2 x 2 4 x 8 0 is
Dr. John Chung's SAT II Math Level 2 Test 11
377
2 33. (B)
2 2 3 4 n 1 2 3 n 1 lim 2 2 2 2 2 lim lim n n n n n n n n n2
n(1 n) 2 n2
n(1 n) n n2 1 lim n 2n 2 n 2n 2 2 lim
34. (C)
Let three number be a d , a, and a d , where d 0. (a d ) a (a d ) 30 a 10 (a d )a(a d ) (10 d )(10)(10 d ) 640 100 d 2 64 d 2 36 Since d 6 , the smallest number is 10 6 4. 2
g ( x) 22log 4 x 2log4 x 2
log 4 x 2
35. (B)
f
36. (E)
Choice (E): If x 2, then y 2. (Vertical line test fails.)
37. (A)
tan
2log2 x x
x 0
m2 m1 1 m1m2
If m1 1 and m2 3 , then tan
Or,
tan 1 1 1 45
2
1
3 1 2 tan 1 (2) 63.43494882 63.4 1 (3)(1) tan 2 3 2 tan 1 3 71.56505118 71.565
180 (45 71.565) 63.435 63.4
38. (D)
Because cos 2 1 2sin 2 and sin 2 2sin cos , sin 2 2sin cos 2sin cos cos cot 2 1 cos 2 1 (1 2sin ) sin 2sin 2
39. (E)
cos( ) cos cos sin sin cos 0.61
40. (E)
x2 y 2 1 a 2 b2
x2 y 2 1 9 16
a 3 and b 4
Since c 32 42 5 , the foci are at ( 5, 0) and (5, 0) . 41. (D)
x t2 x0 x t 2 t x 2 2 y ( x 2 2) 5 y x 2 7
42. (A)
378
Diagonal AD 32 52 7 2 83 and DE 32 52 34 . 34 34 ADF cos 1 cos ADF 50.2 83 83
2 43. (C)
44. (D)
45. (B)
2 Since x r cos and y r sin , x 2 y 2 4 x 0 r 2 4(r cos ) 0 . Therefore, r 4cos 0 . 2 0 1 2 3 2 1 4 3 1 4 5 3 2 Determinant : (3)(5) (2)(4) 23 4 5 x 2 y 2 z 2 4 x 6 z 0 ( x 2)2 y 2 ( z 3)2 13 Since r 13 , the surface area is 4 r 2 4 (13) 52 .
46. (D)
The plane is parallel to xy-plane: z 5 z 5 O x y
47. (D)
x 2 px q 5 , x 2 px q must have a factor of ( x 1) to have a finite limit x 1 x 1
Since lim
value. f (1) 1 p q 0 q 1 p x 2 px q x 2 px 1 p x 2 px 1 p x 2 1 p( x 1) ( x 1)( x 1) p ( x 1) ( x 1)( x 1 p) x 2 px q ( x 1)( x 1 p ) x 1 p lim lim 2 p. x 1 x 1 x 1 x 1 ( x 1) 1 2 p 5 p 3.
Therefore, lim
48. (C)
The graph of f is as follows. x , f ( x) 2,
x 2 x 2 y
2
O
2
x
The minimum of y 0 .
Dr. John Chung's SAT II Math Level 2 Test 11
379
2 49. (C)
2 ( x 5)( x 2) 0 ( x 1)( x 5)( x 2) 0 and x 1 x 1 Method 1: Graphic solution 2
1
5
The positive integers are 2, 3, 4, and 5.
( x 5)( x 2) (3)(4) 0 0 (ok) x 1 1 Therefore, there are 4 positive integers, 2, 3, 4, and 5 in 1, 5 . Method 2: Test sign: Test at x 2 in 1, 5
50. (C)
General term:
12 Cr
x
2 12 r
r
1 x
12 Cr x
For r 8 , it is constant. Therefore,
END
380
24 2 r r
12 C8 x
x
0
12 Cr x 24 3r
495 .
Test 12 Dr. John Chung’s SAT II Mathematics Level 2
Dr. John Chung's SAT II Math Level 2 Test 12
381
2
2
MATHEMATICS LEVEL 2 TEST REFERENCE INFORMATION
THE FOLLOWING INFORMATION IS FOR YOUR REFERENCE IN ANSWERING SOME OF THE QUESTIONS IN THIS TEST 1 Volume of a right circular cone with radius r and height h : V r 2 h 3 1 Lateral Area of a right circular cone with circumference of the base c and slant height : S c 2
Volume of a sphere with radius r : V 4 r 3 3 Surface Area of a sphere with radius r : S 4 r 2 1 Volume of a pyramid with base area B and height h : V 3 Bh
382
2
2 Dr. John Chung’s SAT II Math Level 2
Answer Sheet 01 02
A
B
C
D
E
A
B
C
D
E
03
A
B
C
D
E
B
C
D
04 05
A A
B
C
06
A
B
07
A
08
26 27
A
B
C
D
E
A
B
C
D
E
A
B
C
D
E
E
28 29
A
B
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D
E
D
E
30
A
B
C
D
E
C
D
E
A
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C
D
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B
C
D
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31 32
A
B
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E
A
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33
A
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D
E
09
A
B
C
D
E
34
A
B
C
D
E
10 11
A
B
C
D
E
A
B
C
D
E
A
B
C
D
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35 36
A
B
C
D
E
12
A
B
C
D
E
37
A
B
C
D
E
13 14
A
B
C
D
E
B
C
D
E
A
B
C
D
E
38 39
A
A
B
C
D
E
15
A
B
C
D
E
40
A
B
C
D
E
16 17
A
B
C
D
E
A
B
C
D
E
A
B
C
D
E
41 42
A
B
C
D
E
18
A
B
C
D
E
B
C
D
E
19 20
B
C
D
E
43 44
A
A
A
B
C
D
E
A
B
C
D
E
A
B
C
D
E
A
B
C
D
E
A
B
C
D
E
21 22
45 46
A
B
C
D
E
47
A
B
C
D
E
23
A
B
C
D
E
48
A
B
C
D
E
24
A
B
C
D
E
49
A
B
C
D
E
25
A
B
C
D
E
50
A
B
C
D
E
The number of right answers:
The number of wrong answers:
# of correct
Dr. John Chung's SAT II Math Level 2 Test 12
1 4 # of wrong
Raw score
383
2
2 Score Conversion Table 6
Scaled Score 480
630
5
470
26
620
4
470
800
25
620
3
460
46
800
24
610
2
460
45
800
23
610
1
450
44
800
22
600
0
450
43
800
21
600
42
800
20
590
41
800
19
590
40
780
18
580
39
760
17
570
38
750
16
560
37
740
15
550
36
720
14
540
35
710
13
530
34
700
12
520
33
690
11
510
32
680
10
500
31
670
9
490
30
660
8
490
29
650
7
480
28
Scaled Score 640
800
27
48
800
47
50
Scaled Score 800
49
Raw Score
384
Raw Score
Raw Score
383
2
2
MATHEMATICS LEVEL 2 TEST For each of the following problems, decide which is the BEST of the choices given. If the exact numerical value is not one of the choices, select the choice that best approximates this value. Then fill in the corresponding circle on the answer sheet Note: (1) A scientific or graphing calculator will be necessary for answering some (but not all) of the questions in this test. For each question you will have to decide whether or not you should use a calculator. (2) For some questions in this test you may have to decide whether your calculator should be in the radian mode or the degree mode. (3) Figures that accompany problems in this test are intended to provide information useful in solving the problems. They are drawn as accurately as possible EXCEPT when it is stated in a specific problem that its figure is not drawn to scale. All figures lie in a plane unless otherwise indicated. (4) Unless otherwise specified, the domain of any function f is assumed to be the set of all real numbers x for which f ( x) is a real number. The range of f is assumed to be the set of all real numbers f ( x) , where x
is in the domain of f . (5) Reference information that may be useful in answering the questions in this test can be found on the page preceding Question 1. USE THIS SPACE FOR SCRATCHWORK 1.
If 3 x a (A) 4
2.
81x12 , then a b
(B) 5
(C) 6
If 3a b 2 a b , then (A) 3
3.
b
(B) 2
If 2 x1 8 , then (A) (B) (C) (D) (E)
(D) 7
(E) 8
ab ab
(C) 1
(D) 2
(E) 3
23 x 22 x 2
20 40 48 64 80
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Dr. John Chung's SAT II Math Level 2 Test 12
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2
2 MATHEMATICS LEVEL 2 TEST - Continued USE THIS SPACE FOR SCRATCHWORK.
4.
What is the smallest integer value of x to satisfy the x 2x 1 inequality 1 ? 4 2
(A) 1
(B) 0
(C) 1
(D) 2
(E) 3 m
5.
In Figure 1, when the rectangle with dimensions 4 and 10 is rotated about line m , what is the volume of the resulting solid? (A) (B) (C) (D) (E)
6.
10
100 120 160 320 640
2
Figure 1
log3 16 log 2 9 (A) (B) (C) (D) (E)
7.
4
8 log 6 144 24 log5 25
36
If sin
1 and is acute angle, what is the value of 2
sin 2 ? (A) (B) (C) (D) (E)
8.
0.20 0.42 0.50 0.87 0.95
Which of the following is an asymptote of f ( x) tan 2 ? (A)
8
(B)
4
(C)
2
(D)
2 5 (E) 3 3
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386
2
2 MATHEMATICS LEVEL 2 TEST - Continued USE THIS SPACE FOR SCRATCHWORK.
9.
In Figure 2, if triangle ABC has BC 10 , A 30 , and
B
C 70 , then what is the area of the triangle? (A) (B) (C) (D) (E)
10.
92.5 97.3 112.5 125.8 135.1
10
A
30
70
C
Figure 2
Which of the following is true for the graph of f ( x) sin 2 x 3 ? (A) (B) (C) (D)
symmetric with respect to the y-axis symmetric with respect to the x-axis symmetric with respect to the origin symmetric with respect to the point (3, 0)
(E) symmetric with respect to the point (0, 3)
11.
2 2 The graph of x y 2 x 4 y 5 0 is which of the following?
(A) (B) (C) (D) (E)
12.
A point A circle An ellipse A hyperbola A parabola
If is a positive acute angle and sin 2
1 , then 2
sin cos 2 (A) (B) (C) (D) (E)
1 1.5 2.2 2.5 2.8
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2
2 MATHEMATICS LEVEL 2 TEST - Continued USE THIS SPACE FOR SCRATCHWORK.
13.
In which quadrant is the graph of an ellipse represented by 2 2 the equation x 6x 4 y 16 y 17 0 located?
(A) (B) (C) (D) (E) 14.
I II III IV III and IV
A cannonball is launched from a height of 80 feet. If the height of the cannonball in feet, h , is defined by the 2
equation h(t ) 18t 72t 80 , where t is time in seconds , how long does the rocket remain at or above 134 feet from the ground , in seconds? (A) 1
(B) 1.2
(C) 1.5
(D) 2
(E) 2.3 5
15.
1 What is the fourth term in the expansion of x 2 2 ? x (A) 5x 4 (B) 10x 2 (C) 15x 3 (D) 10x 2 (E) 15x 3
16.
What is the solution of the equation 2log 9 (5 x) 3 ? (A) (B) (C) (D) (E)
17.
3 5 5.4 6 7.2
Find the sum of the first 30 terms of the recursive sequence defined as a1 3 and an an 1 5 . (A) (B) (C) (D) (E)
148 1680 2265 2340 3120 GO ON TO THE NEXT PAGE
388
2
2 MATHEMATICS LEVEL 2 TEST - Continued USE THIS SPACE FOR SCRATCHWORK.
18.
If two forces of 30 newton and 40 newton acting on a body form an angle of 50 , what is the magnitude of the resultant force? (A) (B) (C) (D) (E)
19.
2
2
If a b (a b)i 10 5i , where a and b are real numbers, what is the value of a ? (A) (B) (C) (D) (E)
20.
50.25 newton 54.78 newton 63.58 newton 76.45 newton 81.68 newton
2 2.5 3 3.5 4
1 lim 1 n n
2n
(A) 1 (B) e (C) e 2 (D) e 2 (E) 0
21.
What is the period of the function defined by x f ( x) 5 2cos 2 ? 3
(A) (B) (C) (D) (E)
1 2 3 6 8
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2
2 MATHEMATICS LEVEL 2 TEST - Continued USE THIS SPACE FOR SCRATCHWORK.
22.
Which of the following is not an odd function? (A) y sin 2 x (B) y x5 3 x 3 (C) y x3 2 x 1
(D) y x 3 (E) y
23.
1 1 x
What is the value of sin Arc tan a , where a 0 ?
(A) (B) (C) (D)
1 a2 a a2 1 a a
1 a2 a
1 a2 a (E) 1 a2
24.
x 1 1 If f ( x) e , what is f ( x) ?
(A) ln ( x 1) (B) ln ( x 1)
x (C) ln 2 x (D) ln e (E) ln ex
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390
2
2 MATHEMATICS LEVEL 2 TEST - Continued USE THIS SPACE FOR SCRATCHWORK.
25.
f If f ( x) 2 x 1 and g ( x) x 3 , then (5) g (A) (B) (C) (D) (E)
26.
5.5 10 17 22 25
What is the domain of the function f ( x) 8 2 x x 2 ? (A) , 4 (B) , 5 (C) 4, 2 (D) 1, 4 (E) All real
27.
What is the sum of the infinite geometric series 0.5 0.25 0.125 ? (A) (B) (C) (D) (E)
28.
0 1 2 3 4
Six students are to be seated in a row of 6 chairs. If three of these students must be seated together, how many ways could this be accomplished? (A) (B) (C) (D) (E)
24 48 120 144 210
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2
2 MATHEMATICS LEVEL 2 TEST - Continued USE THIS SPACE FOR SCRATCHWORK.
29.
Suppose the graph of f ( x) ( x 1) 2 2 is translated 2 units left and 3 units up. if the resulting graph represents g ( x), what is the value of g (2.5)? (A) (B) (C) (D) (E)
30.
29.25 14.62 1.25 14.65 18.75
If sin 2 x sin x cos 2 x , then what is the smallest positive value of x ?
(A) (B)
3
2 7 (C) 6 4 (D) 3 11 (E) 6
31.
2 If f ( x) 2 x x 2 and g ( x) x 2 , then what is the
domain of ( g f )( x) ? (A) (B) (C) (D)
x0 x0 0 x2 x 2 and x 0
(E) All real x
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392
2
2 MATHEMATICS LEVEL 2 TEST - Continued USE THIS SPACE FOR SCRATCHWORK.
32.
If the function f ( x) is continuous, what is the value of k ? x2 4 f ( x) x 2 k
(A) (B) (C) (D) (E)
33.
if x 2
1 2 3 4 Undefined
A radioactive nuclide has a half-life of 10 days. At what rate does the substance decay each day? (A) (B) (C) (D) (E)
34.
if x 2
5% 6.7% 8.5% 10% 12.5%
The graph of f ( x) x 5 2 is translated 6 units left and 3 units down. If the resulting graph is g ( x) , then g (2) is (A) (B) (C) (D) (E)
35.
0 1 2 3 4
1 cot 2 2 sin
(A) (B) (C) (D) (E)
1 sin cos tan cot
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2
2 MATHEMATICS LEVEL 2 TEST - Continued USE THIS SPACE FOR SCRATCHWORK.
36.
37.
If
x 10 a b , then a 2 x 4 x2 x2
(A) (B) (C) (D) (E)
3 2 1 1 2
If a die is rolled three times, what is the probability that all three numbers are different? (A) (B) (C) (D) (E)
1 3 1 2 4 9 5 9 2 3 y
38.
m
In Figure 3, if line m is tangent to the circle at point P , which of the following is the equation of line m ? (A) x 2 y 26
P(8, 9) 5
(B) 2 x 2 y 34 (C) 3x 4 y 60 (D) 4 x 3 y 59 (E) 5 x 2 y 58
O
x
5
Note:Figure not drawn to scale. Figure 3
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394
2
2 MATHEMATICS LEVEL 2 TEST - Continued USE THIS SPACE FOR SCRATCHWORK.
39.
The graph in Figure 4 shows a portion of a function. Which of the following could be the function of f ? 3 2 I. f ( x) x ax bx 3
y
5 4 3 2 II. f ( x) x ax bx cx dx 5
III. f ( x) x7 ax6 cx5 dx4 cx3 dx2 ex 5 (A) (B) (C) (D) (E)
I only II only I and II only II and III only I, II, and III
x
O
Figure 4 40.
5log x If f ( x) 2 2 , then what is the smallest integral value of
x such that f ( x) 100 ? (A) (B) (C) (D) (E)
41.
2 3 50 100 1000
If log
3
k log3 2 log3 (k 4) , then what is the value of
k?
(A) (B) (C) (D) (E)
42.
2 or 4 2 or 5 4 only 4 or 8 Undefined
If triangle ABC with dimensions 10 and 16 is rotated about
A
16
AB , what is the surface area of the resulting solid? (A) (B) (C) (D) (E)
314.2 592.8 634.7 906.9 5026.6
B
C 10 Note: Figure not drawn to scale. Figure 5
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2
2 MATHEMATICS LEVEL 2 TEST - Continued USE THIS SPACE FOR SCRATCHWORK.
43.
What is the measure of the angle between a (1, 3) and b (2, 1) ?
(A) 22.5 (B) 30 (C) 45 (D) 60 (E) 75 44.
Which of the following is the graph of the polar equation r 2csc ? (A) (B) (C) (D) (E)
45.
A point A line A circle A parabola An ellipse
If sin sin 2 ?
(A) (B) (C) (D) (E) 46.
0.71 0.50 0.31 0.31 0.71
Figure 6 shows an isosceles trapezoid. What is the length of its diagonal? (A) (B) (C) (D) (E)
47.
5 3 and , then what is the value of 2 13
3 5
6.5 7.4 8.1 8.7 9.3
5 10 Figure 6
If 5 6i a 2i , then a could be (A) 7.9
(B) 8.6
(C) 11.3
(D) 8.1 (E) 7.5
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396
2
2 MATHEMATICS LEVEL 2 TEST - Continued USE THIS SPACE FOR SCRATCHWORK.
48.
1 , then what is 3 the probability of at most 2 days of rain during the next 4
If the probability of rain on any given day is
days? (A) (B) (C) (D) (E) 49.
0.5 0.54 0.71 0.89 0.95
If the equation of a parabola is given by y 1
1 x 2 2 , 4
then what is its focus? (A) (0, 2) (B) (1, 2) (C) (2, 2) (D) (2, 3) (E) (2, 2) 50.
If a hyperbola has the equation y 2 25 x 2 25 , what are the equations of its asymptotes? 1 x 5 2 (B) y x 5
(A) y
(C) y 5 x (D) y 10 x (E) y 25 x
STOP IF YOU FINISH BEFORE TIME IS CALLED, YOU MAY CHECK YOUR WORK ON THIS TEST ONLY. DO NOT TURN TO ANY OTHER TEST IN THIS BOOK.
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No Test Material on This Page
398
2
2
TEST 12 # 1 2 3 4 5 6 7 8 9 10
answer D E B B D A D B A E
ANSWERS # 11 12 13 14 15 16 17 18 19 20
answer A B D D B C C C D C
# 21 22 23 24 25 26 27 28 29 30
answer C E C D A C B D C B
# 31 32 33 34 35 36 37 38 39 40
answer C D B A A E D C D B
# 41 42 43 44 45 46 47 48 49 50
answer C D C B A B E D C C
Explanations: Test 12 1. (D)
3x
a b
81x12 3b x ab 34 x12
Therefore, b 4 , ab 12 , and a 3. The answer is a b 7 . 2. (E)
3a b 2a b a 2b a b 2b b 3b 3 a b 2b b b
3. (B)
2 x 1 8 2 x 1 23 x 2 23 x 2 2 x 2 6 2 4 25 23 32 8 40 2 2
4. (B)
5. (D)
x 2x 1 4 1(4) 2 4 The smallest integer is 0.
x 2(2 x 1) 4 3x 2 x
2 3
V (6) 210 (2) 210 320 m
6
2
Dr. John Chung's SAT II Math Level 2 Test 12
10
399
2 6. (A)
2 log a b
log b log a log16 log 9 4log 2 2log 3 8 log 3 log 2 log 3 log 2
log3 16 log 2 9 7. (D)
Method 1: 1 3 30 sin 2 sin 60 0.8660254038 0.87 sin 2 2 Method 2: 3 In Quadrant I : cos 2 3 1 3 sin 2 2sin cos 2 2 2 2
8. (B)
Period of tan 2 is
2
.
Therefore, the asymptotes are at x
9. (A)
Area of a triangle
80o
10. (E)
30o
3 5 , , 4 4
The law of sines: sin 80 sin 30 10sin 80 b b 10 sin 30
10
1 10sin 80 Area 10 sin 70 92.54165784 92.5 2 sin 30
70o C
b
Point of symmetry:
4
,
ab sin C : 2 B
A
y
(0, 3) O
x
2
11. (A)
x 2 y 2 2 x 4 y 5 0 ( x 1) 2 ( y 2) 2 0 x 1 and y 2. Since the length of the radius is zero, it represents a point (1, 2) .
12. (B)
sin cos 2 1 2sin cos 1 sin 2 1 sin 2 1
400
1 3 2 2
2 13. (D)
2 x 2 6 x 4 y 2 16 y 17 0
x 2 6 x 4( y 2 4 y ) 17 ( x 3) 2 ( y 2) 2 1 8 2
( x 3)2 4( y 2) 2 8
a 2 8 a 2 2 , b 2 2 b 2 , and center at (3, 2) . The graph is as follows. y
3
x
O
2
14. (D)
2.82
1.41
h(t ) 18t 2 72t 80 134 18t 2 72t 54 0 t 2 4t 3 0 (t 3)(t 1) 0 t 3, 1 Therefore, 3 1 2 seconds.
feet
h
1
3
y 134
80 O
seconds
5
t
x
15. (B)
2 1 x 2 x
16. (C)
2 log 9 (5 x) 3 log 9 (5 x)
The fourth term = 5 C3 x 2
2
2 3
3
x 17. (C)
27 5
10 x 4 x 6 10x 2
3 5 x 9 2 5 x 32 2
3 2
27
(Or, use a calculator: 9 (1.5) 2 )
an an 1 5 an an 1 5 : Common difference is 5 and the first term is 3. a30 a1 (n 1)d 3 29 5 148 Sum of the arithmetic sequence: S30
Dr. John Chung's SAT II Math Level 2 Test 12
30(3 148) 2265 2
401
2 18. (C)
2 The law of cosines: The magnitude of the resultant 302 402 2(30)(40) cos130 63.58215365 63.58
30
R 50
o
130 o
30
40
19. (D)
a 2 b 2 (a b)i 10 5i a 2 b 2 10, a b 5 (a b)(a b) 10 and a b 5 a b 2 . Therefore, a 3.5. n
20. (C)
1 Since lim 1 e , n n
1 lim 1 n n 21. (C)
22. (E)
23. (C)
2n
1 lim 1 n n
Since the period of cos 3
2n
e2
2 6 x is 6 , the period of cos 2 x is 3 . 3 2 3
1 1 1 1 and f ( x) 1 1 . x x x Since f ( x) f ( x) , f ( x) is not an odd function. f ( x)
y sin Arc tan a : Let X Arctan a . a tan X and y sin X
sin X
1
X
a
a 1 a2
1 a2
24. (D)
y e x 1 f 1 ( x) : x e y 1 y ln( x) 1 y ln
25. (A)
f f (5) 11 Quotient of functions: (5) 5.5 g (5) 2 g
402
x e
2
2
26. (C)
Domain: 8 2 x x 2 0 x 2 2 x 8 0 ( x 4)( x 2) 0 Therefore, the domain is 4 x 2 .
27. (B)
The first term is 0.5 and the common ratio is 0.5. 0.5 a S 1 1 r 1 0.5 Permutation:
28. (D)
4
P4 4!
3 P3 3!
Therefore, 3! 4! 144 . 29. (C)
g ( x) ( x 2) 1 2 3 ( x 1) 2 1 g (2.5) (2.5 1) 2 1 1.25
30. (B)
sin 2 x sin x cos 2 x sin 2 x sin x 1 sin 2 x 2sin 2 x sin x 1 0 (2sin x 1)(sin x 1) 0 1 Therefore, sin x or sin x 1 . The smallest positive value of x is . 2 2
31. (C)
g f ( x) x 2 2 x 2
2
The domain of ( g f )( x) is 0 x 2 . From above, it might appear that the domain of the composition is all real numbers. However, this is not true because the domain of f is 0 x 2 , the domain of g f is 0 x 2 . 32. (D)
x2 4 ( x 2)( x 2) ( x 2) lim lim 4 x2 x 2 x2 x2 ( x 2) 1 In order to be continuous, k should be equal to 4.
lim
t
33. (B)
1 10 Radioactive decay: P P 2 1
1 10 The rate each day: P P 0.9330329915 P 2 Therefore, 1 0.9330329915 0.0669670085 6.7% decay.
34. (A)
Shift 6 left and down 3 f ( x) x 5 2 g ( x ) x 6 5 2 3 g ( x) x 1 1
g (2) 2 1 1 0
35. (A)
1 1 cos 2 1 cos 2 sin 2 2 cot 1 sin 2 sin 2 sin 2 sin 2 sin 2
Dr. John Chung's SAT II Math Level 2 Test 12
403
2 36. (E)
2 x 10 a b x 10 a( x 2) b( x 2) 2 x 2 x 2 x 4 x2 x2 x2 4 x 10 (a b) x 2a 2b x 2 x 2 x2 4 Therefore, a b 1 and 2a 2b 10 . a 2
37. (D)
The number of all possible outcomes (sample space) is 6 6 6 216 . If the first roll is a 1, the second and third time cannot rolls cannot be 1. If the second roll is a 2, the third cannot be a 2. Therefore, 6 5 4 120 . 120 5 . Probability P 216 9
38. (C)
The tangent is perpendicular to the diameter of the circle. y
95 4 85 3 3 The slope of line m : 4 The equation of line m is 3 x y 9 x 8 4 or 3 x 4 y 60
m
The slope of line :
P(8, 9) (5,5)
5 O
39. (D)
5
The function has one positive real root, a positive y-intercept, rises to the left, and falls to the right. I. f ( x) x3 ax 2 bx 3 Negative y-intercept II. f ( x) x5 ax 4 bx3 cx 2 dx 5 7
6
5
4
3
2
Four imaginary roots and one real root
III. f ( x) x ax cx dx cx dx ex 5
Six imaginary roots and one real
root 40. (B)
5
f ( x) 25log2 x 100 2log2 x x5 1
x 5 100 x 100 5 2.5118864 The smallest integer value of x is 3.
41. (C)
log a b log a n b n : log
3
k log 3 2 log 3 (k 4) log 3 k 2 log 3 2 log 3 (k 4)
log3 k 2 log3 2k 8 k 2 2k 8 k 2 2k 8 0
Therefore, k 4 k 2 0 . k 4 or k 2. From log
404
3
k , k cannot be less than or equal to 0. The value of k is only 4.
2 42. (D)
2 Lateral surface area is rs . s 356
The Surface area: rs r 2
16
(10)( 356) (10)2 906.9137817 906.9
43. (C)
3 1 tan 1 3 1 1 1 2 tan 1 tan 2 2 2 1 1 2 tan 1 3 tan 1 45 2 tan 1
2 sin
44. (B)
r 2 csc
45. (A)
is in Quadrant III.
r
2 y r
r 10
(1,3)
1
(1, 2)
2
O
y2
5 13 12 cos 13
sin 12 5
13
5 12 120 0.7100591716 0.71 sin 2 2sin cos 2 13 13 169 46. (B)
From the figure below: 3 5
A
C 52 3.52
5 6.5
3.5
The length of the diagonal AC 6.52 52 3.52 7.416198487 7.4
Dr. John Chung's SAT II Math Level 2 Test 12
405
2 47. (E)
2 5 6i 52 (6) 2 61 a 2i a 2 4 a 2 4 61 a 2 4 61 a 2 57 a 7.549834435 7.5
48. (D)
At most two rainy days: 0 days, I day, and 2 days 0
4
1
3
2
2
1 2 1 2 1 2 P 4 C0 4 C1 4 C2 0.88888 3 3 3 3 3 3 Or 1 binomcdf 4, , 2 0.888888 0.89 3 49. (C)
1 1 x 2 2 y x 2 2 1 p 1 4 4p Vertex 2,1 , p 1 and directrix y 0 . y 1
The coordinates of the focus is (2, 1 1) 2, 2 . y
(2, 2)
x
50. (C)
y 2 25 x 2 25
y 2 x2 1 1 52
a 1 and b 5 . 5 b Asymptotes: y x x 5 x 1 a y
y 5 x
y 5x
5
O
5
END
406
x