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English Pages [8] Year 2023
1.
Find the solution to (a)
2.
If
Ans: A 3.
1 3 1 2 i (b) i 2 2 2 2
Ans: A
Attempt all questions (3hrs) (c)
2 1 i 2 2
(d)
1 3 i 2 2
5 2 2x 4 y x y find x and y respectively 1 x y y x 1 3 1 1 2 2 1 (a) , (b) , (c) , 2 2 2 3 3 3
(d)
1 1 , 3 2
Let z1 and z2 be two complex numbers such that z1 iz2 0 and arg( z1 z2 ) . Then find arg( z1 )
(a)
Ans: A 4.
x x 1 0 2
3 4
(b)
(c)
(d)
2
2
Find the solution to the system of linear equations below
x1 x2 x3 10 x2 x3 3
2 x1 2 x2 x3 5 (a) ( 7,12, 15) 5.
Ans: D
(b) (7, 12, 15)
(c) (7,12, 15)
(d)
Let z1 and z2 be two complex numbers such that z1 z2 z1 z2 . Find arg( z1 ) arg( z2 )
(a)
(b)
(c) 0
(d)
Ans: C
4 0 1 6. Find the eigenvalues of 2 1 0 2 0 1 a. 1,1, 2 (b). 1, 2, 2 (c). 1, 2, 3 Ans: C 7.
8.
(7, 12,15)
Solve the equation 2 z 2 2iz 5 0, z
(d).
2
1, 1, 2
3 1 3 1 1 3 1 3 z i (b) z i (c) z i (d ) z i 2 2 2 2 2 2 2 2 Ans: A 4i z , . Given that z is a real number, find the possible values of . 1 i 1 a. 1 (b) 2 (c) 4 (d) 4
Ans: B
a.
1 2 0 1 9. Given the matrix A 2 4 1 4 , Determine the number of independent columns in the matrix 3 6 3 9 Ans: A
(a). 2
(b) 1
(c) 3
x y 10. Determine the value of 5 x 4 y a.
x x 4x 2x
10 x 8 y 8 x
0
(b)
x3
(d) 4
(c)
3x
y3
(d) x
Ans: B 11. For rank of a matrix: I. It is the number of linearly independent matrix rows. II. It is the number of linearly dependent matrix rows. III. Perform row operations and choose the number of nonzero rows. (a). I only (b). II only (c). III only (d). I and III only Ans. D
9 40i . (b). 5 4i and -5-4i (c). 3 4i and 3-4i
12. Find the square root of the complex number (a). 5 4i Answer B
and 5-4i
(d).
3 4i and 3+4i
13. Find the value of sin i cos sin i cos . 8 8 8 8 8
(a). -1 Answer A
14. Given that
2 9 (a). 1 6
Answer A
(b). 0
(c). 1
8
(d). 2i
3 2 A . Find the inverse matrix of A. 3 4 1 2 1 2 9 9 9 9 (b). (c). 1 1 1 1 6 6 6 6
1 9 2 6
1 cos 2 i sin 2 1 cos 2 i sin 2 (a). cos 60 i sin 60 (b). cos 60 i sin 60
15. Simplify
1 9 2 6
30
(c). -2i
Answer B
16. Find all the eigenvalues and associated eigenvectors of the matrix (a).
2 9 D. 1 6
1 1 1 3, 1 and 2 2, 2 1 1 4
(b).
1 4 A . 1 2
(c). 2i
1
1
1 3, 1 and 2 2, 2 1 1 4
1 1 (c). 1 3, 1 and 2 2, 2 1 1 4
D. None
Answer C
1 0 0 17. Given that matrix P 0 1 0 0 0 1 I. Matrix P is a row Echelon form. II. Matrix P is a reduced row Echelon form. III. Matrix P is a Gauss-Jordan method. IV. Matrix P is a Gaussian elimination method.
(a). P is I only. (b). P is II only. (c). P is I and III only. Answer B
(d). P is II and IV only.
5 2 1 0 18. Given that M 1 1 1 0 .The solution to matrix M is: 4 2 3 0 (a). Homogeneous Answer B
(b). Consistent
(c). Inconsistent (d). None
2 4 . If AB 12 , find x and y. 6 y (b). x 2 and y 6 (c). x 2 and y 6 (d). x 2 and y 6.
1 x 3 19. Given that A and B = 2 1 1 (a). x 2 and y 6 Answer D
2 1 3 20. Given that A 1 2 0 . Find the adj A. 3 2 1 2 7 6 2 7 6 7 3 3 (a). 1 (b). 1 7 8 7 3 8 7 3 Answer C
2 7 6 (c). 1 7 3 8 7 3
21. Write 3i into polar form
3 3 i sin (b) 3 cos i sin 2 2 2 2
(a) 3 cos
5 4 7 and AB B 2 2 3 1 3 11 a. (b). 2 7 27
22. Suppose
Ans: D
3 . Find A 1 17 41
(c).
D. None
i sin (d) 2 2
(c) 3 cos
13 43 4 18
(d).
3 3 i sin 2 2
3 cos
3 13 8 27
4 2 3 A . A is equal to 1 1 (a). 5 A 6 I (b). 19 A 30 I
23. Let
(c) 2 A 3 I Ans: B 24. The relative position of Michael to Tom, in meters is 10 i What is the relative position of Anita to Tom? (a).
4 i 19 j
(b).
4 i 19 j
Ans: C
25. A constant force of magnitude 6N in a direction
(d). 3 A 2 I
15 j . The relative position of Anita to Michael is 6 i 4 j .
(c). 16 i
11 j (d) None
i 2 j k , displaces a particle from the point (1,0,1) to the point (3,4,-
1). Calculate the amount of work done by the force. (a). 10 Ans: B
6J
(b).
8 6J
(c). 8J
26. Find the cross product of u i j and (a). 1
Ans: C
(b). 4
(c).
3 i 3 j 3k
(d) 48J
v 2 i j 3k (d).
3 i 3k
27. Determine the area of a parallelogram ABCD whose position vectors are
a 2i j
(a)
26
Ans: A
,
b i 2 j k c 3 i 4 j 2k d 4 i 3 j k (b)
1
(c).
,
3 (d). 3 3
28. Determine the angle between the vectors
,
u i 2 j k and v 2 i 3 j k
(a). 49.8 (b) 40.2 (c) 0 (d) None Ans: B 29. If Rank ( A) 2 and Rank ( B) 3 then the Rank Rank (a). 6 Ans: D
30. If
(c). 3
(b). 5
8 5 121 120 A then the value of A A 7 6
(a). 120
(b). 1
(c). 0
Ans: C 31. The condition for which the eigenvalues of the matrix (a).
k
Ans: B
1 2
(b).
0
Ans: D
1 2
(c). k 2
(d). 121
2 1 A are positive is 1 k
(d) k 0
, do the simultaneous equation 2 x 3 y 1 , 4 x 6 y (b). 1 (c). 2 (d). 2
32. For what values of (a).
k
( AB) is: (d). 2
have infinite solutions
1 3 0 2 6 0 33. Given that the determinant of the matrix A 2 6 4 is -12, determine the determinant of B 4 12 8 1 0 2 2 0 4 (a). -96
(b). -24
Ans: A
(c). 24
34. The lowest eigen value of the matrix (a). 1 (b). 2 Ans: B
(c). 5
(d). 3
(a). m n
(b). n n
(d). 96
4 2 1 3 is
35. If A is m n such that AB and BA both are defined, then B is a matrix of order Ans: D
(c). m m
(d). n m
2 3 A if the eigenvalues of A are 4 and 8, then x y (a). x 4, y 10 (b). x 4, y 10 (c). x 5, y 8 (d). x 3, y 9
36. Consider the matrix
Ans: B
37. A set of linear equations is represented by the matrix equation Ax b . The necessary condition for the existence of solution for the system is (a). A must be invertible (b) b must be linearly depended on the columns of A (c). b must be linearly independent of the columns of A (d). None 38. If A and B are square matrices of size n n , then which of the following statement is not true? (a). AB A B
(b). kn k
n
(c). A B A B
A
(d).
AT
Ans: C 39. The expression of complex number a.
sin 1 i 2 1 cos 2
b.
Ans: C 40.
If a.
a.
1 sin i 2 2 1 cos
1 i x 2i 2 3i y i i then
3,1
Ans: B If z
41.
1 in the form a bi is: 1 cos i sin
3 i
3i
b.
3, 1
c.
x, y
3,1
4 3i 1 then z is: 5 3i
11 27 i 25 25
b.
11 27 i 25 25
c.
c.
d.
11 27 i 25 25
1 1 i tan 2 2 2
3, 1
d.
11 27 i 25 25
d.
1 A1
1 1 tan i 2 2 2
Ans: A 42.
z1 , z2 are two complex numbers such that arg z1 z2 0 and Im z1 z2 0 , then
If
z1 z2
a.
Ans: C
43.
Let
z1 z2
c.
z1 z2
d.None of these
6 1 A . Which of the following matrices is similar to A ? 3 4
7 0 0 3
a.
b.
6 0 0 4
b.
Ans: A
c.
1 0 0 3
d. None of the above
1 44. Which of the following is a unit vector in the same direction as v 2 ? 1 6 6 3 6
a.
Ans: B
6 6 b. 6 3 6 6
1 6 c. 1 3 1 6
d.
v is already a unit vector
a b 2 45. If a and b are two vectors such that and a.b 1 , then the angle between a and b is:
3
a.
Ans C
46.
If
a.
c.
2 3
d. None of these
ab
c. a b
d. None of these.
a b . ac If a 2i 3 j k , b i 2 j 4 k and c i j k , then is:
a.
Then
4
, then:
b.
Ans: B
Ans; B 48.
a b a b
a b
a.
47.
b.
74
b. -74
c. 52
d. -52
a i j b 3 i 4 k The vector is to be written as the sum of a vector parallel to and a vector perpendicular to a . is:
3 i j 2
Ans: A
2 i j b. 3
1 i j c. 2
1 i j d. 3
a i j k b i 2 j 2 k c If , and i 2 j k , the unit vector normal to vectors a b and b c is:
49. a.
50.
a.
Ans: C
i
Ans: A If
b.
j
c.
a b 4 a.b 2
6
b. 2
,
then
d. None of those
2 2 a b d. 8
is:
d 3 i j 2 k d i 3 j 4k . The area of the 1 The diagonals of a parallelogram are represented by the vectors and 2
51. parallelogram is: a.
c. 20
k
7 3sq.units
Ans: B
b. 5 3sq.units
c. 3 5 sq.units d. None of these.
52. If arg( z ) 0 , then arg( z ) arg( z ) is (a).
(b).
(c).
Ans: A
(d). 2 2
53. If the cube roots of unity are 1, , 2 then the roots of the equation ( x 1) 3 8 0 are (a). 1, 1 2 , 1 2 2 Ans: C
(b). 1, 1, 1 (c). 1,1 2 ,1 2 2 (d). 1,1 2 ,1 2 2
54. The modulus of 5 4i is (a). 41
(b). 41
55. The numbers
a bi and a bi are said to be?
Ans: C
a.
d. 56.
Factor of each other
Conjugate of each other
57.
b. Additive inverse of each other
c. Multiplicative inverse of each other
z a bi , if i is replaced by i , then another complex number obtained is said to be?
Additive inverse of z Ans: C
z.z b. a 2 b 2
Ans: A
c. complex conjugate of z
b. prime factor of z
The absolute value of the complex number a.
(d). 41
41
Ans: D In
a.
(c).
c.
ab
z a bi is: d.
z. z
d. multiplicative inverse of z
58.
The argument of the complex number a.
59.
450
Ans: C
b. 90
c. 180
0
The 4-th root of 1 are: a.
i, i
Ans: B
b. 1, 1, i, i
If z x yi and
60. a.
4 10i 3
Ans: A
c. 1, i
(1 i )4 is: d. 135
0
0
d. 1, 1
3x 3x y i 4 6i then z
b.
4 10i 3
c.
4 10i 3
d.
4 10i 3