# Linear Algebra

Explore the foundations of Linear Algebra through our comprehensive online workbook. Dive into essential concepts such a

179 88 487KB

English Pages [8] Year 2023

##### Citation preview

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

14. Given that

2 9 (a).  1  6

(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

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

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

3i

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 A1

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

  ab

c. a  b

 

d. None of these.



       a b . ac 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.

ab

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