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[ASK/EM-I/MCQ] August 30, 2012

Engineering Mathematics –I

MCQ for Phase-I

UNIT: I – MATRICES

1. The rank of the matrix A =

(a) 4 (b) 2 (c) 0 (d) 1

2. The value of λ for which the matrix A =

will be of rank one is

(a) λ = -3 (b) λ = 3 (c) λ = 2 (d) λ = -2

3. For what value of k is the rank of matrix equal to 2.

A =

(a)any row number (b) 1 (c) 3 (d) 2

4. A be the matrix of order m × n then the rank of matrix A is

(a) rank A ≤ min. of m and n (b) rank A ≥ min. of m and n

(c) rank A < min. of m and n (d) rank A = min. of m and n

5. A matrix ‘A’ has a nonzero minor r then

(a) ρ(A)< r (b) ρ(A)≤r (c) ρ(A)> r (d) ρ(A) ≥ r

6. If A is a m×n matrix of rank r such that PAQ =

, then

(a) P & Q are singular matrices

(b) P is non-singular & Q is singular matrix

(c) both P & Q must be a non-singular matrices

(d) P-1 = Q

7. The system of equations is said to be consistent if

(i) ρ(A) = ρ(A:B) = n (ii) ρ(A:B) < n

(a) Both (i) & (ii) are true (b) only (i) is true

(c) (i) is true but (ii) is false (d) (i) is false but (ii) is true


8. The condition on λ for which the system of equations

, has a unique solution is

(a) λ ≠ -5 (b) λ ≠ 5 (c) λ = 5 (d) λ = -5

9. The eigen values of the matrix A =

is

(a) a, b, g (b) a, b, 0 (c) a, 0, 0 (d)a, b, c

10. If λ is an eigen value of matrix A, then the eigen value of A-1 is

(a)

(b) λ (c) λ-1 (d) λ+1

11. If A = [ aij] is a square matrix of order n , then trace of A is

(a) product of diagonal elements (b) sum of diagonal elements

(c) sum of row elements (d) sum of column elements

12. The characteristic equation of matrix A =

is

(a) λ3 + 6 λ2 - 9 λ + 4=0 (b) λ3 - 6 λ2 - 9 λ - 4=0

(c) λ3 - 6 λ2 + 9 λ - 4=0 (d) λ3 + 6 λ2 - 12 λ - 4=0

13. If A =

then the eigen value of A3+5A+8I are

(a) -1, 27, -8 (b) -1, 3, -2 (c) 2, 50, -10 (d)2,50,10

14. The matrix A is defined by A =

. The eigen values of A2 are

(a) -9,-4,-1 (b) -1,-3,2 (c) 1,3,-2 (d)1,9,4.

15. The vectors (1,-1,1), (2,1,1), & (3,0,2)

(a) Linearly independent (b) Linearly dependent

(c) only two are dependent (d) only two are independent


16. Normal form of the matrix A =

is

(a) [ I1,o] (b) [I3] (c)[I1] (d)

17. For a non-singular matrix A, there exist two non-singular matrices P &

Q such that PAQ is in normal form, then A-1 is equal to

(a) PQ (b)P-1Q-1 (c) QP (d) Q-1P-1

18. Homogeneous system of linear equations

(a) is always consistent (b) is always inconsistent

(c) has always infinite solution (d) has no solution

19. Given system of linear equations 3x+2y+z = 0, x+4y+z = 0, 2x+y+4z = 0

has (a) no solution (b) only trivial solution

(b) Infinite solution (d) None of these

20. Given system of linear equations x + y + z =1, x+2y+4z = 2, x+4y+10z=4

(a) no solution (b) unique solution

(c) Infinitely many solution (d) n-r solutions

21. For what value of λ, the system of linear equations x + y + z = 6,

X+2y+3z = 10, x+2y+λz = 10 has infinitely many solutions?

(a) λ = 1 (b) λ = 3 (c) λ = -3 (d) λ = 10

22. For what value of b the matrix A =

is an orthogonal?

(a) ± 5 (b) ± 13 (c) ± 12 (d) ± 16

23. If the characteristic of the matrix A of order 3×3 is λ3 -3 λ2 +3 λ – 1 = 0

Then by Caley Hamilton theorem A-1 is equal to

(a)A3 - 3A2 + 3A – I (b) A2 - 3A -3 I

(c) 3A2 - 3A – I (d) A2 - 3A + 3 I


24. For the linear transformation Y = AX if = 0 then the transformation

is (a) non-singular (b) orthogonal (c) singular d) none of these

25. For the linear transformation Y = AX if ≠ 0 then the transformation

is (a) non-singular (b) orthogonal (c) singular d) none of these

26. For an orthogonal matrix A =

, A-1 is

(a)

(b)

(c)

(d)

27. For the transformation

=

coordinates (y1, y2, y3)

in Y corresponding to (-1,3, 0) in X are

(a) (-1,-2,-1) (b) (1,2,-1) (c) (1,1,1) (d) (-1,-2,1)

28. If λ1, λ2, λ3 are eigen values of the matrix A of order 3 then eigen values

of matrix Am are

(a)

,

,

(b)

(b) (d)

29. If the characteristic equation of the matrix A is λ3 -6 λ2 +11λ – 6 = 0

Then the eigen values of the matrix A are

(A) 1,2,3 (b) -1, -2,-3 (c) 1,2,-3 (d) none of these

30. If the characteristic equation of the matrix A of order 3×3 is

λ3 -5 λ2 +9λ – 1 = 0 then by Caley Hamilton theorem

(a) A3 - 5A2 - 9A – I = 0 (b) A3 + 5A2 + 9A + I = 0

(c) A3 - 5A2 + 9A – I = 0 (d) 5A2 - 9A – I = 0


31. using Caley Hamilton theorem, A-1 for the matrix A

A =

is calculated from

(a)

(-A-4I) (b)

(A-4I) (c)

(A+4I) (d)

(4I-A)

32. A linear transformation AX = Y is regular if

(a) = 0 (b) 0 (c) A is singular (d) none of these

33. If the form of augmented matrix is [A:B] =

1 2 3: 1

0 0 5 : 2

0 0 6 : 4

then the

corresponding linear has

(a) unique solution (b) infinitely many solutions

(c) no solution (d) none of these

34. The column vectors of the identity matrix are

(a) Linearly dependent (b) Linearly independent

(c) Both (a) & (b) (c) none of these

35. A homogeneous system of linear equations AX = 0 in n variables has

Non-trivial solution if rank A is

(a) n (b) > n (c) < n (d) = n

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UNIT: II – COMPLEX NUMBERS

1. If then imaginary part of z is

(a) 2

z z (b)

2

z z

i

(c) zz (d)

2

z z

2. If 1z i then its exponential form is

(a) 42i

e

(b) 42i

e

(c) 42i

e

(d) 42i

e

3. The value of 100i is

(a) 1 (b) -1 (c) i (d) i

4. If , 0, 0z x iy wherex y then arg z is

(a) 1tany

x

(b) 1tany

x (c) 1tan

y

x

(d) 0

5. Which of the following is correct?

(a) The area of parallelogram having sides z1 and z2 is 1 2z z

(b) Arg (0) is not defined

(c) 1 2z z or

2 1z z has no meaning in c

(d) 1 2 1 2z z z z

(e) All are correct.

6. Which of the following is the locus of 2 3 1z i

2 2

2 2

2 2

2 2

( ) 1

( )( 2) 1

( )( 2) ( 3) 1

( ) 1

a x y

b x y

c x y

d x y

7. If cos sinx i , then the value of 100

100

1x

x is

(a) 2cos100

(b) 2cos100

(c) cos

(d) 1

cos2

8. The fourth roots of unity are

(a) 1 (b) , 1i (c) 1, 1, -1, -1 (d) 1, -i

9. If then the polar form of z is

(a)

(b)

(c)

(d)


10. The locus of z satisfying is

(a) (b)

(c) (d)


(a) (b)

(c) (d)


(a) interior of (b) interior of

(c) exterior of (d) exterior of

13. All nth roots of unity form a

(a) arithmetic progression (b) geometric progression

(c) mean (d) none of these

14. The sum of all nth roots of unity is

(a) 0 (b) 1

(c) -1 (d) n

15. By rotating the vector 2OA i in anticlockwise through an angle3

, we get

(a) 3 1

1 32 2

i

(b) 3 1

1 32 2

i

(c) 3 1

1 32 2

i

(d) 3 1

1 32 2

i

16. Using DeMoivre’s theorem simplified form of 8 81 (1 )i i is equal to

(a) 82 (b)

42 cos4

(c)

52 (d) 82 cos

8


17. The roots of the equation 3 1 0x are

(a) 2 1 (2 1)

cos sin3 3

k ki

,k = 0,1,2

(b) 2 1 (2 1)

cos sin , 0,1,23 3

k ki k

(c) 2 2

cos sin , 0,1,23 3

k ki k

(d) None of these

18. For any complex numbers 1 2,z z 1

2

argz

z

is equal to

(a) 1 2arg argz z (b) 1 2arg argz z

(c) 1

2

arg

arg

z

z (d) 1 2arg argz z

19. The value of 3

22

i is ---------

(a) 3 1

2 2i (b)

3 1

2 2i

(c) 3 1

2 2i (d) (a) and (b)

20. The value of 2i is --------------

(a) (0,1) (b) (0,-1) c) (-1,0) (d) (1,0)

21. The real part of ze is

(a) cosxe y (b) sinxe y

(c) xe (d) cos y

22. If cos sinie i then, i ie e

(a) 2sin (b) 2 sini (c) 2 cosi (d) 2cos


23. The argument of complex number (1 cos ) sini is -------

(a) 4 2

(b)

2 2

(c) 2

(d)

2 4


1 2 1 2

1 1

2 2

1 2 1 2

1 2 1 2

( ) arg( ) (arg )(arg )

arg( )arg

arg

( )arg( ) arg arg

( )arg( ) arg arg

a z z z z

z zb

z z

c z z z z

d z z z z

25. Which of the following is incorrect?

1 2 1 2

11 2

2

1 2 1 2

( )arg( ) (arg ) (arg )

( )arg arg arg

( )arg( ) arg arg

a z z z z

zb z z

z

c z z z z

(d) All above are incorrect.

26. If

(a)

(b)

(c)

(d) none of these

27. If

(a)

(b)

(c)

(d)


(a) (b)

(c) 11 2

2

zz z

z (d) All above are correct


29. The Value of (sin cos )ni is-----

(a) sin cosn i n (b) cos sinn i n

(c) cos( ) sin( )2 2

n nn i n

(d) sin( ) cos( )

2 2

n nn i n

30. The a ib form for 1

1

i

i

is ----

1

( )2

ia

1( )

2

ib

1

( )2

ic

1

( )2

id

31. The exponential form for 1z i is ---

(a) 42i

e

(b) 42i

e

(c) 2 (d) 62i

e

32. If 1 21 , 3z i z z i which of the following is correct.

(a) 1 2z z (b)

1 2z z

(c) 1 2z z (d) 1 2z z

33. 2 3z then locus of z is

(a) circle (b) straight line

(c) ellipse (d) parabola

34. If 2(5 3 )iz e then real part of z is

(a) 15 cos30e (b)

16 cos30e

(c) 16 cos30e (d) none of these

35. The complex conjugate of 5 2z i is ----

(a) -5-2i (b) 5-2i (c) -5+2i (d) none of these


37. Using DeMoivres theorem, simplified form of 8 8(1 ) (1 )i i is equal to

(a) 82 (b)

42 cos4

(c)

52 (d) 82 cos

8

38. The roots of the equation 3 1 0x are ----

(a) (2 1) (2 1)

cos sin , 0,1,23 3

k ki k

(b) (2 1) (2 1)

cos sin , 0,1,23 3

k ki k

(c) 2 2

cos sin , 0,1,23 3

k ki k

(d) none of these

39. The hyperbolic sinh( )x is defined as -----

(a) 2

x xe e (b)

2

x xe e

i

(c)

2

x xe e (d) none of these

40. The hyperbolic is ------

(a) x x

x x

e e

e e

(b)

2 2

2 2

x x

x x

e e

e e

(c)

1x xe e

(d) 2 2

2

x xe e

41. The hyperbolic is -------

(a) 2

x xe e (b)

2

x xe e (c)

2

x xe e

i

(d)

2

x xe e

42. Which of the following is correct ?

(a) 1 2sinh ( ) log( 1)x x x (b) 1 1 2sinh ( ) cosh ( 1)x x

(c) 12

i

(d) 1 2cosh ( ) log( 1)x x x

43. The imaginary part of cosh( )x iy is ------

(a) sin sinh( )x y (b) sin sinh( )x y (c) sinh( )sinx y (d) sinh( )sinh( )x y

44. Which of the following is the period of coth( )x ?

(a) 2 i (b) 2 (c) (d) i


45. The principal logarithm of log( )i is ----

(a) 12

i

(b) 12

i

(c) 2

i

(d) 2

i

46. If log log5z i then z -------

(a) 5 (b) log 5 (c) 5 (d) 5

47. Which of the following is true ?

(a) sin( ) sin cosh cos sinhx iy x y i x y (b) sin( ) sin cosh cos sinhx iy i x y x y

(c) sin( ) sin cosh cos sinhx iy x y i x y (d) sin( ) sin cosh cos sinhx iy i x y x y

48. If sin( )i x iy then 2 2

2 2cosh sinh

x y

is equal to

(a) 1 (b) 0 (c) -1 (d)

49. The value of ‘a’ in the result of 1 3cos

4

ia ib

is ----

(a) 2

(b) (c) (d) none of these

50. The number ii is ----

(a) purely imaginary (b) complex no.

(c) natural no. (d) Rational no.

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