Untitled Document

Question Bank No: 2

1. If w is an imaginary cube root of unity, then $| \begin{matrix} 2 & 2 w & - w^{2} \\ 1 & 1 & 1 \\ 1 & - 1 & 0 \end{matrix} | is$

a)		1
b)		-1
c)		0
d)		none of these

2. $Δ = | \begin{matrix} \frac{1}{a} & 1 & bc \\ \frac{1}{b} & 1 & ca \\ \frac{1}{c} & 1 & ab \end{matrix} | is equal to$

a)		abc
b)		0
c)		$\frac{1}{abc}$
d)		none of these

3. The value of $| \begin{matrix} 3 i & 2 i & 2 i \\ 5 & 4 & - 3 i \\ i & 2 i & 7 \end{matrix} | is$

a)		2i+12
b)		2i-12
c)		-2i-12
d)		-2i+12

4. If $\propto = | \begin{matrix} a - b & b - c & c - a \\ b - c & c - a & a - b \\ c - a & a - b & b - c \end{matrix} | then$

a)		$\propto = 1$
b)		$\propto = 0$
c)		$\propto = - 1$
d)		none of these

5. The value of the determinant $| \begin{matrix} 1 & a & b + c \\ 1 & b & c + a \\ 1 & c & a + b \end{matrix} | is$

a)		a+b+c
b)		0
c)		1
d)		abc

6. If A and B are square matrix of same order, then ${(A + B)}^{2} = A^{2} + 2 AB + B^{2} if$

a)		AB=BA
b)		A=-B
c)		A=A'
d)		2A=-B

7. If a matrix is symmetric as well as skew symmetric then

a)		A is a diagonal matrix
b)		A is a null matrix
c)		A is a unit matrix
d)		A is a triangular matrix

8. If A is square matrix then A-A' is

a)		unit matrix
b)		null matrix
c)		A
d)		a skew symmetric matrix

9. If A is square matrix then A+A' is

a)		unit matrix
b)		null matrix
c)		A
d)		symmetric matrix

10. If A= $[\begin{matrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ a & b & - 1 \end{matrix}] then A^{2} is equal to$

a)		unit matrix
b)		null matrix
c)		A
d)		-A

11. If the matrix AB is zero then

a)		A=0 or B=0
b)		A=0 and B=0
c)		It is not necessary that either A=0 or B should be zero
d)		All the above statements are wrong

12. If A and B are arbitrary square matrices of same order then

a)		${(AB)}^{1} = A^{1} B^{1}$
b)		${(A}^{1})^{1} {(B}^{1})^{1} = B^{1} A^{1}$
c)		${(A + B)}^{1} = A^{1} - B^{1}$
d)		${(AB)}^{1} = B^{1} A^{1}$

13. If A and B are two matrics of the same type, then ${(A + B)}^{1} is equal to$

a)		$A^{1} + B^{1}$
b)		$A^{1} + B$
c)		A+ $B^{1}$
d)		A+B

14. If A= $[\begin{matrix} 1 & 2 \\ 3 & - 5 \end{matrix}] then A^{- 1} is$

a)		$[\begin{matrix} - 5 & - 2 \\ - 3 & - 1 \end{matrix}]$
b)		$[\begin{matrix} \frac{5}{11} & \frac{2}{11} \\ \frac{3}{11} & \frac{- 1}{11} \end{matrix}]$
c)		$[\begin{matrix} \frac{- 5}{11} & \frac{- 2}{11} \\ \frac{- 3}{11} & \frac{1}{11} \end{matrix}]$
d)		$[\begin{matrix} 5 & 2 \\ 3 & - 1 \end{matrix}]$

15. The order of the matrix A is 3 $\times$ 5 and that of B is 2 $\times$ 3. The order of the matrix BA is

a)		2 $\times$ 3
b)		3 $\times$ 2
c)		2 $\times$ 5
d)		5 $\times$ 2

16. If A = $[\begin{matrix} x & 1 \\ 1 & 0 \end{matrix}] and A^{2} = I then X =$

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

17. If for the matrix A, $A^{5} = I Then A^{- 1}$

a)		$A^{2}$
b)		$A^{3}$
c)		A
d)		none of these

18. If A= $[\begin{matrix} 3 & 4 \\ 5 & 7 \end{matrix}] then A (adjA) =$

a)		I
b)		$\| A \|$
c)		$\| A \| I$
d)		none of these

19. If A = $[\begin{matrix} 4 & 2 \\ 1 & 1 \end{matrix}] then (A - 2 I) (A - 3 I) =$

a)		A
b)		I
c)		0
d)		5I

20. If A= $[\begin{matrix} 1 & 0 \\ 0 & 1 \end{matrix}] then A^{4} =$

a)		$[\begin{matrix} 1 & 0 \\ 0 & 1 \end{matrix}]$
b)		$[\begin{matrix} 1 & 1 \\ 0 & 0 \end{matrix}]$
c)		$[\begin{matrix} 0 & 0 \\ 1 & 1 \end{matrix}]$
d)		$[\begin{matrix} 0 & 1 \\ 1 & 0 \end{matrix}]$

21. If D= $| \begin{matrix} 1 & 1 & 1 \\ 1 & 1 + x & 1 \\ 1 & 1 & 1 + y \end{matrix} |$ for xy $\neq 0$ , then D is divisible by

a)		both x and y
b)		x but not y
c)		y but not x
d)		neither x nor y

22. The value of $| \begin{matrix} \cos (α + β) & - \sin (α + β) & \cos 2 β) \\ \sin α & \cos α & \sin β \\ - \cos α & \sin α & \cos β \end{matrix} |$ is independent of

a)		$α$
b)		$β$
c)		$α, β$
d)		none of these

23. If the equations $a (y + z) = x, b (z + x) = y, c (x + y) = z$ have nontrivial solutions, then $\frac{1}{1 + a}$ + $\frac{1}{1 + b}$ + $\frac{1}{1 + c}$ =

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

24. If x= $\frac{a}{b - c}, y = \frac{b}{c - a}, z = \frac{c}{a - b}$ then

a)		xy+yz+zx=0
b)		xy+yz+zx=-1
c)		xy+yz+zx=1
d)		x+y+z=1

25. If A = $[\begin{matrix} 4 & 1 & 0 \\ 1 & - 2 & 2 \end{matrix}], B = [\begin{matrix} 2 & 0 & - 1 \\ 3 & 1 & x \end{matrix}], C = [\begin{matrix} 1 \\ 2 \\ 1 \end{matrix}]$ and D= $[\begin{matrix} 15 + x \\ 1 \end{matrix}]$ such that (2A-3B)C=D, then x=

a)		3
b)		-4
c)		-6
d)		6

26. If A= $[\begin{matrix} 1 & - 1 \\ 2 & - 1 \end{matrix}], B = [\begin{matrix} x & 1 \\ y & - 1 \end{matrix}]$ and ${(A + B)}^{2} = A^{2} + B^{2}$ , then x+y=

a)		2
b)		3
c)		4
d)		5

27. $[\begin{matrix} 1 & x & 1 \end{matrix}] [\begin{matrix} 1 & 3 & 2 \\ 2 & 5 & 1 \\ 15 & 3 & 2 \end{matrix}] [\begin{matrix} 1 \\ 2 \\ x \end{matrix}]$ =0 if x=

a)		-7
b)		-11
c)		-4
d)		-14

28. The homogeneous system of equations $[\begin{matrix} 2 & a + b + c + d & ab + cd \\ a + b + c + d & 2 (a + b) (c + d) & ab (c + d) + cd (a + b) \\ ab + cd & ab (c + d) + cd (a + b) & 2 abcd \end{matrix}]$ $[\begin{matrix} x \\ y \\ z \end{matrix}]$ = 0 has nontrivial solutions only if

a)		a+b+c+d=0
b)		ab+cd=0
c)		ab(c+d)+cd(a+b)=0
d)		for any a,b,c,d

29. If A= $[\begin{matrix} 1 & 2 \\ 3 & 4 \end{matrix}],$ then $A^{4} - 5 A^{3} - A^{2} - 4 A - I =$

a)		0
b)		I
c)		A
d)		A+I

30. If A= $[\begin{matrix} - 4 & - 3 & - 3 \\ 1 & 0 & 1 \\ 4 & 4 & 3 \end{matrix}]$ , then $A^{- 1} =$

a)		$A^{T}$
b)		$A^{3}$
c)		$A^{2} + A - I$
d)		$A^{2} - A - I$

31. If the matrix $[\begin{matrix} 1 & 2 & x \\ 4 & - 1 & 9 \\ 2 & 3 & - 6 \end{matrix}]$ is singular, then x=

a)		-5
b)		-4
c)		$\frac{9}{2}$
d)		$- \frac{9}{2}$

32. If 3A= $[\begin{matrix} 1 & 2 & 2 \\ 2 & 1 & - 2 \\ x & 2 & y \end{matrix}]$ and $A^{T} A = A A^{T} = I,$ then xy=

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

33. If A= $[\begin{matrix} 2 & 1 \\ - 4 & - 2 \end{matrix}],$ then I+2A+3 $A^{2} + ..... \infty$

a)		$[\begin{matrix} 4 & 1 \\ - 4 & 0 \end{matrix}]$
b)		$[\begin{matrix} 3 & 1 \\ - 4 & - 1 \end{matrix}]$
c)		$[\begin{matrix} 5 & 2 \\ - 8 & - 3 \end{matrix}]$
d)		$[\begin{matrix} 5 & 2 \\ - 3 & - 8 \end{matrix}]$

34. If A = $[\begin{matrix} 1 & 2 & - 2 \\ - 2 & 2 & 1 \\ 2 & 1 & 1 \end{matrix}]$ , then $A^{- 1} =$

a)		A
b)		$A^{T}$
c)		$\frac{1}{9} A$
d)		$\frac{1}{9} A^{T}$

35. If A= $[\begin{matrix} - 1 & \frac{3}{2} \\ - \frac{1}{2} & \frac{1}{2} \end{matrix}]$ , then I+A+ $A^{2} + ..... \infty =$

a)		$[\begin{matrix} 1 & - 3 \\ 1 & 4 \end{matrix}]$
b)		$\frac{2}{7} [\begin{matrix} 1 & - 3 \\ - 1 & 4 \end{matrix}]$
c)		$\frac{2}{7} [\begin{matrix} 1 & 3 \\ - 1 & 4 \end{matrix}]$
d)		undefined

36. If $A^{2} - 3 A + 2 I = 0,$ then

a)		A is singular
b)		$A^{- 1} = \frac{3 I + A}{2}$
c)		$A^{- 1} = \frac{I - 3 A}{2}$
d)		$A^{- 1} = \frac{3 I - A}{2}$

37. If A= $[\begin{matrix} - 1 & \frac{3}{2} \\ - \frac{1}{2} & \frac{1}{2} \end{matrix}],$ then $A^{3} =$

a)		$\frac{A}{4}$
b)		$\frac{A}{8}$
c)		$\frac{I}{4}$
d)		$\frac{I}{8}$

38. Let A= $[\begin{matrix} 1 & 2 \\ 3 & 4 \end{matrix}] and B = [\begin{matrix} a & 0 \\ 0 & b \end{matrix}]$ , a, b $\in N .$ Then

a)		there exists exactly one B such that AB=BA
b)		there exists infinitely many B's such that AB=BA
c)		there cannot exist any B such that AB=BA
d)		there exist one than I but finite bumebr of B's such that AB=BA

39. If A and B are 3 $\times 3$ matrices such that $A^{2} - B^{2} = (A - B) (A + B),$ then

a)		either A or B is zero matrix
b)		either A or B is unit matrix
c)		A=B
d)		AB=BA

40. If $a_{1}, a_{2}, a_{3} .......$ are in G.P. then $Δ = | \begin{matrix} \log a_{n} & \log a_{n + 1} & \log a_{n + 2} \\ \log a_{n + 3} & \log a_{n + 4} & \log a_{n + 5} \\ \log a_{n + 6} & \log a_{n + 7} & \log a_{n + 8} \end{matrix} | = 0$

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

41. If A= $[\begin{matrix} 1 & 0 \\ 1 & 1 \end{matrix}]$ , then $A^{n} =$

a)		$2^{n - 1} A - (n - 1) I$
b)		$nA - (n - 1) I$
c)		$2^{n - 1} A + (n - 1) I$
d)		$nA + (n - 1) I$

42. The system of equations $α x + y + z = α - 1$ , $x + α y + z = α - 1$ , $x + y + α z = α - 1$ has no solution if $α$ is

a)		-2 or 1
b)		-2
c)		1
d)		-1

43. If $a^{2} + b^{2} + c^{2} = - 2$ and $f (x) = | \begin{matrix} 1 + a^{2} x & (1 + b^{2}) x & (1 + c^{2}) x \\ (1 + a^{2}) x & 1 + b^{2} x & (1 + c^{2}) x \\ (1 + a^{2}) x & (1 + b^{2}) x & 1 + c^{2} x \end{matrix} |$ , then $f (x)$ is a polynomial of degree

a)		0
b)		1
c)		2
d)		3

44. If $A^{2} - A + I = 0, then A^{- 1}$ =

a)		A
b)		A+I
c)		I-A
d)		A-I

45. If A= $[\begin{matrix} 1 & - 1 & 1 \\ 2 & 1 & - 3 \\ 1 & 1 & 1 \end{matrix}]$ and $A^{3} = \frac{1}{10} [\begin{matrix} 4 & 2 & 2 \\ - 5 & 0 & α \\ 1 & - 2 & 3 \end{matrix}]$ , then $α =$

a)		-2
b)		5
c)		2
d)		-1

46. If A= $[\begin{matrix} 0 & 0 & - 1 \\ 0 & - 1 & 0 \\ - 1 & 0 & 0 \end{matrix}]$ , Then

a)		A is zero matrix
b)		$A^{2} = I$
c)		$A^{- 1}$ does not exist
d)		A=(-1)I

47. If 1, $ω, ω^{2}$ are the cube root of unity. then $Δ = | \begin{matrix} 1 & ω^{n} & ω^{2 n} \\ ω^{n} & ω^{2 n} & 1 \\ ω^{2 n} & 1 & ω^{n} \end{matrix} | =$

a)		0
b)		1
c)		$ω$
d)		$ω^{2}$

48. If A= $[\begin{matrix} a & b \\ b & a \end{matrix}]$ and $A^{2} = [\begin{matrix} α & β \\ β & α \end{matrix}]$ ,then

a)		$α = a^{2} + b^{2}, β = ab$
b)		$α = a^{2} + b^{2}, β = 2 ab$
c)		$α = a^{2} + b^{2}, β = a^{2} - b^{2}$
d)		$α = 2 ab, β = a^{2} + b^{2}$

49. If the system of equations $x + 2 ay + az = 0, x + 3 by + bz = 0, x + 4 cy + cz = 0$ has non zero solution, then a, b, c are in

a)		A.P.
b)		G.P.
c)		H.P.
d)		none of these

50. If $| \begin{matrix} a & a^{2} & 1 + a^{3} \\ b & b^{2} & 1 + b^{3} \\ c & c^{2} & 1 + c^{3} \end{matrix} | = 0$ and the vectors (1, a, $a^{2}), (1, b, b^{2}), (1, c, c^{2})$ are noncoplanar, then abc=

a)		2
b)		-1
c)		1
d)		0