# Matrices & Determinants MCQs Part II

**1) If the system of equations x + ay + az = 0, bx + y + bz = 0, cx + cy +
z = 0 Where a, b and c are non-zero and non-unity, has a non- trivial solution,
then the value of **

**a)**zero

**b)**1

**c)**-1

**d)**

**2) If
then the value of is **

**a)**1

**b)**2

**c)**4

**d)**3

**3) If in
then
is**

**a)**0

**b)**-D

**c)**D

**d)**

**4) Inverse of is ****a)** **b)** **c)** **d)** none of these

**5) ****a)** 1**b)** 0**c)** 3**d)** a + b+ c

**6) A and B are square matrices of order
is equal to **

**a)**

**b)**

**c)**

**d)**

**7) If
Then
is equal to**

**a)**0

**b)**1

**c)**100

**d)**-100

**8) If ,
then **

**a)**5A

**b)**10A

**c)**16 A

**d)**32 A

**9) If
the value of x which satisfies the equation
is **

**a)**x = a

**b)**x = b

**c)**x = c

**d)**x=0

**10) If is
cube root of unity , then **

**a)**

**b)**

**c)**

**d)**

**11) If the system of equation x + 2 y – 3 z = 2, ( K+3) z = 3, (2K+1)y+z=2 is inconsistent , then K is****a)** -2**b)** **c)** 1**d)** 2

**12) If the equations x- Ky – z = 0, Kx- y – z = 0, x+ y – z = 0 has a non zero solution, then the possible values of K are****a)** -1, 2**b)** 1, 2**c)** 0, 1**d)** -1, 1

**13) The parameter on which the values of the determinantdoes not depend upon****a)** a**b)** p**c)** d**d)** x

**14) If A, B, C are the angles of a triangle ABC, then the determinant
is equal to**

**a)**1

**b)**-1

**c)**sin A + sin B + sin C

**d)**none of these

**15) If a, b, c are different, then the value of x satisfying ****a)** a**b)** b**c)** c**d)** 0

**16) The value of the determinant ****a)** k (a + b)(b + c) (c + a)**b)** **c)** k(a – b)(b – c)(c – a)**d)** k(a + b – c)(b + c – a)(c + a – b)

**17) If are
any four real numbers, then the determinant **

**a)**1

**b)**0

**c)**2

**d)**3

**18) If B is a non-singular matrix and A is a square matrix, then
is equal to**

**a)**det (B)

**b)**det (A)

**c)**

**d)**

**19) If
and nN,
then
is equal to
**

**a)**

**b)**

**c)**n A

**d)**none of these

**20) If A =is
a scalar matrix of order n x n such that _{
}= k for all I, then_{ }is
equal to**

**a)**n k

**b)**n + k

**c)**

**d)**

**21) If A is an orthogonal matrix, then is
equal to**

**a)**A

**b)**

**c)**

**d)**None of these

**22) If A is an orthogonal matrix, then****a)** _{}**b)** _{}**c)** _{}**d)** none of these

**23) If =then
t is equal to**

**a)**33

**b)**21

**c)**31

**d)**23

**24) If for AX = B, and
then
X is equal to**

**a)**

**b)**

**c)**

**d)**

**25) ****a)** **b)** **c)** **d)** none of these

**26) The value of determinant is
equal to**

**a)**

**b)**

**c)**

**d)**

**27) The value of determinantis
equal to**

**a)**

**b)**

**c)**

**d)**

**28) ,
then is
equal to**

**a)**

**b)**

**c)**

**d)**9 tan x sec x

**29) If,
then **

**a)**

**b)**

**c)**

**d)**

**30) If then is
equal to**

**a)**

**b)**

**c)**

**d)**

**31) The matrix A satisfying the equationA
=is**

**a)**

**b)**

**c)**

**d)**None of these

**32) If A is an orthogonal matrix, then equals****a)** A**b)** **c)** **d)** None of these

**33) ****a)** D**b)** **c)** **d)** None of these

**34) If A and B are two invertible matrices, then the inverse of AB is equal to****a)** AB**b)** BA**c)** **d)**

**35) If A,B,C are invertible matrices, then
is equal to**

**a)**

**b)**

**c)**

**d)**

**36) If
And then
n equals**

**a)**4

**b)**6

**c)**8

**d)**none of these

**37) If A=diag (****a)** **b)** **c)** A**d)** None of these

**38) If A = [a _{ij}]
is a skew-symmetric matrix of order n, then is**

**a)**0 for some i

**b)**

**c)**1 for some i

**d)**

**39) If A and B are matrices of the same order,then
is possible, if**

**a)**AB=I

**b)**BA = I

**c)**AB=BA

**d)**None of these

**40) ****a)** 17**b)** 25**c)** 3**d)** 12

**41) If A is a skew-symmetric matrix, then trace of A is****a)** 1**b)** -1**c)** 0**d)** None of these

**42) If A,B are square matrices of the same order,then adj(AB) is equal to****a)** (adj A) (adj B)**b)** (adj B) (adj A)**c)** adj A+adj B**d)** adj A-adj B

**43) If A is a square matrix of order and
k is a scalar, then adj(KA) is equal to**

**a)**k adj A

**b)**

**c)**

**d)**

**44) If A is a square matrix or order then
adj(adjA) is equal to**

**a)**

**b)**

**c)**

**d)**

**45) and
**

**a)**sinx cosx

**b)**1

**c)**2

**d)**3

**46) If ****a)** **b)** **c)** n A**d)** none of these

**47) ****a)** **b)** **c)** **d)**

**48) If
then the value of k is**

**a)**3

**b)**5

**c)**7

**d)**-7

**49) If A,B are two square matrics such that AB= A and BA= B, then****a)** A,B are idempotent**b)** Only A is idempotent**c)** Only B is idempotent**d)** None of these

**50) The inverse of a symmetric matrix is ****a)** symmetric**b)** skew-symmetric**c)** diagonal matrix**d)** none of these

**51) If
is a scalar matrix of order such
that **

**a)**nk

**b)**n+k

**c)**

**d)**

**52) If
is a square matrix of order and
k is scalar, then **

**a)**

**b)**

**c)**

**d)**none of these

**53) Let F (a)
, Where
Then
equals**

**a)**

**b)**

**c)**

**d)**none of these

**54) If F(x)
then**

**a)**

**b)**

**c)**

**d)**

**55) ****a)** a=1, b=1**b)** **c)** **d)** none of these

**56) If matrix A is such that
then
is equal to**

**a)**

**b)**

**c)**

**d)**none of these

**57) A and B be
matrices. Then AB=0 implies**

**a)**A=0 and B=0

**b)**

**c)**

**d)**A=0 or B=0

**58) Which of the following is incorrect****a)** **b)** **c)** **d)**

**59) If A is an invertible matrix, then which
of following is correct**

**a)**

**b)**

**c)**

**d)**

**60) If for a matrix
where I is the identity matrix, then A equals**

**a)**

**b)**

**c)**

**d)**

**61) Let A be a skew-symmetric matrix of odd order, then ****a)** 0**b)** 1**c)** -1**d)** None of these

**62) Let A be a non-singular square matrix. Then****a)** **b)** **c)** **d)** none of these

**63) Let
be a square matrix,and let
be cofactor of **

**a)**

**b)**

**c)**

**d)**none of these

**64)
be a matrix such that **

**a)**rank (A) >1

**b)**rank (A)=1

**c)**rank (A)=m

**d)**rank (A)=n

**65) The rank of the matrix****a)** 1**b)** 2**c)** 3**d)** 4

**66) If
is a matrix of rank r, then**

**a)**r=min (m,n)

**b)**r < min (m,n)

**c)**

**d)**none of these

**67) is
a matrix of rank r and B is a square submatrix of order r+1, then**

**a)**B is invertible

**b)**B is not invertible

**c)**B may or may not be invertible

**d)**None of these

**68) Let A be a skew-symmetric matrix of even order, then ****a)** is a square**b)** is not a square**c)** is always zero**d)** none of these

**69) If A is a non-singular square matrix of order n, then the
rank of A is**

**a)**equal to n

**b)**less than n

**c)**greater than n

**d)**none of these

**70) If A is a non-zero column matrix of order and
B is non-zero row matrix of order
then rank of AB equals **

**a)**m

**b)**n

**c)**1

**d)**none of these

**71) If A and B are two matrices such that AB=B and BA=A, then
**

**a)**2AB

**b)**2BA

**c)**A+B

**d)**AB

**72) If ****a)** **b)** **c)** **d)**

**73) If ****a)** **b)** **c)** **d)** none of these

**74) If ****a)** 3**b)** 5**c)** 2**d)** 4

**75) The system of linear equations x+y+z=2, 2x+y-z=3, 3x+2y+kz=4
has a unique solution if**

**a)**

**b)**-1

**c)**-2

**d)**k=0

**76) If A and B are square matrices of order 3 such that****a)** -9**b)** -81**c)** -27**d)** 81

**77) If A is
and B=
then AB is said to be a square matrix of order**

**a)**

**b)**

**c)**

**d)**

**78) In an upper triangular matrix
the element
for**

**a)**

**b)**

**c)**

**d)**

**79) If A and B are arbitrary square matrices of same order, then****a)** **b)** **c)** **d)**

**80) Let A be a square matrix. Then which of the following is not a symmetric
matric**

**a)**

**b)**

**c)**

**d)**

**81) If A is of order
and B is of order ,
then AB is of order**

**a)**

**b)**

**c)**

**d)**

**82) Let
Where
then
is equal to**

**a)**i

**b)**

**c)**

**d)**-i

**83) If
then**

**a)**

**b)**

**c)**

**d)**none of these

**84) If
then x equals**

**a)**1, 1, 0

**b)**0, -1, 1

**c)**1, -1, 3

**d)**0, 0, 3

**85) If ,
one value of x which satisfies the equation. **

**a)**x = a

**b)**x = b

**c)**x = c

**d)**x = 0

**86) Which one of the following determinant has its value as zero?****a)** **b)** **c)** **d)**

**87) The value of ****a)** a + b + c – 3 abc**b)** 3 (a +b) (b + c ) (c + a)**c)** (a – b) (b – c) (c – a )**d)** (a – b ) (b – c ) (c – a ) (a +b + c )

**88) The value of satisfying
**

**a)**

**b)**

**c)**

**d)**

**89) The repeated factor of the determinant ****a)** (x – y)**b)** (y – z)**c)** (z – x)**d)** none of these

**90) The value of ****a)** **b)** **c)** **d)**

**91) If p, q, r are negative distinct real numbers, then the determinant ****a)** < 0**b)** **c)** 0**d)** > 0

**92) The value of ****a)** 20**b)** -2**c)** 0**d)** 5

**93) One of the roots of ****a)** 6**b)** 0**c)** 3**d)** -3

**94) If
is an imaginary cube root of unity, then **

**a)**1

**b)**-1

**c)**0

**d)**none of these

**95) The value of det ****a)** **b)** 0**c)** **d)** none of these

**96) The value of the determinant****a)** 2 (a+b+c)**b)** **c)** ab+bc+ca**d)** 2 abc (ab+bc+ca)

**97) If is
the transpose of a square matrix A, then**

**a)**

**b)**

**c)**

**d)**

**98) If a square matrix A has two identical rows or columns, then det A
is**

**a)**0

**b)**1

**c)**-1

**d)**none of these

**99) Let ****a)** **b)** **c)** **d)** none of these

**100) If A is square matrix such that
then det (A) equals**

**a)**0 or 1

**b)**-2 or 2

**c)**-3 or 3

**d)**none of these

**Answers** **Ans 1)** c*For the non-trivial solution, we must have
Divide through out by (a-1) (b – 1) (c -1)
*

**For the non-trivial solution, we must have**

Ans Desc 1)

Ans Desc 1)

Divide through out by (a-1) (b – 1) (c -1)

**Ans 2)** c

Ans Desc 2)

**Ans 3)** a

Ans Desc 3)

**Ans 4)** c

Ans Desc 4)

**Ans 5)** b

Ans Desc 5)

**Ans 6)** d

Ans Desc 6)

**Ans 7)** a

Ans Desc 7)

**Ans 8)** c

Ans Desc 8)

**Ans 9)** d

if (clearly)

Ans Desc 9)

if (clearly)

**Ans 10)** d*Operate
*

**Operate**

Ans Desc 10)

Ans Desc 10)

**Ans 11)** b*Since the system of equations
is inconsistent.
*

**Since the system of equations**

Ans Desc 11)

Ans Desc 11)

is inconsistent.

**Ans 12)** d*The system of equations
has a non-zero solution if
*

**The system of equations**

Ans Desc 12)

Ans Desc 12)

has a non-zero solution if

**Ans 13)** b

*The value of the determinant
which is independent of p.*

Ans Desc 13)

Ans Desc 13)

The value of the determinant

which is independent of p.

**Ans 14)** d*Its value is 0***Ans Desc 14)** Its value is 0

**Ans 15)** d*Since for x = 0, the determinant reduces to the determinant of a skew symmetric
matrix of odd order which is always zero. Hence x =0 is the solution of
the given equation.*

**Since for x = 0, the determinant reduces to the determinant of a skew symmetric**

Ans Desc 15)

Ans Desc 15)

matrix of odd order which is always zero. Hence x =0 is the solution of

the given equation.

**Ans 16)** c

= k (a – b) ( b – c ) ( c – a )

Ans Desc 16)

= k (a – b) ( b – c ) ( c – a )

**Ans 17)** b*The given determinant =
*

**The given determinant =**

Ans Desc 17)

Ans Desc 17)

**Ans 18)** b*=
=
=
=det (I) . det A
=1. det A = det A*

**=**

Ans Desc 18)

Ans Desc 18)

=

=

=det (I) . det A

=1. det A = det A

**Ans 19)** b

and so on therefore

Ans Desc 19)

and so on therefore

**Ans 20)** d

Ans Desc 20)

**Ans 21)** b*Since A is orthogonal matrix
Therefore
Therefore *

**Since A is orthogonal matrix**

Ans Desc 21)

Ans Desc 21)

Therefore

Therefore

**Ans 22)** b

Ans Desc 22)

**Ans 23)** b*Putting,
we shall get
=1(0-9)+3(4+6)
=-9+30=21*

**Putting,**

Ans Desc 23)

Ans Desc 23)

we shall get

=1(0-9)+3(4+6)

=-9+30=21

**Ans 24)** a*If AX = B, then
*

**If AX = B, then**

Ans Desc 24)

Ans Desc 24)

**Ans 25)** a

=0+0+1=1

Also

is divisible

by x

Ans Desc 25)

=0+0+1=1

Also

is divisible

by x

**Ans 26)** b

(By taking

common from R_{2})

(By taking

common from C_{2})

Operate R_{3}+R_{2
}

Ans Desc 26)

Ans Desc 26)

(By taking

common from R_{2})

(By taking

common from C_{2})

Operate R_{3}+R_{2
}

**Ans 27)** d

Operate R_{1}-R_{2
}

Operate C_{3}+C_{1
}

Â

Ans Desc 27)

Ans Desc 27)

Operate R_{1}-R_{2
}

Operate C_{3}+C_{1
}

Â

**Ans 28)** b

Operate,

we get

=

=

Ans Desc 28)

Operate,

we get

=

=

**Ans 29)** c

Multiply C_{1} by x, C_{2} by y, C_{3} by z ,we get

**
**Operate C1+C2+C3, we get

Ans Desc 29)

Ans Desc 29)

Multiply C_{1} by x, C_{2} by y, C_{3} by z ,we get

**
**Operate C1+C2+C3, we get

**Ans 30)** c*We have,
*

**We have,**

Ans Desc 30)

Ans Desc 30)

**Ans 31)** c*We have,
*

**We have,**

Ans Desc 31)

Ans Desc 31)

**Ans 32)** b*By definition A is orthogonal
*

**By definition A is orthogonal**

Ans Desc 32)

Ans Desc 32)

**Ans 33)** b

*Let D=
Now, cofactor of
Cofactor of
And, cofactor of
= Diag
*

Ans Desc 33)

Ans Desc 33)

Let D=

Now, cofactor of

Cofactor of

And, cofactor of

= Diag

**Ans 34)** d

Ans Desc 34)

**Ans 35)** d*We Have
*

**We Have**

Ans Desc 35)

Ans Desc 35)

**Ans 36)** d*Since
*

*[Since K is variable ]
[Since C*

_{2}– C

_{1}, C

_{3}– C

_{1}] But

**Since**

Ans Desc 36)

Ans Desc 36)

[Since K is variable ]

[Since C_{2} – C_{1} , C_{3} – C_{1} ]

But

**Ans 37)** b*We have,
Proceeding in this manner, we have
*

**We have,**

Ans Desc 37)

Ans Desc 37)

Proceeding in this manner, we have

**Ans 38)** b*Since
is a skew-symmetric matrix. Therefore
*

**Since**

Ans Desc 38)

Ans Desc 38)

is a skew-symmetric matrix. Therefore

**Ans 39)** c*We have
*

**We have**

Ans Desc 39)

Ans Desc 39)

**Ans 40)** a*We know that, if
is a square matrix of order ,
then tr(A)=.*

**We know that, if**

Ans Desc 40)

Ans Desc 40)

is a square matrix of order ,

then tr(A)=.

**Ans 41)** c*Since diagonal element of a skew-symmetric matrix are all zero
*

**Since diagonal element of a skew-symmetric matrix are all zero**

Ans Desc 41)

Ans Desc 41)

**Ans 42)** b*We Know that
We have(AB) (adjB. Adj A)=A(B adj B)adj A
=
From (i) and (ii)
(AB) (adj AB) = (AB) (adjB. adj A)
*

**We Know that**

Ans Desc 42)

Ans Desc 42)

We have(AB) (adjB. Adj A)=A(B adj B)adj A

=

From (i) and (ii)

(AB) (adj AB) = (AB) (adjB. adj A)

**Ans 43)** c*We have
*

**We have**

Ans Desc 43)

Ans Desc 43)

**Ans 44)** c*For any square matrix X, we have
Taking X=adj A,we get
*

**For any square matrix X, we have**

Ans Desc 44)

Ans Desc 44)

Taking X=adj A,we get

**Ans 45)** b

*We know that
It is given that A (adjA) =kI. Therefore, k=1.*

Ans Desc 45)

Ans Desc 45)

We know that

It is given that A (adjA) =kI. Therefore, k=1.

**Ans 46)** b

Ans Desc 46)

**Ans 47)** b*We have
*

**We have**

Ans Desc 47)

Ans Desc 47)

**Ans 48)** b*We know that ***Ans Desc 48)** We know that

**Ans 49)** a*We have, AB =A and BA= B
are indempotent
matrices.*

**We have, AB =A and BA= B**

Ans Desc 49)

Ans Desc 49)

are indempotent

matrices.

**Ans 50)** a*Let A be a symmetric matrix. Then
is a
symmetric matrix*

**Let A be a symmetric matrix. Then**

Ans Desc 50)

Ans Desc 50)

is a

symmetric matrix

**Ans 51)** d*We have, ***Ans Desc 51)** We have,

**Ans 52)** a*We have
therefore
*

**We have**

Ans Desc 52)

Ans Desc 52)

therefore

**Ans 53)** a*We have
*

**We have**

Ans Desc 53)

Ans Desc 53)

**Ans 54)** c*We have,
*

**We have,**

Ans Desc 54)

Ans Desc 54)

**Ans 55)** b*We have
Or,
Or,
or,
*

**We have**

Ans Desc 55)

Ans Desc 55)

Or,

Or,

or,

**Ans 56)** a*We have
*

**We have**

Ans Desc 56)

Ans Desc 56)

**Ans 57)** c*We have ***Ans Desc 57)** We have

**Ans 58)** a*We have,
So,
(a) is not true*

**We have,**

Ans Desc 58)

Ans Desc 58)

So,

(a) is not true

**Ans 59)** d*Since A is invertible, therefore
Thus, (d) is correct*

**Since A is invertible, therefore**

Ans Desc 59)

Ans Desc 59)

Thus, (d) is correct

**Ans 60)** b*Obviously,
satisfies the equation
*

**Obviously,**

Ans Desc 60)

Ans Desc 60)

satisfies the equation

**Ans 61)** a*Let A be a skew-symmetric matrix of odd order (2 n+1) say.Since
A is skew-symmetric,therefore
*

**Let A be a skew-symmetric matrix of odd order (2 n+1) say.Since**

Ans Desc 61)

Ans Desc 61)

A is skew-symmetric,therefore

**Ans 62)** b*Since A is non-singular, therefore
exists and
)*

**Since A is non-singular, therefore**

Ans Desc 62)

Ans Desc 62)

exists and

)

**Ans 63)** b*Since A is non-singular matrix, therefore
*

**Since A is non-singular matrix, therefore**

Ans Desc 63)

Ans Desc 63)

**Ans 64)** b*Let A denote the matrix every element of which is unity.Then
all the 2-rowed minors of A obviously vanish. But A is a non-null
matrix. Hence,
rank of A is 1.*

**Let A denote the matrix every element of which is unity.Then**

Ans Desc 64)

Ans Desc 64)

all the 2-rowed minors of A obviously vanish. But A is a non-null

matrix. Hence,

rank of A is 1.

**Ans 65)** c

We observe that the leading minor of the third order of this

matrix i.e.

Ans Desc 65)

We observe that the leading minor of the third order of this

matrix i.e.

**Ans 66)** c*It is a direct consequence of the definition of rank***Ans Desc 66)** It is a direct consequence of the definition of rank

**Ans 67)** b*By def. Of rank,
and so it is not invertible.*

**By def. Of rank,**

Ans Desc 67)

Ans Desc 67)

and so it is not invertible.

**Ans 68)** a*Letbe
a skew symmetric matrix.Then
*

**Letbe**

Ans Desc 68)

Ans Desc 68)

a skew symmetric matrix.Then

**Ans 69)** a*Since A is non-singular matrix,therefore
*

**Since A is non-singular matrix,therefore**

Ans Desc 69)

Ans Desc 69)

**Ans 70)** c

be two non-zero column and row matrices respectively. We have

Since A and B are non-zero matrices, therefore the matrix AB

will also be a

non-zero matrix. The matrix AB will have at least one non-zero

element

Obtained by multiplying corresponding non-zero elements of A

and B.

All the two-rowed minors of A obviously vanish. But A is a non-zero

Matrix.Hence rank (A) =1

Ans Desc 70)

be two non-zero column and row matrices respectively. We have

Since A and B are non-zero matrices, therefore the matrix AB

will also be a

non-zero matrix. The matrix AB will have at least one non-zero

element

Obtained by multiplying corresponding non-zero elements of A

and B.

All the two-rowed minors of A obviously vanish. But A is a non-zero

Matrix.Hence rank (A) =1

**Ans 71)** c*We have AB=B and BA=A therefore,
*

**We have AB=B and BA=A therefore,**

Ans Desc 71)

Ans Desc 71)

**Ans 72)** a*We have
So
*

**We have**

Ans Desc 72)

Ans Desc 72)

So

**Ans 73)** d

matrices in (a),(b),(c) do not tally with

Ans Desc 73)

matrices in (a),(b),(c) do not tally with

**Ans 74)** b*Since the given matrix is symmetric, therefore
*

**Since the given matrix is symmetric, therefore**

Ans Desc 74)

Ans Desc 74)

**Ans 75)** a*The given system of equation has unique solution if
*

**The given system of equation has unique solution if**

Ans Desc 75)

Ans Desc 75)

**Ans 76)** a*We have
*

**We have**

Ans Desc 76)

Ans Desc 76)

**Ans 77)** b*AB of type ***Ans Desc 77)** AB of type

**Ans 78)** b*By Definition.***Ans Desc 78)** By Definition.

**Ans 79)** d

Ans Desc 79)

**Ans 80)** d

is not symmetric

Ans Desc 80)

is not symmetric

**Ans 81)** b*Order of ***Ans Desc 81)** Order of

**Ans 82)** a

*Operate
*

Ans Desc 82)

Ans Desc 82)

Operate

**Ans 83)** b*Operate
we get *

**Operate**

Ans Desc 83)

Ans Desc 83)

we get

**Ans 84)** d*Operate
and taking out 3-x as common from ,
we get
operate
we get
*

**Operate**

Ans Desc 84)

Ans Desc 84)

and taking out 3-x as common from ,

we get

operate

we get

**Ans 85)** d*For x = 0 the given determinant is a skew-symmetric determinant of odd
order
its value = 0*

**For x = 0 the given determinant is a skew-symmetric determinant of odd**

Ans Desc 85)

Ans Desc 85)

order

its value = 0

**Ans 86)** d*Value = 0.
All elts of second row = 0]*

**Value = 0.**

Ans Desc 86)

Ans Desc 86)

All elts of second row = 0]

**Ans 87)** d

Ans Desc 87)

**Ans 88)** d

Ans Desc 88)

**Ans 89)** c*Operate
we get the given value
Operate
*

**Operate**

Ans Desc 89)

Ans Desc 89)

we get the given value

Operate

**Ans 90)** b*The given value
*

**The given value**

Ans Desc 90)

Ans Desc 90)

**Ans 91)** d*Operate
Operate
*

**Operate**

Ans Desc 91)

Ans Desc 91)

Operate

**Ans 92)** b*Operate ***Ans Desc 92)** Operate

**Ans 93)** b*Operate
we shall get x = 0 as one root*

**Operate**

Ans Desc 93)

Ans Desc 93)

we shall get x = 0 as one root

**Ans 94)** c*Operate ,
we get the
*

**Operate ,**

Ans Desc 94)

Ans Desc 94)

we get the

**Ans 95)** c*Value of the det. = ***Ans Desc 95)** Value of the det. =

**Ans 96)** b

Operate

Ans Desc 96)

Operate

**Ans 97)** b

Ans Desc 97)

**Ans 98)** a*Det. A = 0***Ans Desc 98)** Det. A = 0

**Ans 99)** a*Adj ***Ans Desc 99)** Adj

**Ans 100)** a*Since
*

**Since**

Ans Desc 100)

Ans Desc 100)