Matrices & Determinants MCQs Part III

1) The value of the determinant
a)
b)
c)
d)

2) The solution set of the equation
a)
b)
c)
d)

3) If A is square matrix such that
then for any +ve integer n,
is equal to

a) 0
b)
c)
d)

4) The multiplicative inverse of
a)
b)
c)
d)

5) The multiplicative inverse of the matrix
a)
b)
c)
d)

6) If a, b, c are non-zero real numbers, then the inverse of the matrix
a)
b)
c)
d)

7) For any
matrix A, if A (Adj A)

a) 0
b) 10
c) 20
d) 100

8) If
a)
b)
c)
d)

9) If
then the value of

a)
b)
c)
d)

10) For a
matrix A, if det A = 4, then det (Adj A) equals

a) -4
b) 4
c) 16
d) 64

11) If
the entries in a
determinant
are either 0 or 1, then the greatest value of this determinant is

a) 1
b) 2
c) 3
d) 9

12) The inverse of the matrix
a) A
b)
c)
d)

13) If a + b + c = 0, one root of
a) x = 1
b) x = 2
c)
d) x = 0

14) The factors of
a) x – a, x – b and x + a + b
b) x + a, x +b and x +a + b
c) x + a, x + b and x- a- b
d) x – a, x – b and x – a – b

15) If
a)
b)
c)
d)

16) If one of the roots of the equation
a) (2,6)
b) (3,6)
c) (2,7)
d) (3,7)

17) One root of the equation
a)
b)
c)
d)

18) If
then the value of

a) 0
b)
c)
d)

19) If (1 2 3 ) A = (4 5), what is the order of matrix A ?
a)
b)
c)
d)

20) If
a)
b)
c)
d)

21) The value of the determinant
a) n
b) a
c) x
d) none of these

22) If a, b, and c are all different from zero and
then the value of

a) abc
b)
c) -a – b – c
d) -1

23) If
a) 0
b) abc
c) -abc
d) none of these

24) If are
real numbers then

a) -1
b)
c)
d) 0

25) If a, b and c are non-zero real numbers, then
a) abc
b)
c) bc + ca + ab
d) 0

26) If each element of a
matrix is multiplied by 3, then the determinant of the newly formed matrix
is

a) 3 det A
b) 9 det A
c) 27 det A
d)

27) Let
denote the element of the ith row and jth column in a
determinant
and let
for every
and j. Then the determinant has all the principal diagonal elements as

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

28) The value of the determinant
a) 1
b) -1
c) 0
d) none of these

29) The value of the determinant
a) 1
b) 0
c) 2
d) 3

30) If x,y,z are in A.P., then the value of the det
a) 0
b) 1
c) 2
d) none of these

31) If
are the roots of the equation
then value of the

a) p
b) q
c)
d) 0

32) If a,b,c are in A.P., then the value of
a) 3
b) -3
c) 0
d) none of these

33) If a > 0, b > 0, c > 0 are respectively, the pth, qth,
rth terms of a G.P., then the value of the determinant

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

34) If a, b, c are in G.P ., then the value of the determinant
a) 1
b) 0
c) -1
d) none of these

35) Given then
the value of

a) 0
b)
c) 1
d) 2

36) Let A be an invertible matrix. Which of the following is not true?
a)
b)
c)
d) none of these

37) Let Then,

a)
b)
c)
d)

38) The determinant
a)
b)
c)
d)

39) If the system of equations
and
has a non-trivial solution, then the value of
is

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

40) The value of a for which the system equation

has a non- zero solution is

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

41) If
then the value of
is

a) 1
b) 2
c) 4
d) 3

42) If in
then
is

a) 0
b) -D
c) D
d)

43) Inverse of
is

a)
b)
c)
d) none of these

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

a)
b)
c)
d)

45) If
Then
is equal to

a) 0
b) 1
c) 100
d) -100

46) The value of
is

a) 0
b) a + b+ c
c) 4 abc
d) abc

47) If A and B are square matrices of order 3 such that
then

a) -9
b) -81
c) -27
d) 81

48) For any
matrix A, if A (adj. A)

a) 0
b) 10
c) 20
d) 100

49) If
the value of x which satisfies the equation
is

a) x = a
b) x = b
c) x = c
d) x=0

50) If
is cube root of unity , then

a)
b)
c)
d)

51) If every element of a third order determinant of value is multiplied by 5 then the value of the new determinant is
a)
b)
c)
d)

52) If=,
then x=

a) 6
b) 7
c) 8
d) none of these

53) If=0,
then one of the values of x is

a) a+b+c
b) -(a+b+c)
c)
d)

54) Cofactor of 4 in the determinant
is equal to

a) 2
b) -2
c) -5
d) none of these

55) If x+y+z=,
then the value of
is equal to

a) 0
b) 2sinB tanA cosC
c) 1
d) none of these

56) The value of determinant
a) -2
b)
c) 2
d) none of these

57) If one root of the equation
is x=-9, then the other root are

a) 2,6
b) 3,6
c) 2,7
d) 3,7

58) If
is a cube root of unity, then a root of the equation
is

a) x=1
b)
c)
d) x = 0

59) If the value of a third order determinant is 11, then the value of the
square of the determinant formed by the cofactors will be

a) 11
b) 121
c) 1331
d) 14641

60) LetThe

lies in the interval

a) [3,4]
b) [2,4]
c) [1,4]
d) none of these

61) is
equal to

a) m(m+1)
b) 0
c) 1
d) m(m-1).

62) If a,b,c are different and
then

a) a+b+c=0
b) abc=1
c) a+b+c=1
d) ab+bc+ca=0.

63) If
are given determinants, then

a)
b)
c)
d)

64) For the system of equations
if
then the system has

a) more than two solutions
b) one trivial and one non-trivial solution
c) No solution
d) only trivial solution(0,0,0).

65) If the system of linear equationshave
a non-zero solution, then a, b,c are in

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

66) The value of
for which the system of equations 2x-y+3=0; x+y+7=0;
3x+2y-2=0 is consistent is given by

a) – 3
b) – 9
c) -45/13
d) 45/13

67) If are
respectively the cofactors of the elements
of the determinant

a)
b)
c)
d) none of these

68) If
then the real value of x is

a) 2
b) 3
c) 4
d) 6

69) The parameter, on which the value of the determinant
does not depend upon, is

a) a
b) p
c) d
d) x

70) The determinant
if

a) x,y,z are in A.P.
b) x,y,z are in G.P.
c) x,y,z are in H.P.
d) xy,yz,zx are in A.P

71) Let
where p is a constant. Then
at x=0 is

a) p
b)
c)
d) Independent of p.

72) If x is a positive integer, then
is equal to

a)
b)
c)
d)

73) If
then

a) –1/3
b) 1/4
c) 1/2
d) none of these

74) If
then

a) a is one of the cube roots of unity
b) b is one of the cube roots of unity
c) (a/b)is one of the cube roots of unity
d) none of these

75)
a) 0
b) a
c) 3
d) 2a

76)
a) p + q + r
b) 0
c) p – q – r
d) – p + q + r

77) If
then

a) x = 3, y = 1
b) x = 1, y = 3
c) x = 0, y = 3
d) x = 0, y = 0

78) The determinant for all if
a) x, y, z are in AP
b) x, y, z are in GP
c) x, y, z are in HP
d) xy, yz, zx are in AP

79) The determinant
then

a) d=0
b) a + d = 0
c) d = 0 or a + d = 0
d) none of these

80) The value of the determinant,
where a, b, c are the pth, qth and rth terms of a HP, is

a) ap + bq + cr
b) (a+b+c)(p+q+r)
c) 0
d) none of these

81) The sum of two non-integral roots of
is

a) 5
b) -5
c) -18
d) none of these

82) If x, y, z are integers in AP, lying between 1 and 9, and x51, y41 and z31 are
three-digit numbers, then the value of
is

a) x + y + z
b) x – y + z
c) 0
d) none of these

83) The value of
is equal to

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

84) Eliminating a, b, c from
we get

a)
b)
c)
d) none of these

85) The system of equations
has

a)
b)
c)
d)

86)
a) abc > 1
b) abc > -8
c) abc < -8
d) abc > -2

87)
a) Zero
b) positive
c) negative
d)

88)
a)
b)
c)
d) none of these

89) Given that xyz = -1, the value of the determinant
is

a) 0
b) positive
c) negative
d) none of these

90)
a)
b)
c) 1
d) none of these

91) The value of the determinant
is

a)
b)
c)
d) none of these

92)
a)
b)
c)
d) none of these

93) Let
then the value of

a) 0
b) a + b + c
c) ab + bc + ca
d) none of these

94)
is
a polynomial of degree

a) 2
b) 3
c) 4
d) none of these

95)
then
equals

a)
b)
c)
d) none of these

96) If
then k equals

a) 1
b) -1
c)
d)

97) If
is an imaginary cube root of unity, then the value of

a) 0
b)
c)
d)

98)
a)
b)
c)
d) none of these

99)
a)
b)
c)
d)

100) The largest value of a third order determinant whose elements are 0 or 1,
is

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

Answers
Ans 1) b
Operate
we get the given determinant


Operate


Ans Desc 1)
Operate
we get the given determinant


Operate

Ans 2) d
Operate
we get






Ans Desc 2)
Operate
we get





Ans 3) c


Ans Desc 3)

Ans 4) b




Ans Desc 4)



Ans 5) d




Ans Desc 5)



Ans 6) a




Ans Desc 6)



Ans 7) b




Ans Desc 7)



Ans 8) c




Ans Desc 8)



Ans 9) b


Ans Desc 9)

Ans 10) c





Ans Desc 10)




Ans 11) b
Greatest value = 2
Ans Desc 11)
Greatest value = 2

Ans 12) a

Adj A:


Ans Desc 12)

Adj A:

Ans 13) d
Operate , we shall get


Ans Desc 13)
Operate , we shall get

Ans 14) a




Ans Desc 14)



Ans 15) d

[All others will vanish]

Ans Desc 15)

[All others will vanish]

Ans 16) c

Operate we get







Ans Desc 16)

Operate we get






Ans 17) b
Operate
we get


Ans Desc 17)
Operate
we get

Ans 18) a








Ans Desc 18)







Ans 19) a


Ans Desc 19)

Ans 20) b

Ans Desc 20)

Ans 21) a
Operate






,
Which is independent of n.

Ans Desc 21)
Operate






,
Which is independent of n.

Ans 22) d
We know that





Ans Desc 22)
We know that




Ans 23) b


Ans Desc 23)

Ans 24) d

= (0) (0) = 0

Ans Desc 24)

= (0) (0) = 0

Ans 25) d
Perform


Operate
and taking out ab+bc+ca as common


Ans Desc 25)
Perform


Operate
and taking out ab+bc+ca as common

Ans 26) c

Ans Desc 26)

Ans 27) c

for a skew-symmetric matrix, all diagonal elts = 0

Ans Desc 27)

for a skew-symmetric matrix, all diagonal elts = 0

Ans 28) a
The given determinant

Operate

Ans Desc 28)
The given determinant

Operate

Ans 29) b





Ans Desc 29)




Ans 30) a




Ans Desc 30)



Ans 31) d
Since
are the roots of



= 0

Ans Desc 31)
Since
are the roots of



= 0

Ans 32) c

given determinant will have all elements, in the first row as zero

Ans Desc 32)

given determinant will have all elements, in the first row as zero

Ans 33) b
The given determinant




Ans Desc 33)
The given determinant



Ans 34) b
Operate



Ans Desc 34)
Operate


Ans 35) c

Given value





Ans Desc 35)

Given value




Ans 36) a
L.H.S
is a matrix and R.H.S. is a number.

Ans Desc 36)
L.H.S
is a matrix and R.H.S. is a number.

Ans 37) c



Ans Desc 37)


Ans 38) a
Operate


Ans Desc 38)
Operate

Ans 39) c
Since given system of equations has a non-trivival solution

Operate


Dividing by ( 1 -a) ( 1 – b ) (1 – c), we get


Ans Desc 39)
Since given system of equations has a non-trivival solution

Operate


Dividing by ( 1 -a) ( 1 – b ) (1 – c), we get

Ans 40) c
The system of linear equation will have non-zero solution if






Ans Desc 40)
The system of linear equation will have non-zero solution if





Ans 41) c



Ans Desc 41)


Ans 42) a

Ans Desc 42)

Ans 43) c





Ans Desc 43)




Ans 44) d


Ans Desc 44)

Ans 45) a





Ans Desc 45)




Ans 46) c






Ans Desc 46)





Ans 47) b


Ans Desc 47)

Ans 48) b

Ans Desc 48)

Ans 49) d




if (clearly)

Ans Desc 49)




if (clearly)

Ans 50) d
Operate









Ans Desc 50)
Operate








Ans 51) d
Since the multiplication of each element of a row will multiply the value of det.by5. So if all elements of all 3 rows are multiplied by 5.5.5=125.
Ans Desc 51)
Since the multiplication of each element of a row will multiply the value of det.by5. So if all elements of all 3 rows are multiplied by 5.5.5=125.

Ans 52) a

From the given equation, we have
4=3x-2(x+6)=2x-8
x=6


Ans Desc 52)

From the given equation, we have
4=3x-2(x+6)=2x-8
x=6

Ans 53) b
Apply
Ans Desc 53)
Apply

Ans 54) b
-4 is in the place
in
Therefore, cofactor of
is

Ans Desc 54)
-4 is in the place
in
Therefore, cofactor of
is

Ans 55) a
Since x+y+z=,


Ans Desc 55)
Since x+y+z=,

Ans 56) a
By
and

we have
By =
==2(2-3)
= – 2

Ans Desc 56)
By
and

we have
By =
==2(2-3)
= – 2

Ans 57) c

=
=(x+9)(x-2)(7-x)
Other roots are (2,7)

Ans Desc 57)

=
=(x+9)(x-2)(7-x)
Other roots are (2,7)

Ans 58) d


=

Ans Desc 58)


=

Ans 59) b

Ans Desc 59)

Ans 60) b
=


lies in [2,4]

Ans Desc 60)
=


lies in [2,4]

Ans 61) c
=
which
is given in (c). =1

Ans Desc 61)
=
which
is given in (c). =1

Ans 62) b
We have

Ans Desc 62)
We have

Ans 63) b

Ans Desc 63)

Ans 64) a

equations will have infinite number of solutions

Ans Desc 64)

equations will have infinite number of solutions

Ans 65) c
The system has non-zero i.e. non-trivial solution if the determinant of coefficients.
Now

a,b,c
are in H.P.

Ans Desc 65)
The system has non-zero i.e. non-trivial solution if the determinant of coefficients.
Now

a,b,c
are in H.P.

Ans 66) c
System is consistent if
Ans Desc 66)
System is consistent if

Ans 67) a









(a)is correct

Ans Desc 67)









(a)is correct

Ans 68) c


only
real value of x is 4

Ans Desc 68)


only
real value of x is 4

Ans 69) b

cos dx gives


which is independent of p

Ans Desc 69)

cos dx gives


which is independent of p

Ans 70) b

gives



i.e. x,y,z are in G.P

Ans Desc 70)

gives



i.e. x,y,z are in G.P

Ans 71) d

which
is independent of p

Ans Desc 71)

which
is independent of p

Ans 72) c



Ans Desc 72)


Ans 73) a


Ans Desc 73)

Ans 74) d


or(a/b)is
one of the cube roots of –1
But a0
so (a/b)must be one of the cube roots of –1,
which is given in (d).

Ans Desc 74)


or(a/b)is
one of the cube roots of –1
But a0
so (a/b)must be one of the cube roots of –1,
which is given in (d).

Ans 75) a

Ans Desc 75)

Ans 76) b

Ans Desc 76)

Ans 77) d

Ans Desc 77)

Ans 78) b

Ans Desc 78)

Ans 79) c

Ans Desc 79)

Ans 80) c


Ans Desc 80)

Ans 81) b
Here
or
or or
The non-integral roots are the roots of

Ans Desc 81)
Here
or
or or
The non-integral roots are the roots of

Ans 82) c

=

Ans Desc 82)

=

Ans 83) b

=
=

Ans Desc 83)

=
=

Ans 84) d
Here Eliminating
a,b,c, we get

Ans Desc 84)
Here Eliminating
a,b,c, we get

Ans 85) c

For existence of nontrivial solutions,

If
is the only solution

Ans Desc 85)

For existence of nontrivial solutions,

If
is the only solution

Ans 86) b





Ans Desc 86)




Ans 87) c






Ans Desc 87)





Ans 88) b





Ans Desc 88)




Ans 89) a








 


Ans Desc 89)








 

Ans 90) d
Let A, B and C be three numbers such that





Ans Desc 90)
Let A, B and C be three numbers such that




Ans 91) a




Ans Desc 91)



Ans 92) a




Ans Desc 92)



Ans 93) a



=2 x 0 = 0

Ans Desc 93)



=2 x 0 = 0

Ans 94) d
Since f(x),g(x) and h(x) are polynomial of degree 3.



Ans Desc 94)
Since f(x),g(x) and h(x) are polynomial of degree 3.


Ans 95) d
We have
=
0 x 0 = 0.

Ans Desc 95)
We have
=
0 x 0 = 0.

Ans 96) c

Ans Desc 96)

Ans 97) d

Ans Desc 97)

Ans 98) c



Ans Desc 98)


Ans 99) c

Ans Desc 99)

Ans 100) c

Since each element of
is either 1 or 0, therefore the value of the determinant cannot exceed 3. Clearly
the value of
is maximum when the value of each term in first bracket is 1 and the value of
each term in the second bracket is Zero. But
implies that every element of the determinant
is 1 and in that case
Thus, we may have

Ans Desc 100)

Since each element of
is either 1 or 0, therefore the value of the determinant cannot exceed 3. Clearly
the value of
is maximum when the value of each term in first bracket is 1 and the value of
each term in the second bracket is Zero. But
implies that every element of the determinant
is 1 and in that case
Thus, we may have

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