Processing math: 100%
MCQExams
0:0:1
CBSE
JEE
NTSE
NEET
Practice
Homework
×
CBSE Questions for Class 12 Commerce Maths Determinants Quiz 1 - MCQExams.com
CBSE
Class 12 Commerce Maths
Determinants
Quiz 1
The number of distinct real roots of the equation,
|
cos
x
sin
x
sin
x
sin
x
cos
x
sin
x
sin
x
sin
x
cos
x
|
=
0
in the interval
[
−
π
4
,
π
4
]
is/are :
Report Question
0%
3
0%
2
0%
1
0%
4
Explanation
The given determinant is :
|
cos
x
sin
x
sin
x
sin
x
cos
x
sin
x
sin
x
sin
x
cos
x
|
=
0
Taking
cos
3
x
common in the above determinant, we get,
cos
3
x
|
1
t
a
n
x
t
a
n
x
t
a
n
x
1
t
a
n
x
t
a
n
x
t
a
n
x
1
|
=
0
cos
3
x
[
1
(
1
−
tan
2
x
)
−
t
a
n
x
(
t
a
n
x
−
t
a
n
2
x
)
+
t
a
n
x
(
t
a
n
2
x
−
t
a
n
x
)
]
=
0
cos
3
x
[
1
−
3
t
a
n
2
x
+
2
t
a
n
3
x
]
=
0
We notice that if
cos
3
x
=
0
→
x
=
π
2
This is not satisfied by the given interval
Thus,
x
=
π
2
is not a solution.
Now,
1
−
3
t
a
n
2
x
+
2
t
a
n
3
x
=
0
It is obvious from the above equation that
t
a
n
x
=
1
is a solution.
Thus,
x
=
π
4
is a solution.
We just found out that
(
t
a
n
x
−
1
)
is a factor of the polynomial
1
−
3
t
a
n
2
x
+
2
t
a
n
3
x
=
0
Thus, dividing the polynomial by
(
t
a
n
x
−
1
)
, we get the quotient as
2
t
a
n
x
2
−
t
a
n
x
−
1
and remainder
0
Thus, this can be written as
(
2
t
a
n
x
+
1
)
(
t
a
n
x
−
1
)
=
0
Or,
t
a
n
x
=
−
1
2
or
t
a
n
x
=
1
We already know that
t
a
n
x
=
1
is a solution to the above equation.
Thus,
t
a
n
x
=
−
1
2
Or,
x
=
t
a
n
−
1
(
−
1
2
)
This also lies in the interval
[
−
π
4
,
π
4
]
Thus there are two distinct real roots to the above equation.
The points
(
0
,
8
3
)
,
(
1
,
3
)
and
(
82
,
30
)
:
Report Question
0%
form an obtuse angled triangle.
0%
form a right angled triangle.
0%
lie on a straight line.
0%
form an acute angled triangle.
Explanation
Given
(
0
,
8
3
)
,
(
1
,
3
)
and
(
82
,
30
)
|
0
8
3
1
1
3
1
82
30
1
|
=
216
−
216
=
0
Hence, the given points are collinear.
If
A
=
[
2
−
3
−
4
1
]
, then adj
(
3
A
2
+
12
A
)
is equal to.
Report Question
0%
[
72
−
84
−
63
51
]
0%
[
51
63
84
72
]
0%
[
51
84
63
72
]
0%
[
72
−
63
−
84
51
]
Explanation
It is given that
A
=
[
2
−
3
−
4
1
]
Hence,
A
2
=
[
2
−
3
−
4
1
]
[
2
−
3
−
4
1
]
=
[
16
−
9
−
12
13
]
Now,
3
A
2
=
[
48
−
27
−
36
29
]
12
A
=
[
24
−
36
−
48
12
]
Consider,
3
A
2
+
12
A
=
[
48
−
27
−
36
29
]
+
[
24
−
36
−
48
12
]
=
[
72
−
63
−
84
51
]
∴
3
A
2
+
12
A
=
[
72
−
63
−
84
51
]
Hence,
adj
(
3
A
2
+
12
A
)
=
[
51
84
63
72
]
T
=
[
51
63
84
72
]
Let
P
=
[
a
i
j
]
be a 3
×
3 matrix and let
Q
=
[
b
i
j
]
, where
b
i
j
=
2
i
+
j
a
i
j
for
1
≤
i
,
j
≤
3
. If the determinant of P is 2, then the determinant of the matrix Q is
Report Question
0%
2
10
0%
2
11
0%
2
12
0%
2
13
Explanation
Given,
[
P
]
=
a
i
j
is a order 3 matrix.
Let
Q
=
[
2
2
a
11
2
3
a
12
2
4
a
13
2
3
a
21
2
4
a
22
2
5
a
23
2
4
a
31
2
5
a
32
2
6
a
33
]
|
Q
|
=
|
2
2
a
11
2
3
a
12
2
4
a
13
2
3
a
21
2
4
a
22
2
5
a
23
2
4
a
31
2
5
a
32
2
6
a
33
|
⇒
Q
=
2
2
2
3
2
4
[
a
11
a
12
a
13
2
a
21
2
a
22
2
a
23
2
2
a
31
2
2
a
32
2
2
a
33
]
⇒
|
Q
|
=
2
9
×
2
×
2
2
|
a
11
a
12
a
13
a
21
a
22
a
23
a
31
a
32
a
33
|
⇒
|
Q
|
=
2
12
|
P
|
⇒
|
Q
|
=
2
13
Consider three points
P
=
(
−
sin
(
β
−
α
)
,
−
cos
β
)
,
Q
=
(
cos
(
β
−
α
)
,
sin
β
)
and
R
=
(
cos
(
β
−
α
+
θ
)
,
sin
(
β
−
θ
)
)
, where
0
<
α
,
β
,
θ
<
π
4
. Then
Report Question
0%
P
lies on the line segment
R
Q
0%
Q
lies on the line segment
P
R
0%
R
lies on the line segment
Q
P
0%
P
,
Q
,
R
are non-collinear
Explanation
P
≡
(
−
sin
(
β
−
α
)
,
−
cos
β
)
≡
(
x
1
,
y
1
)
Q
≡
(
cos
(
β
−
α
)
,
sin
β
)
≡
(
x
2
,
y
2
)
and
R
≡
(
x
2
cos
θ
+
x
1
sin
θ
,
y
2
cos
θ
+
y
1
sin
θ
)
We see that
T
≡
(
x
2
cos
θ
+
x
1
sin
θ
cos
θ
+
sin
θ
,
y
2
cos
θ
+
y
1
sin
θ
cos
θ
+
sin
θ
)
and
P
,
Q
,
T
are collinear
⇒
P
,
Q
,
R
are non-collinear.
The number of
A
in
T
p
such that
A
is either symmetric or skew-symmetric or both, and det
(
A
)
divisible by
p
, is
Report Question
0%
(
p
−
1
)
2
0%
2
(
p
−
1
)
0%
(
p
−
1
)
2
+
1
0%
2
p
−
1
Explanation
Given
A
=
[
a
b
c
a
]
,
a
,
b
,
c
∈
{
0
,
1
,
2
,
.
.
.
,
p
−
1
}
If A is skew -symmetric matrix, then a =0, b=-c
∴
|
A
|
=
−
|
b
|
2
Thus, p divides
|
A
|
only when
b
=
0
...(1)
Again, if A is symmetric matrix, then
v
=
c
and
|
A
|
=
a
2
−
b
2
Thus p divides
|
A
|
is either p divides
(
a
−
b
)
or o divides
(
a
+
b
)
p divides
(
a
−
b
)
only when
a
=
b
i.e
a
=
b
∈
{
0
,
1
,
2
,
.
.
.
,
(
p
−
1
)
}
i.e p choice
p divides
(
a
+
b
)
⇒
p choice including
a
=
b
=
0
in (1)
Therefore total number of choices are
(
p
+
p
+
1
)
=
2
p
−
1
Let k be a positive real number and let
A
=
[
2
k
−
1
2
√
k
2
√
k
2
√
k
1
−
2
k
−
2
√
k
2
k
−
1
]
and
B
=
[
0
2
k
−
1
√
k
1
−
2
k
0
2
√
k
−
√
k
−
2
√
k
0
]
.
If det
(
a
d
j
A
)
+
d
e
t
(
a
d
j
B
)
=
10
6
,
t
h
e
n
[
k
]
is equal to
[Note: adj M denotes the adjoint of a square matrix M and [k] denotes the largest integer less than or equal to k].
Report Question
0%
3
0%
4
0%
5
0%
6
Explanation
|
A
|
=
(
2
k
+
1
)
3
,
|
B
|
=
0
(Since
B
is a skew-symmetric matrix of order 3)
⇒
det
(
a
d
j
A
)
=
|
A
|
n
−
1
=
(
(
2
k
+
1
)
3
)
2
=
10
6
⇒
2
k
+
1
=
10
⇒
2
k
=
9
[
k
]
=
4
.
The value of
|
U
|
is
Report Question
0%
3
0%
−
3
0%
3/2
0%
2
Explanation
A
=
[
1
0
0
2
1
0
3
2
1
]
|
A
|
=
1
A
d
j
A
=
[
1
0
0
−
2
1
0
1
−
2
1
]
⇒
A
−
1
=
[
1
0
0
−
2
1
0
1
−
2
1
]
A
U
1
=
[
1
0
0
]
⇒
U
1
=
A
−
1
[
1
0
0
]
=
[
1
0
0
−
2
1
0
1
−
2
1
]
[
1
0
0
]
=
[
1
−
2
1
]
A
U
2
=
[
2
3
0
]
⇒
U
2
=
A
−
1
[
2
3
0
]
=
[
1
0
0
−
2
1
0
1
−
2
1
]
[
2
3
0
]
=
[
2
−
1
−
4
]
A
U
3
=
[
2
3
1
]
⇒
U
3
=
A
−
1
[
2
3
1
]
=
[
1
0
0
−
2
1
0
1
−
2
1
]
[
2
3
1
]
=
[
2
−
1
−
3
]
∴
U
=
[
1
2
2
−
2
−
1
−
1
1
−
4
−
3
]
U
−
1
=
1
3
[
−
1
−
2
0
−
7
−
5
−
3
9
6
3
]
∴
Sum of elements of
U
−
1
is
0
.
Hence, option B.
Hence
U
=
[
1
2
2
−
2
−
1
−
1
1
−
4
−
3
]
⇒
|
U
|
=
3
A
=
[
1
2
3
4
]
,
B
=
[
2
1
3
4
]
then
|
(
B
T
A
T
)
−
1
|
is equal to
Report Question
0%
10
0%
1
10
0%
1
0%
−
1
Explanation
A
=
[
1
2
3
4
]
B
=
[
2
1
3
4
]
B
T
A
T
=
[
2
3
1
4
]
[
1
3
2
4
]
=
[
2
+
6
6
+
12
1
+
8
3
+
16
]
=
[
8
18
9
19
]
d
e
t
(
B
T
A
T
)
=
8
×
19
−
9
×
18
=
152
−
162
=
−
10
(
B
T
A
T
)
−
1
=
1
d
e
t
(
B
T
A
T
)
a
d
j
(
B
T
A
T
)
=
1
−
10
[
19
−
18
−
9
8
]
d
e
t
(
(
B
T
A
T
)
−
1
)
=
19
×
8
−
18
×
9
−
10
=
1
If three points
(
k
,
2
k
)
,
(
2
k
,
3
k
)
,
(
3
,
1
)
are collinear, then
k
is equal to:
Report Question
0%
−
2
0%
1
0%
1
2
0%
−
1
2
Explanation
If given points are collinear, then
|
k
2
k
1
2
k
3
k
1
3
1
1
|
=
0
⇒
|
k
2
k
1
k
k
0
3
1
1
|
=
0
,
R
2
→
R
2
−
R
1
⇒
1
(
k
−
3
k
)
+
1
(
k
2
−
2
k
2
)
=
0
⇒
k
2
+
2
k
=
0
⇒
k
=
0
,
−
2
|
4
sin
2
θ
cos
2
θ
3
sec
2
θ
cosec
2
θ
|
=
Report Question
0%
8
sin
2
θ
cos
2
θ
0%
4
sin
2
θ
cos
2
θ
0%
1
0%
4
cos
3
θ
−
3
cos
θ
Explanation
The value of
|
4
sin
2
θ
cos
2
θ
3
sec
2
θ
cosec
2
θ
|
is
=
4
sin
2
θ
.
cosec
2
θ
−
3
sec
2
θ
.
cos
2
θ
=
4
−
3
=
1
|
0
c
o
s
α
c
o
s
β
c
o
s
α
0
c
o
s
γ
c
o
s
β
c
o
s
γ
0
|
=
Report Question
0%
cos
α
+
cos
β
+
cos
γ
0%
cos
α
cos
β
cos
γ
0%
2
cos
α
cos
β
cos
γ
0%
2
∑
cos
α
cos
β
Explanation
Let
A
=
|
0
c
o
s
α
cos
β
cos
α
0
c
o
s
γ
c
o
s
β
cos
γ
0
|
Expanding the determinant,
A
=
−
c
o
s
α
[
−
c
o
s
β
c
o
s
γ
]
+
c
o
s
β
[
c
o
s
α
cos
γ
]
=
2
cos
α
cos
β
cos
γ
A
=
{
8
9
10
11
}
, then cofactor of
a
12
is:
Report Question
0%
11
0%
10
0%
-11
0%
-10
Explanation
By property of cofactor (2),
A
=
[
8
9
10
11
]
minor of
a
12
=
M
12
=
10
So, cofactor of
a
12
=
M
12
(
−
1
)
1
+
2
=
−
M
12
=
−
10
If P =
[
1
4
2
6
]
,then adj (P)
Report Question
0%
[
1
4
2
6
]
0%
[
6
−
4
−
2
1
]
0%
[
6
−
2
−
4
1
]
0%
[
2
1
6
4
]
Explanation
Cofactor of the martix is
[
M
11
−
M
12
−
M
21
M
22
]
Where
M
11
=
6
M
12
=
2
M
21
=
4
M
22
=
1
Substituting all value we get
Cofactor of the matrix as
[
6
−
2
−
4
1
]
The adjoint of the matrix is transpose of cofactor of the given matrix
∴
Adjoint of the matrix is
[
6
−
4
−
2
1
]
Hence the answer is option B.
|
x
2
+
3
x
−
1
x
+
3
x
+
3
−
2
x
x
−
4
x
−
3
x
+
4
3
x
|
=
p
x
4
+
q
x
3
+
r
x
2
+
s
x
+
t
,
then
t
=
Report Question
0%
72
0%
0
0%
24
0%
−
48
Explanation
On expanding the determinant we get,
(
x
2
+
3
)
(
−
6
x
2
−
x
2
+
16
)
−
(
x
−
1
)
(
3
x
2
+
9
x
−
x
2
+
7
x
−
12
)
+
(
x
+
3
)
(
x
2
+
7
x
+
12
−
2
x
2
+
6
x
)
Let us consider only the constant terms,
since
t
is constant term
∴
(
16
)
(
3
)
−
(
−
1
)
(
−
12
)
+
3
(
12
)
=
72
Hence,
t
=
72
Maximum value of a second order determinant whose every element is either 0,1 or 2 only is:
Report Question
0%
0
0%
1
0%
2
0%
4
Explanation
So,
A
=
[
a
b
c
d
]
Given a,b,c & D can only be 0,1,2
det A= ad-bc
So for max. value of A,
a=2 and d=2 and
b
,
c
ϵ
0
,
0
So, Max value of det
A
=
[
2
0
0
2
]
=
4
If
|
c
o
s
(
A
+
B
)
−
s
i
n
(
A
+
B
)
c
o
s
2
B
s
i
n
A
c
o
s
A
s
i
n
B
−
c
o
s
A
s
i
n
A
c
o
s
B
|
=0 then B=
Report Question
0%
(2n+1)
π
2
0%
n
π
0%
(2n+1)
π
0%
2n
π
Explanation
⇒
c
o
s
(
A
+
B
)
(
c
o
s
A
c
o
s
B
−
s
i
n
A
s
i
n
B
)
+
s
i
n
2
(
A
+
B
)
+
c
o
s
2
B
+
0
⇒
c
o
s
2
(
A
+
B
)
+
s
i
n
2
(
A
+
B
)
+
c
o
s
2
B
=
0
⇒
c
o
s
(
2
B
)
=
−
1
2
B
=
(
2
n
+
1
)
π
B
=
(
2
n
+
1
)
π
/
2
so
(
2
n
+
1
)
π
/
2
If A
=
[
0
c
−
b
−
c
0
a
b
−
a
0
]
then
(
a
2
+
b
2
−
c
2
)
|
A
|
=
Report Question
0%
a
b
c
0%
a
+
b
+
c
0%
(
a
3
+
b
3
+
c
3
)
0%
0
Explanation
|
A
|
=
0
×
(
a
2
)
−
c
(
−
a
b
)
−
b
(
a
c
)
=
0
+
a
b
c
−
a
b
c
=
0
Find x if it is given that:
det
[
2
0
0
4
3
0
4
6
x
]
=
42
Report Question
0%
8
0%
7
0%
6
0%
21
/
4
Explanation
Given
d
e
t
(
2
0
0
4
3
0
4
6
x
)
=
42
By operation of matrix (5)
x
(
6
)
=
42
⇒
x
=
7
I
f
|
1
0
0
2
3
4
5
−
6
x
|
=
45
t
h
e
n
x
=
Report Question
0%
4
0%
7
0%
−
5
0%
−
7
Explanation
Given
|
1
0
0
2
3
4
5
−
6
x
|
=
45
By operation of matrix (5),
1
(
3
x
+
24
)
=
45
3
x
=
21
⇒
x
=
7
If A is a singular matrix, then A (adj A) is a
Report Question
0%
scalar matrix
0%
zero matrix
0%
identity matrix
0%
orthogonal matrix
Explanation
Given
A
is a singular matrix.
⇒
|
A
|
=
0
A
(
a
d
j
A
)
=
|
A
|
I
=
0
I
=
O
∴
A
(
a
d
j
A
)
is a zero matrix.
Hence, option B.
If A is a singular matrix, then adj A is
Report Question
0%
n
o
n
−
s
i
n
g
u
l
a
r
0%
singular
0%
symmetric
0%
not defined
Explanation
Given
|
A
|
=
0
We know
|
a
d
j
A
|
=
|
A
|
n
−
1
∴
|
a
d
j
A
|
=
0
Hence, adj A is singular
If the value of the determinant
|
m
2
−
5
7
|
is
31
, find
m
.
Report Question
0%
3
0%
8
0%
6
0%
2
Explanation
Given,
|
m
2
−
5
7
|
⇒
7
m
−
(
−
10
)
=
31
⇒
7
m
+
10
=
31
⇒
7
m
=
21
⇒
m
=
3
If
A
is a
3
×
3
matrix and
det
(
3
A
)
=
k
(
det
A
)
, then
k
=
Report Question
0%
9
0%
6
0%
1
0%
27
Explanation
The non zero determinant of a scalar multiple of a n×n matrix is given by the following property.
|
k
A
|
=
k
n
|
A
|
⟹
|
3
A
|
=
3
3
|
A
|
=
27
|
A
|
k
=
27
If A and B are similar matrices such that
d
e
t
(
A
B
)
=
0
,
then
Report Question
0%
d
e
t
(
A
)
=
0
and
d
e
t
(
B
)
=
0
0%
d
e
t
(
A
)
=
0
and
d
e
t
(
B
)
≠
0
0%
A
=
0
and
B
=
0
0%
A
=
0
or
B
=
0
Explanation
Given. two n
×
n square matrices A and B
Given,
B
=
P
−
1
A
P
d
e
t
(
B
)
=
d
e
t
(
P
−
1
A
P
)
d
e
t
(
B
)
=
d
e
t
(
P
−
1
A
P
)
d
e
t
(
B
)
=
(
d
e
t
(
P
)
)
−
1
(
d
e
t
(
A
)
)
(
d
e
t
(
P
)
)
d
e
t
(
B
)
=
(
d
e
t
(
P
)
)
−
1
(
d
e
t
(
A
)
)
(
d
e
t
(
P
)
)
d
e
t
(
B
)
=
d
e
t
(
A
)
d
e
t
(
A
B
)
=
(
d
e
t
(
A
)
)
(
d
e
t
(
B
)
)
If
(
d
e
t
(
A
)
)
=
0
then
(
d
e
t
(
B
)
)
also zero.
Let a, b, c be three complex numbers, and let
z
=
|
0
−
b
−
c
b
0
−
a
c
a
0
|
then z equal
Report Question
0%
0
0%
purely imaginary
0%
a
b
c
0%
none of these
Explanation
z
=
|
0
−
b
−
c
b
0
−
a
c
a
0
|
=
0
(
0
+
a
2
)
+
b
(
0
+
a
c
)
−
c
(
a
b
−
0
)
=
0
Two
n
×
n
square matrices A and B are said to be similar if there exists a non-singular matrix P such that
P
−
1
A
P
=
B
.
If A and B are similar matrices such that
d
e
t
(
A
)
=
1
, then
Report Question
0%
d
e
t
(
B
)
=
1
0%
d
e
t
(
A
)
=
−
d
e
t
(
B
)
0%
d
e
t
(
B
)
=
−
1
0%
none of these
Explanation
Given, two n
×
n square matrices A and B
Given,
B
=
P
−
1
A
P
d
e
t
(
B
)
=
d
e
t
(
P
−
1
A
P
)
d
e
t
(
B
)
=
(
d
e
t
(
P
−
1
)
)
(
d
e
t
(
A
)
)
(
d
e
t
(
P
)
)
d
e
t
(
B
)
=
(
d
e
t
(
P
)
)
−
1
(
d
e
t
(
A
)
)
(
d
e
t
(
P
)
)
[
∵
|
A
|
|
A
−
1
|
=
|
A
A
−
1
|
=
|
I
|
=
1
]
d
e
t
(
B
)
=
d
e
t
(
A
)
=
1
If A is a square matrix of order
n
then adj
(
a
d
j
A
)
is equal to
Report Question
0%
|
A
|
n
A
0%
|
A
|
n
−
1
A
0%
|
A
|
n
−
2
A
0%
|
A
|
n
−
3
A
Explanation
A
−
1
=
A
d
j
A
d
e
t
A
=
I
A
=
A
d
j
A
|
A
|
A
(
a
d
j
A
)
=
|
A
|
Replacing
A
=
a
d
j
A
a
d
j
A
(
A
d
j
A
d
j
A
)
=
|
A
d
j
A
|
⇒
A
d
j
(
A
d
j
A
)
=
|
A
d
j
(
A
)
|
|
A
|
×
A
=
|
A
|
n
−
1
|
A
|
×
A
=
|
A
|
n
−
2
×
A
A
d
j
(
A
d
j
A
)
=
|
A
|
n
−
2
×
2
n
→
order
Option C is correct
If
|
−
12
0
λ
0
2
−
1
2
1
15
|
=
−
360
, then the value of
λ
,is
Report Question
0%
−
1
0%
−
2
0%
−
3
0%
4
Explanation
|
−
12
0
λ
0
2
−
1
2
1
15
|
=
−
360
Expanding along first column
⇒
(
−
12
)
(
31
)
−
4
λ
=
−
360
⇒
λ
=
−
3
If
A
is any skew-symmetric matrix of odd order then
|
A
|
equals
Report Question
0%
−
1
0%
0
0%
1
0%
none of these
Explanation
if
A
is skew symmetric matrix
then
A
=
−
A
T
Therefore,
|
A
|
=
−
|
A
T
|
=
−
|
A
|
⇒
2
|
A
|
=
0
⇒
|
A
|
=
0
Ans: B
If
|
x
y
4
2
|
=
7
and
|
2
3
y
x
|
=
4
then
Report Question
0%
x
=
−
3
,
y
=
−
5
2
0%
x
=
−
5
2
,
y
=
−
3
0%
x
=
−
3
,
y
=
5
2
0%
x
=
−
5
2
,
y
=
3
Explanation
|
x
y
4
2
|
=
7
⇒
2
x
−
4
y
=
7
|
2
3
y
x
|
=
4
⇒
2
x
−
3
y
=
4
Therefore,
x
=
−
5
/
2
,
y
=
−
3
Ans: B
If
|
a
−
b
−
c
−
a
b
−
c
−
a
−
b
−
c
|
+
λ
a
b
c
=
0
, then
λ
is equal to
Report Question
0%
2
0%
4
0%
−
2
0%
−
4
Explanation
|
a
−
b
−
c
−
a
b
−
c
−
a
−
b
−
c
|
+
λ
a
b
c
=
0
Substitute
a
=
b
=
c
=
1
|
1
−
1
−
1
−
1
1
−
1
−
1
−
1
−
1
|
+
λ
=
0
(
−
2
−
2
)
+
λ
=
0
⇒
λ
=
4
The cofactors of elements in second row of the determinant
|
2
−
1
4
4
2
−
3
1
1
2
|
are
Report Question
0%
5
,
6
,
4
0%
6
,
0
,
−
3
0%
5
,
1
,
8
0%
6
,
0
,
3
Explanation
|
2
−
1
4
4
2
−
3
1
1
2
|
C
21
=
−
|
−
1
4
1
2
|
=
6
C
22
=
|
2
4
1
2
|
=
0
C
23
=
−
|
2
−
1
1
1
|
=
−
3
A set of points which do not lie on the same line are called as
Report Question
0%
collinear
0%
non-collinear
0%
concurrent
0%
square
Explanation
A set of points which do not lie on the same line are called as non collinear points
A
and
B
are two points and
C
is any point collinear with
A
and
B
. IF
A
B
=
10
,
B
C
=
5
, then
A
C
is equal to:
Report Question
0%
either
15
or
5
0%
necessarily
5
0%
necessarily
16
0%
none of these
Explanation
Since
C
is collinear with
A
and
B
,
C
lies either
(i) to the left of point B or
(ii) to the right of point B
∴
In case (i)
A
C
=
A
B
−
B
C
=
10
−
5
=
5
In case (ii)
A
C
=
A
B
+
B
C
=
10
+
5
=
15
If
A
=
[
3
−
5
−
1
0
]
, then adj.
A
is equal to
Report Question
0%
[
3
5
1
0
]
0%
[
0
−
5
−
1
3
]
0%
[
0
5
1
3
]
0%
[
0
−
1
−
5
3
]
Explanation
co-factor of
A
11
=
0
; co-factor of
A
12
=
5
; co-factor of
A
21
=
1
; co-factor of
A
22
=
3
a
d
j
A
=
[
0
5
1
3
]
Ans: C
The value of the determinant
|
a
b
0
0
a
b
b
0
a
|
is equal to
Report Question
0%
a
3
−
b
3
0%
a
3
+
b
3
0%
0
0%
none of these
Explanation
|
a
b
0
0
a
b
b
0
a
|
=
a
(
a
2
−
0
)
−
b
(
0
−
b
2
)
+
0
=
a
3
+
b
3
The value of the determinant
|
1
2
3
3
5
7
8
14
20
|
is equal to
Report Question
0%
20
0%
10
0%
0
0%
45
Explanation
The value of the determinant is
|
1
2
3
3
5
7
8
14
20
|
=
1
(
100
−
98
)
−
2
(
60
−
56
)
+
3
(
42
−
40
)
=
2
−
8
+
6
=
0
If
|
6
i
−
3
i
1
4
3
i
−
1
20
3
i
|
=
x
+
i
y
, then
Report Question
0%
x
=
3
,
y
=
1
0%
x
=
1
,
y
=
3
0%
x
=
0
,
y
=
3
0%
x
=
0
,
y
=
0
Explanation
|
6
i
−
3
i
1
4
3
i
−
1
20
3
i
|
=
x
+
i
y
⇒
6
i
(
3
i
2
+
3
)
+
3
i
(
4
i
+
20
)
+
1
(
12
−
60
i
)
=
x
+
i
y
⇒
0
=
x
+
i
y
Therefore,
x
=
y
=
0
Ans: D
The points which are not collinear are:
Report Question
0%
(
0
,
1
)
,
(
8
,
3
)
a
n
d
(
6
,
7
)
0%
(
4
,
3
)
,
(
5
,
1
)
a
n
d
(
1
,
9
)
0%
(
2
,
5
)
,
(
−
1
,
2
)
a
n
d
(
4
,
7
)
0%
(
−
3
,
2
)
(
1
,
−
2
)
a
n
d
(
9
,
−
10
)
Explanation
L
e
t
u
s
t
a
k
e
t
h
r
e
e
p
o
i
n
t
s
A
,
B
&
C
.
T
h
r
e
e
p
o
i
n
t
s
A
,
B
&
C
w
i
l
l
b
e
c
o
l
l
i
n
e
a
r
I
f
A
B
+
B
C
=
A
C
.
T
h
r
e
e
p
o
i
n
t
s
A
,
B
&
C
w
i
l
l
n
o
t
b
e
c
o
l
l
i
n
e
a
r
I
f
A
B
+
B
C
=
A
C
.
L
e
t
u
s
i
n
v
e
s
t
i
g
a
t
e
t
h
r
g
i
v
e
n
o
p
t
i
o
n
s
.
O
p
t
i
o
n
A
⟶
T
h
e
p
o
i
n
t
s
a
r
e
A
(
0
,
1
)
B
(
8
,
3
)
&
C
(
6
,
7
)
.
∴
A
B
=
√
(
0
−
8
)
2
+
(
1
−
3
)
2
u
n
i
t
s
=
√
68
=
8.25
u
n
i
t
s
,
B
C
=
√
(
8
−
6
)
2
+
(
3
−
7
)
2
u
n
i
t
s
=
√
28
=
5.29
u
n
i
t
s
,
a
n
d
A
C
=
√
(
0
−
6
)
2
+
(
1
−
7
)
2
u
n
i
t
s
=
√
72
=
8.49
u
n
i
t
s
.
∴
A
B
+
B
C
=
(
8.25
+
5.29
)
u
n
i
t
s
.
=
13.54
u
n
i
t
s
.
S
o
A
B
+
B
C
≠
A
C
.
∴
A
,
B
&
C
a
r
e
n
o
t
c
o
l
l
i
n
e
a
r
.
O
p
t
i
o
n
B
⟶
T
h
e
p
o
i
n
t
s
a
r
e
A
(
4
,
3
)
B
(
5
,
1
)
&
C
(
1
,
9
)
.
∴
A
B
=
√
(
4
−
5
)
2
+
(
3
−
1
)
2
u
n
i
t
s
=
√
5
=
2.24
u
n
i
t
s
,
B
C
=
√
(
5
−
1
)
2
+
(
1
−
9
)
2
u
n
i
t
s
=
√
80
=
8.94
u
n
i
t
s
,
a
n
d
A
C
=
√
(
4
−
1
)
2
+
(
3
−
9
)
2
u
n
i
t
s
=
√
45
=
6.71
u
n
i
t
s
.
∴
A
B
+
A
C
=
(
2.24
+
6.71
)
u
n
i
t
s
=
8.95
u
n
i
t
s
.
S
o
A
B
+
A
C
=
B
C
.
∴
A
,
B
&
C
a
r
e
c
o
l
l
i
n
e
a
r
.
O
p
t
i
o
n
C
⟶
T
h
e
p
o
i
n
t
s
a
r
e
A
(
2
,
5
)
B
(
−
1
,
2
)
&
C
(
4
,
7
)
.
∴
A
B
=
√
(
2
+
1
)
2
+
(
5
−
2
)
2
u
n
i
t
s
=
√
18
u
n
i
t
s
=
4.24
u
n
i
t
s
,
B
C
=
√
(
−
1
−
4
)
2
+
(
2
−
7
)
2
u
n
i
t
s
=
√
50
u
n
i
t
s
=
7.07
u
n
i
t
s
,
a
n
d
A
C
=
√
(
2
−
4
)
2
+
(
5
−
7
)
2
u
n
i
t
s
=
√
8
u
n
i
t
s
=
2.83
u
n
i
t
s
.
∴
A
B
+
A
C
=
(
4.24
+
2.83
)
u
n
i
t
s
=
7.07
u
n
i
t
s
.
S
o
A
B
+
A
C
=
B
C
.
∴
A
,
B
&
C
a
r
e
c
o
l
l
i
n
e
a
r
.
O
p
t
i
o
n
C
⟶
T
h
e
p
o
i
n
t
s
a
r
e
A
(
−
3
,
2
)
B
(
1
,
−
2
)
&
C
(
9
,
−
10
)
.
∴
A
B
=
√
(
−
3
−
1
)
2
+
(
2
+
2
)
2
u
n
i
t
s
=
√
32
u
n
i
t
s
=
5.66
u
n
i
t
s
,
B
C
=
√
(
1
−
9
)
2
+
(
−
2
+
10
)
2
u
n
i
t
s
=
√
128
u
n
i
t
s
=
11.31
u
n
i
t
s
,
a
n
d
A
C
=
√
(
−
3
−
9
)
2
+
(
2
+
10
)
2
u
n
i
t
s
=
√
288
u
n
i
t
s
=
16.97
u
n
i
t
s
.
∴
A
B
+
B
C
=
(
5.66
+
11.31
)
u
n
i
t
s
=
16.97
u
n
i
t
s
.
S
o
A
B
+
B
C
=
A
C
.
∴
A
,
B
&
C
a
r
e
c
o
l
l
i
n
e
a
r
.
A
n
s
−
O
p
t
i
o
n
A
.
If
A
=
[
a
b
b
a
]
, then
|
A
+
A
T
|
equals
Report Question
0%
4
(
a
2
−
b
2
)
0%
2
(
a
2
−
b
2
0%
a
2
−
b
2
)
0%
4
a
b
Explanation
A
+
A
T
=
[
a
b
b
a
]
[
a
b
b
a
]
=
[
2
a
2
b
2
b
2
a
]
|
A
+
A
T
|
=
4
a
2
−
4
b
2
Ans: A
If
Δ
1
=
|
1
0
a
b
|
and
Δ
2
=
|
1
0
c
d
|
then
Δ
2
Δ
1
is equal to
Report Question
0%
a
c
0%
b
d
0%
(
b
−
a
)
(
d
−
c
)
0%
none of these
Explanation
Δ
1
=
|
1
0
a
b
|
=
1.
b
−
0.
a
=
b
Δ
2
=
|
1
0
c
d
|
=
1.
d
−
0.
c
=
d
then
Δ
2
Δ
1
=
b
d
Ans: B
If
A
and
B
are two matrices of same order
3
×
3
, where
A
=
[
1
2
3
2
3
4
5
6
8
]
and
B
=
[
3
2
5
2
3
8
7
2
9
]
The value of
Adj
(
Adj
A
)
is equal to
Report Question
0%
−
A
0%
A
0%
8
A
0%
16
A
Explanation
|
A
|
=
1
(
3
×
8
−
4
×
6
)
−
2
(
2
×
8
−
5
×
4
)
+
3
(
2
×
6
−
5
×
3
)
=
−
1
Use following results
Adj
(
Adj
A
)
=
|
A
|
n
−
2
A
⇒
Adj
(
Adj
A
)
=
|
A
|
n
−
2
A
=
(
−
1
)
1
A
=
−
A
Ans:
A
If
ω
is a cube root of unity and
Δ
=
|
1
2
ω
ω
ω
2
|
, then
Δ
2
is equal to
Report Question
0%
−
ω
0%
ω
0%
1
0%
ω
2
Explanation
Δ
=
ω
2
−
2
ω
2
=
−
ω
2
Δ
2
=
ω
4
=
ω
Ans: B
If
|
x
2
18
x
|
=
|
6
2
3
x
6
|
, then
x
is equal to
Report Question
0%
6
0%
±
6
0%
−
6
0%
0
Explanation
|
x
2
18
x
|
=
|
6
2
3
x
6
|
⇒
x
2
−
36
=
36
−
6
x
⇒
x
2
+
6
x
−
72
=
0
⇒
x
=
−
6
±
√
6
2
+
4
×
72
2
=
−
6
±
√
36
(
1
+
8
)
2
=
−
3
±
9
=
6
o
r
−
12
Which of the following is correct?
Report Question
0%
Determinant is a square matrix
0%
Determinant is a number associated to a matrix
0%
Determinant is a number associated to a square matrix
0%
None of these
Explanation
Determinant is defined only for a square matrix.
and its denotes the value of that square matrix.
If
|
A
|
≠
0
, then
A
is
Report Question
0%
zero matrix
0%
singular matrix
0%
non - singular matrix
0%
diagonal matrix
Explanation
If
|
A
|
≠
0
then by definition,
A
is non-singular matrix
A
=
[
1
2
3
4
]
and
A
(
adj
A
)
=
K
I
, then the value of
K
is:
Report Question
0%
2
0%
−
2
0%
10
0%
−
10
Explanation
Given,
A
=
[
1
2
3
4
]
Adjoint of
A
=
[
4
−
2
−
3
1
]
∴
A
(
adj
A
)
=
K
I
[
1
2
3
4
]
[
4
2
−
3
1
]
=
K
[
1
0
0
1
]
[
−
2
0
0
−
2
]
=
[
K
0
0
K
]
∴
K
=
−
2
What is the determinant of the matrix
[
3
6
−
1
2
]
?
Report Question
0%
0
0%
12
0%
|
0
|
0%
|
6
|
Explanation
Given,
|
3
6
−
1
2
|
Let determinent be
|
d
|
Value of
|
d
|
will be
|
d
|
=
3
×
2
−
(
6
×
−
1
)
=
6
+
6
=
12
If
A
=
[
1
log
b
a
log
a
b
1
]
then
|
A
|
is equal to
Report Question
0%
0
0%
log
a
b
0%
−
1
0%
log
b
a
Explanation
On solving the given matrix,
|
A
|
=
1
−
log
a
b
.
log
b
a
=
1
−
1
=
0
0:0:1
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
0
Answered
1
Not Answered
49
Not Visited
Correct : 0
Incorrect : 0
Report Question
×
What's an issue?
Question is wrong
Answer is wrong
Other Reason
Want to elaborate a bit more? (optional)
Practice Class 12 Commerce Maths Quiz Questions and Answers
<
>
Support mcqexams.com by disabling your adblocker.
×
Please disable the adBlock and continue.
Thank you.
Reload page