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CBSE Questions for Class 12 Commerce Maths Determinants Quiz 9 - MCQExams.com
CBSE
Class 12 Commerce Maths
Determinants
Quiz 9
|
a
b
−
c
c
+
b
a
+
c
b
c
−
a
a
−
b
b
+
a
c
|
=
Report Question
0%
a
b
c
(
a
2
+
b
2
+
c
2
)
0%
a
b
c
(
a
+
b
+
c
)
0%
a
+
b
+
c
(
a
2
+
b
2
+
c
2
)
0%
a
b
c
(
1
+
1
a
+
1
b
+
1
c
)
If
A
=
[
1
2
2
1
]
then
a
d
j
(
A
)
=
?
Report Question
0%
[
1
−
2
−
2
1
]
0%
[
2
1
1
1
]
0%
[
1
−
2
−
2
−
1
]
0%
[
−
1
2
2
−
1
]
Explanation
Given,
A
=
[
1
2
2
1
]
The adjoin of a square matrix of order 2 is obtained by interchanging the diagonal elements & changing signs of off diagonal elements.
adj (A) =
A
=
[
1
−
2
−
2
1
]
To solve
x
+
y
=
3
:
3
x
−
2
y
−
4
=
0
by determinant method find
D
.
Report Question
0%
5
0%
1
0%
−
5
0%
−
1
Explanation
a
1
x
+
b
1
y
−
c
1
=
0
⇒
a
1
x
+
b
1
y
=
c
1
a
2
x
+
b
2
y
−
c
2
=
0
⇒
a
2
x
+
b
2
y
=
c
2
then the solution of
x
and
y
can be obtained by evaluating the following integral :
x
=
|
c
1
c
2
b
1
b
2
|
|
a
1
a
2
b
1
b
2
|
and
y
=
|
a
1
a
2
c
1
c
2
|
|
a
1
a
2
b
1
b
2
|
∴
x
+
y
=
3
3
x
−
2
y
=
4
can be solved using the above method
x
=
|
3
4
1
−
2
|
|
1
3
1
−
2
|
;
y
=
|
1
3
3
4
|
|
1
3
1
−
2
|
x
=
−
6
−
4
−
2
−
3
;
y
=
4
−
9
−
5
x
=
10
−
5
;
y
=
−
5
−
5
x
=
2
;
y
=
1
now the quantity
|
1
3
1
−
2
|
=
D
(determinant)
D
=
|
1
3
1
−
2
|
=
−
5
So, answer is option C.
The number of distinct real roots of the equation,
|
c
o
s
x
s
i
n
x
s
i
n
x
s
i
n
x
c
o
s
x
s
i
n
x
s
i
n
x
s
i
n
x
c
o
s
x
|
=
0
In t interval
[
−
π
4
π
4
]
is/are:
Report Question
0%
3
0%
2
0%
1
0%
4
If the point
(
λ
+
1
,
1
)
,
(
2
λ
+
1
,
3
)
and
(
2
λ
+
2
,
2
λ
)
are collinear then the possible value of
λ
is
Report Question
0%
2
0%
1
/
2
0%
3
0%
1
/
3
Solve
|
1
1
1
1
1
+
x
1
1
1
1
+
y
|
Report Question
0%
1
0%
0
0%
x
0%
x
y
Explanation
Let
A
=
|
1
1
1
1
1
+
x
1
1
1
1
+
y
|
R
1
→
R
1
−
R
3
R
2
→
R
2
−
R
3
A
=
|
0
0
−
y
0
x
−
y
1
1
1
+
y
|
=
−
y
(
0
−
x
)
=
x
y
If
(
k
,
2
−
2
k
)
,
(
−
k
+
1
,
2
k
)
,
(
−
4
−
k
,
6
−
2
k
)
are collinear, then
k
=
Report Question
0%
+
1
0%
−
1
0%
−
2
0%
2
Explanation
w
h
e
r
e
m
o
r
e
t
h
a
n
t
h
r
e
e
p
o
i
n
t
s
a
r
e
c
o
l
l
i
n
e
a
r
s
l
o
p
e
o
f
t
h
e
l
i
n
e
a
r
e
s
a
m
e
S
o
,
s
l
o
p
e
:
[
(
k
,
2
−
2
k
)
,
(
−
k
+
1
,
2
k
)
]
=
[
(
−
k
+
1
,
2
k
)
,
(
−
4
−
k
,
6
−
2
k
)
]
⇒
2
−
2
k
−
2
k
k
+
k
−
1
=
2
k
−
6
+
2
k
−
k
+
1
+
4
+
k
⇒
2
−
4
k
2
k
−
1
=
4
x
−
6
5
⇒
(
1
−
2
k
)
5
=
(
2
k
−
3
)
(
2
k
−
1
)
5
−
10
k
=
4
k
2
−
2
k
−
6
k
+
3
4
k
2
+
2
k
−
2
=
0
2
k
2
+
k
−
1
=
0
2
k
2
+
2
k
−
k
−
1
=
0
2
k
(
k
+
1
)
−
1
(
k
+
1
)
=
0
(
2
k
−
1
)
(
k
+
1
)
=
0
k
=
−
1
o
r
1
2
Hence,option
B
is the correct answer.
If
D
P
=
|
P
15
8
P
2
35
9
P
3
25
10
|
,
then
D
1
+
D
2
+
D
3
+
D
4
+
D
5
is equal to -
Report Question
0%
−
29000
0%
−
25000
0%
25000
0%
none of these
Explanation
Given
D
p
=
|
p
15
8
p
2
35
9
p
3
25
10
|
=
p
(
350
−
225
)
−
15
(
10
p
2
−
9
p
3
)
+
8
(
25
p
2
−
35
p
3
)
D
1
=
1
(
350
−
225
)
−
15
(
10
(
1
)
2
−
9
(
1
)
3
)
+
8
(
25
(
1
)
2
−
35
(
1
)
3
)
=
125
−
15.
(
1
)
+
8
(
−
10
)
=
125
−
80
−
15
=
125
−
95
=
30
D
2
=
2
(
350
−
225
)
−
15
(
10
(
2
)
2
−
9
(
2
)
3
)
+
8
(
25
(
2
)
2
−
35
(
2
)
3
)
=
2.
(
125
)
−
15
(
40
−
72
)
+
8
(
100
−
280
)
=
250
+
15
(
32
)
−
1440
=
730
−
1440
=
−
710
D
3
=
3
(
350
−
225
)
−
15
(
10
(
3
)
2
−
9
(
3
)
3
)
+
8
(
25
(
3
)
2
−
35
(
3
)
3
)
=
3.
(
125
)
−
15
(
90
−
243
)
+
8
(
225
−
945
)
=
375
+
15
(
153
)
−
8
(
720
)
=
375
+
2
,
295
−
5760
=
−
3090
.
D
4
=
4
(
350
−
225
)
−
15
(
10
(
4
)
2
−
9
(
4
)
3
)
+
8
(
25
(
4
)
2
−
35
(
4
)
3
)
=
4.
(
125
)
−
15
(
160
−
576
)
+
8
(
400
−
2240
)
=
500
+
6240
−
8.
(
1840
)
=
5740
−
14720
=
−
8980
D
5
=
5
(
350
−
225
)
−
15
(
10
(
5
)
2
−
9
(
5
)
3
)
+
8
(
25
(
5
)
2
−
35
(
5
)
3
)
=
5
(
125
)
−
15
(
250
−
1125
)
+
8
(
625
−
4375
)
=
625
+
13125
+
(
−
30
,
000
)
=
−
16250
.
∴
D
1
+
D
2
+
D
3
+
D
4
+
D
5
=
30
−
710
−
3090
−
8980
−
16250
=
−
29000
.
The value of determinant
|
b
c
−
a
2
a
c
−
b
2
a
b
−
c
2
a
c
−
b
2
a
b
−
c
2
b
c
−
a
2
a
b
−
c
2
b
c
−
a
2
a
c
−
b
2
|
is
Report Question
0%
always non-negative
0%
always non-positive
0%
always zero
0%
can't say anything
If
A
=
[
2
−
3
−
4
1
]
, then
adj
(
3
A
2
+
12
A
)
is equal to
Report Question
0%
[
51
63
84
72
]
0%
[
51
84
63
72
]
0%
[
72
−
63
−
84
51
]
0%
[
72
−
84
−
63
51
]
Explanation
A
=
[
2
−
3
−
4
1
]
then
A
2
=
[
2
−
3
−
4
1
]
[
2
−
3
−
4
1
]
A
2
=
[
16
−
9
−
12
13
]
3
A
2
=
[
48
−
27
−
36
39
]
3
A
2
+
12
A
=
[
48
−
27
−
36
39
]
+
[
24
−
36
−
48
12
]
=
[
72
−
63
−
84
51
]
a
d
j
(
3
A
2
+
12
A
)
=
[
51
63
84
72
]
If a,b,c are distinct and
|
a
a
2
a
3
−
1
b
b
2
b
3
−
1
c
c
2
c
3
−
1
|
=
0
then
Report Question
0%
a
+
b
+
c
=
1
0%
a
b
+
b
c
+
c
a
=
0
0%
a
+
b
+
c
=
0
0%
a
b
c
=
1
Explanation
|
a
a
2
a
3
−
1
b
b
2
b
3
−
1
c
c
2
c
3
−
1
|
=
0
⇒
a
(
b
2
c
3
−
b
2
−
c
2
b
3
+
c
2
)
−
a
2
(
b
c
3
−
b
−
c
b
3
+
c
)
+
(
a
3
−
1
)
(
b
c
2
−
c
b
2
)
=
0
⇒
a
(
b
2
c
2
(
c
−
b
)
+
c
2
−
b
2
)
−
a
2
(
b
c
(
c
2
−
b
2
)
+
c
−
b
)
+
(
a
3
−
1
)
b
c
(
c
−
b
)
=
0
⇒
a
(
b
2
c
2
(
c
−
b
)
+
(
c
−
b
)
(
c
+
b
)
)
−
a
2
(
b
c
(
c
−
b
)
(
c
+
b
)
+
(
c
−
b
)
)
+
b
c
(
a
3
−
1
)
(
c
−
b
)
=
0
⇒
a
(
c
−
b
)
(
b
2
c
2
+
c
+
b
)
−
a
2
(
c
−
b
)
(
b
c
(
c
+
b
)
+
1
)
+
b
c
(
a
3
−
1
)
(
c
−
b
)
=
0
⇒
(
c
−
b
)
{
a
(
b
2
c
2
+
c
+
b
)
−
a
2
(
b
c
(
c
+
b
)
+
1
)
+
b
c
(
a
3
−
1
)
}
=
0
⇒
(
c
−
b
)
{
a
b
2
c
2
+
a
c
+
a
b
−
a
2
b
c
2
−
a
2
b
2
c
−
a
2
+
a
3
b
c
−
b
c
}
=
0
⇒
(
c
−
b
)
{
a
b
c
(
b
c
−
a
c
−
a
b
+
b
2
)
−
(
b
c
−
a
c
−
a
b
+
a
2
)
}
=
0
⇒
(
c
−
b
)
(
b
c
−
a
c
−
a
b
+
a
2
)
(
a
b
c
−
1
)
=
0
⇒
(
c
−
b
)
(
b
(
c
−
a
)
−
a
(
c
−
a
)
)
(
a
b
c
−
1
)
=
0
⇒
(
c
−
b
)
(
c
−
a
)
(
b
−
a
)
(
a
b
c
−
1
)
=
0
⇒
c
−
b
=
0
or
c
−
a
=
0
or
b
−
a
=
0
⇒
c
=
b
c
=
a
b
=
a
which is not possible since
a
,
b
,
c
are distinct.
∴
a
b
c
−
1
=
0
⇒
a
b
c
=
1
If
[
1
α
3
1
3
3
2
4
4
]
is the adjoint of a
3
×
3
matrix
A
and
|
A
|
=
4
, then
α
is equal to:
Report Question
0%
4
0%
11
0%
5
0%
0
Explanation
|
1
α
3
1
3
3
2
4
4
|
=
1
(
3
×
4
−
3
×
4
)
−
α
(
1
×
4
−
3
×
2
)
+
3
(
1
×
4
−
3
×
2
)
=
2
α
−
6
|
1
α
3
1
3
3
2
4
4
|
=
|
a
d
j
.
A
|
⇒
(
2
α
−
6
)
=
|
a
d
j
.
A
|
Now,
|
a
d
j
.
A
|
=
|
A
|
3
−
1
=
4
2
=
16
∴
2
α
−
6
=
16
⇒
2
α
=
16
+
6
⇒
α
=
22
2
=
11
If f(x), g(x), h(x) are polynomials in x of degree 2 and F(x)=
|
f
g
h
f
′
g
′
h
′
f
"
g
"
h
"
|
,
, then F(x) is equal to
Report Question
0%
1
0%
0
0%
-1
0%
f
(
x
)
.
g
(
x
)
.
h
(
x
)
Explanation
L
e
t
,
f
(
x
)
=
a
x
2
+
b
x
+
c
f
′
(
x
)
=
2
a
x
+
b
f
″
(
x
)
=
2
a
g
(
x
)
=
a
1
x
2
+
b
1
x
+
c
1
g
′
(
x
)
=
2
a
1
x
+
b
1
g
″
(
x
)
=
2
a
1
a
n
d
,
h
(
x
)
=
a
2
x
2
+
b
2
x
+
c
2
h
′
(
x
)
=
2
a
2
x
+
b
2
f
(
x
)
=
|
a
x
2
+
b
x
+
c
a
1
x
2
+
b
1
x
+
c
1
a
2
x
2
+
b
2
x
+
c
2
2
a
x
+
b
2
a
1
x
+
b
1
2
a
2
x
+
b
2
2
a
2
a
1
2
a
2
|
f
′
(
x
)
=
|
2
a
x
+
b
2
a
1
x
+
b
1
2
a
2
x
+
b
2
2
a
x
+
b
2
a
1
x
+
b
1
2
a
2
x
+
b
2
2
a
2
a
1
2
a
2
|
+
0
+
0
=
0
Hence, the option
B
is the correct answer.
If
|
1
sin
x
sin
2
x
1
cos
x
cos
2
x
1
tan
x
tan
2
x
|
=
0
,
x
∈
[
0
,
2
π
]
,
then number of possible values of
x
is.
Report Question
0%
4
0%
5
0%
6
0%
None of these
Explanation
|
1
1
1
s
i
n
x
c
o
s
x
t
a
n
x
sin
2
x
cos
2
x
tan
2
x
|
=
0
(
R
1
→
R
1
−
R
2
R
2
→
R
2
−
R
3
)
⇒
|
0
0
1
s
i
n
x
−
c
o
s
x
c
o
s
x
−
t
a
n
x
t
a
n
x
sin
2
x
−
cos
2
x
cos
2
x
−
tan
2
x
tan
2
x
|
=
0
⇒
(
s
i
n
x
−
c
o
s
x
)
(
c
o
s
x
−
t
a
n
x
)
|
0
0
1
1
1
t
a
n
x
s
i
n
x
+
c
o
s
x
c
o
s
x
+
t
a
n
x
tan
2
x
|
=
0
Expanding along colomn I we get
⇒
(
c
o
s
x
−
t
a
n
x
)
(
s
i
n
x
−
c
o
s
x
)
(
c
o
s
x
+
t
a
n
x
−
s
i
n
x
−
c
o
s
x
)
=
0
⇒
(
c
o
s
x
−
t
a
n
x
)
(
s
i
n
x
−
c
o
s
x
)
(
t
a
n
x
−
s
i
n
x
)
=
0
∴
c
o
s
x
−
t
a
n
x
=
0
or
s
i
n
x
=
c
o
s
x
or
t
a
n
x
=
s
i
n
x
or
c
o
s
x
=
t
a
n
x
;
s
i
n
x
=
c
o
s
x
;
t
a
n
x
=
s
i
n
x
for
x
ϵ
[
0
,
2
π
]
c
s
o
x
=
t
a
n
x
from graph we see there are
2
solutions.
for
x
ϵ
[
0
,
2
π
]
s
i
n
x
=
c
o
s
x
from graph we have two solutions
for
x
ϵ
[
0
,
2
π
]
from graph we have solutions
So a total of
2
+
2
+
3
=
7
solutions
i.e Answer : D.
If
|
x
−
4
2
x
2
x
2
x
x
−
4
2
x
2
x
2
x
x
−
4
|
=
(
A
+
B
x
)
(
x
−
A
)
2
,
then the ordered pair
(
A
,
B
)
is equal to:
Report Question
0%
( -4 , -5 )
0%
( -4 , 3 )
0%
( -4 , 5 )
0%
( 4 , 5 )
Explanation
|
x
−
4
2
x
2
x
2
x
x
−
4
2
x
2
x
2
x
x
−
4
|
=
(
x
−
4
)
[
(
x
−
4
)
2
−
4
x
2
]
−
2
x
[
2
x
(
x
−
4
)
−
4
x
2
]
+
2
x
[
4
x
2
−
2
x
(
x
−
4
)
]
=
(
x
−
4
)
[
x
2
+
16
−
8
x
−
4
x
2
]
−
2
x
[
2
x
2
−
8
x
−
4
x
2
]
+
2
x
[
4
x
2
−
2
x
2
+
8
x
]
=
(
x
−
4
)
[
16
−
8
x
−
3
x
2
]
−
2
x
[
−
8
x
−
2
x
2
]
+
2
x
[
2
x
2
+
8
x
]
=
(
x
−
4
)
(
16
−
8
x
−
3
x
2
)
+
4
x
[
2
x
2
+
8
x
]
=
16
x
−
8
x
2
−
3
x
3
−
64
+
32
x
+
12
x
2
+
8
x
3
+
32
x
2
=
5
x
3
+
36
x
2
+
48
x
−
64
(
A
+
B
x
)
(
x
−
A
)
2
=
(
A
+
B
x
)
(
x
2
+
A
2
−
2
x
A
)
=
A
x
2
+
A
3
−
2
x
A
2
+
B
x
3
+
B
A
2
x
−
2
A
B
x
2
=
B
x
3
+
(
A
−
2
A
B
)
x
2
+
(
B
A
2
−
2
x
)
x
+
A
3
B
=
5
A
−
2
A
B
=
36
A
(
1
−
2
B
)
=
36
A
(
1
−
10
)
=
36
A
=
−
4
∴
(
A
,
B
)
=
(
−
4
,
5
)
.
If
A
=
|
10
2
30
6
|
then
|
A
|
=
Report Question
0%
0
0%
10
0%
12
0%
60
Explanation
A
=
|
10
2
30
6
|
=
10
×
6
−
30
×
2
=
60
−
60
=
0
If
|
(
a
d
j
A
)
|
=
81
,
for
3
×
3
matrix, then det
A
is equal to
Report Question
0%
1
0%
2
0%
4
0%
9
Explanation
Given for a square matrix
A
of order
3
.
|
a
d
j
A
|
=
|
A
|
n
−
1
, where
n
is the order of matrix.
We have,
|
a
d
j
(
A
)
|
=
|
A
|
(
3
−
1
)
or,
|
A
|
2
=
81
or,
|
A
|
=
9
. [ Taking the positive value]
For what value of x, will the points (-1,x),(-3,2) and (-4,4) lie on a line?
Report Question
0%
-3
0%
3
0%
-2
0%
2
Explanation
A
(
−
1
,
x
)
,
B
(
−
3
,
2
)
,
C
(
−
4
,
4
)
lie on a line
∴
1
2
|
x
1
(
y
2
−
y
3
)
+
x
2
(
y
3
−
y
1
)
+
x
3
(
y
1
−
y
2
)
|
=
0
1
2
|
−
1
(
2
−
4
)
+
−
3
(
4
−
x
)
+
−
4
(
x
−
2
)
|
=
0
1
2
|
−
1
(
−
2
)
−
3
(
4
−
x
)
−
4
(
x
−
2
)
|
=
0
1
2
|
2
−
3
×
4
+
3
x
−
4
x
+
4
×
2
|
=
0
1
2
|
2
−
12
+
3
x
−
4
x
+
8
|
=
0
1
2
|
−
2
−
x
|
=
0
−
2
−
x
=
0
x
=
−
2
.
|
1
a
a
2
b
c
1
b
b
2
c
a
1
c
c
2
a
b
|
=
Report Question
0%
abc
0%
a+b+c
0%
0
0%
4abc
Explanation
Given,
|
1
a
a
2
b
c
1
b
b
2
c
a
1
c
c
2
a
b
|
=
1
a
det
(
b
2
c
a
c
2
a
b
)
−
a
2
det
(
1
b
c
a
1
c
a
b
)
+
b
c
det
(
1
b
b
2
1
c
c
2
)
=
1
a
(
a
b
3
−
a
c
3
)
−
a
2
⋅
0
+
b
c
(
c
2
b
−
b
2
c
)
=
b
3
−
c
3
−
0
+
b
c
(
c
2
b
−
b
2
c
)
=
b
3
+
c
3
−
b
3
−
c
3
=
0
The point
(
x
1
,
y
1
)
,
(
x
2
,
y
2
)
,
(
x
1
,
y
2
)
&
(
x
2
,
y
1
)
are always
Report Question
0%
Collinear
0%
Concyclic
0%
Vertices of a square
0%
Vertices of a rhombus.
Explanation
Let the coordinates be denoted as
A
(
x
1
,
y
2
)
,
B
(
x
2
,
y
2
)
,
C
(
x
2
,
y
1
)
and
D
(
x
1
,
y
1
)
Plot the given points on a graph as above,
It is not necessary that
|
x
2
−
x
1
|
=
|
y
2
−
y
1
|
With
(
x
2
,
y
1
)
and
(
x
1
,
y
2
)
as ends of diameter
∠
A
B
C
=
90
∘
and
∠
A
D
C
=
90
∘
∴
A
B
C
D
are concyclic.
So,
B
is the correct option.
Find the value of
|
5
3
−
7
−
4
|
Report Question
0%
-1
0%
-41
0%
41
0%
1
Explanation
|
5
3
−
7
−
4
|
=
5
×
(
−
4
)
−
3
×
(
−
7
)
=
−
20
+
21
=
1
Hence, the correct answer is 1.
If
a
,
b
,
c
are
p
t
h
,
q
t
h
and
r
t
h
terms of a GP, then
|
log
a
p
1
log
b
q
1
log
c
r
1
|
is equal to
Report Question
0%
0
0%
1
0%
log
a
b
c
0%
none of these
Explanation
It is given that the given terms are in G.P
∴
p
t
h
term
=
a
,
q
t
h
term
=
b
and
r
t
h
term
=
c
⇒
a
=
A
R
p
−
1
⇒
log
a
=
log
A
(
R
)
p
−
1
=
log
A
+
(
p
−
1
)
log
R
⇒
b
=
A
R
q
−
1
⇒
log
b
=
log
A
(
R
)
q
−
1
=
log
A
+
(
q
−
1
)
log
R
⇒
c
=
A
R
q
−
1
⇒
log
c
=
log
A
(
R
)
r
−
1
=
log
A
+
(
r
−
1
)
log
R
Now,
Δ
=
|
log
a
p
1
log
b
q
1
log
c
r
1
|
=
|
log
A
+
(
p
−
1
)
log
R
p
1
log
A
+
(
q
−
1
)
log
R
q
1
log
A
+
(
r
−
1
)
log
R
r
1
|
C
2
→
C
2
−
C
3
=
|
log
A
+
(
p
−
1
)
log
R
p
−
1
1
log
A
+
(
q
−
1
)
log
R
q
−
1
1
log
A
+
(
r
−
1
)
log
R
r
−
1
1
|
C
1
→
C
1
−
(
log
A
)
C
3
−
(
log
R
)
C
2
=
|
0
p
−
1
1
0
q
−
1
1
0
r
−
1
1
|
=
0
|
A
|
=
6
and
A
is
3
×
3
matrix then
d
e
t
(
2
a
d
j
(
2
(
A
−
1
)
T
)
)
=
Report Question
0%
162
0%
36
0%
216
0%
512
If
Δ
(
x
)
=
|
e
x
sin
2
x
tan
x
2
l
n
(
1
+
x
)
cos
x
sin
x
cos
x
2
e
x
−
1
sin
x
2
|
=
A
+
B
x
+
C
x
2
+
.
.
.
.
.
then B is equal to
Report Question
0%
0
0%
1
0%
2
0%
4
If
A
=
[
5
5
x
x
0
x
5
x
0
0
5
]
and
|
A
2
|
=
25
, then
|
x
|
is equal to?
Report Question
0%
1
5
0%
5
0%
5
2
0%
1
Explanation
Given
A
=
[
5
5
x
x
0
x
5
x
0
0
5
]
and
|
A
2
|
=
25
|
A
2
|
=
|
A
|
2
=
25
|
A
|
=
5
[
5
x
−
0
]
+
0
=
25
x
⇒
(
25
x
)
2
=
25
⇒
x
2
=
1
25
⇒
|
x
|
=
1
5
.
If a determinant of order
3
×
3
is formed by using the numbers
1
or
−
1
, then the minimum value of the determinant is?
Report Question
0%
−
2
0%
−
4
0%
0
0%
−
8
If
α
,
β
are the roots of
x
2
+
x
+
1
=
0
then
|
y
+
1
β
α
β
y
+
α
1
α
1
y
+
β
|
=
?
Report Question
0%
y
2
−
1
0%
y
(
y
2
−
1
)
0%
y
2
−
y
0%
y
3
|
sin
2
θ
cos
2
θ
−
cos
2
θ
sin
2
θ
|
=
Report Question
0%
cos
2
θ
0%
1
2
(
1
+
cos
2
2
θ
)
0%
1
2
(
1
−
sin
2
2
θ
)
0%
1
2
sin
2
2
θ
Explanation
we know that
(
s
i
n
2
x
+
c
o
s
2
x
)
2
=
s
i
n
4
x
+
c
o
s
4
x
+
2
s
i
n
2
x
c
o
s
2
x
|
sin
2
θ
cos
2
θ
−
cos
2
θ
sin
2
θ
|
=
sin
4
θ
+
cos
4
θ
=
1
−
2
sin
2
θ
cos
2
θ
=
1
−
1
2
sin
2
2
θ
=
1
−
1
2
(
1
−
cos
2
2
θ
)
=
1
2
(
1
+
cos
2
2
θ
)
The sum of the real roots of the equation
|
x
−
6
−
1
2
−
3
x
x
−
3
−
3
2
x
x
+
2
|
=
0
is equal to
Report Question
0%
6
0%
1
0%
0
0%
−
4
Explanation
Given
|
x
−
6
−
1
2
−
3
x
x
−
3
−
3
2
x
x
+
2
|
=
0
By expansion, we get
x
(
−
3
x
2
+
6
x
)
−
(
−
6
)
(
2
x
+
4
−
3
x
+
3
)
+
(
−
1
)
(
4
x
+
9
x
)
⇒
−
5
x
3
+
30
x
−
30
+
5
x
=
0
⇒
−
5
x
3
+
35
x
−
30
=
0
⇒
x
3
−
7
x
+
6
=
0
⇒
(
x
−
1
)
(
x
−
2
)
(
x
−
3
)
⇒
x
=
1
,
2
,
−
3
Therefore, All roots are real
So, sum of roots
=
0
|
a
+
i
b
c
+
i
d
−
c
+
i
d
a
−
i
b
|
=
?
Report Question
0%
(
a
2
+
b
2
−
c
2
−
d
2
)
0%
(
a
2
−
b
2
+
c
2
−
d
2
)
0%
(
a
2
+
b
2
+
c
2
+
d
2
)
0%
n
o
n
e
o
f
t
h
e
s
e
For square matrices
A
and
B
of the same order, we have
a
d
j
(
A
B
)
=
?
Report Question
0%
(
a
d
j
A
)
(
a
d
j
B
)
0%
(
a
d
j
B
)
(
a
d
j
A
)
0%
|
A
B
|
0%
None of these
|
cos
70
o
sin
20
o
sin
70
o
cos
20
o
|
=
?
Report Question
0%
1
0%
0
0%
cos
50
o
0%
sin
50
o
Explanation
Let us consider
Δ
=
|
cos
70
o
sin
20
o
sin
70
o
cos
20
o
|
Δ
=
cos
70
0
cos
20
0
−
sin
70
0
sin
20
0
=
cos
(
70
0
+
20
0
)
=
cos
90
0
Δ
=
0
Option
B
.
Evaluate :
|
sin
23
o
−
sin
7
o
cos
23
o
cos
7
o
|
Report Question
0%
√
3
2
0%
1
2
0%
sin
16
o
0%
cos
16
o
If
A
=
[
a
b
c
d
]
then
a
d
j
A
=
?
Report Question
0%
[
d
−
c
−
b
a
]
0%
[
−
d
b
c
−
a
]
0%
[
d
−
b
−
c
a
]
0%
[
−
d
−
b
c
a
]
If
A
is a
3
−
r
o
w
e
d
square matrix and
|
A
|
=
5
then
|
a
d
j
A
|
=
?
Report Question
0%
5
0%
25
0%
125
0%
None of these
Explanation
|
a
d
j
A
|
=
|
A
|
(
n
−
1
)
=
|
A
|
2
=
5
2
=
25
.
|
cos
15
o
sin
15
o
sin
15
o
cos
15
o
|
=
?
Report Question
0%
1
0%
1
2
0%
√
3
2
0%
n
o
n
e
o
f
t
h
e
s
e
Explanation
Let us consider
Δ
=
|
cos
15
o
sin
15
o
sin
15
o
cos
15
o
|
=
Δ
=
cos
2
15
0
−
sin
2
15
0
=
cos
30
0
.........
(
∵
cos
2
θ
=
cos
2
θ
−
sin
2
θ
)
=
√
3
2
Hence
|
cos
15
o
sin
15
o
sin
15
o
cos
15
o
|
=
√
3
2
Option
C
.
The roots of the equation
|
3
x
2
x
2
+
x
cos
θ
+
cos
2
θ
x
2
+
x
sin
θ
+
sin
2
θ
x
2
+
x
cos
θ
+
cos
2
θ
3
cos
2
θ
1
+
sin
2
θ
2
x
2
+
x
sin
θ
+
sin
2
θ
1
+
cos
2
θ
2
3
sin
2
θ
|
=
0
are
Report Question
0%
sin
θ
,
cos
θ
0%
sin
2
θ
,
cos
2
θ
0%
sin
θ
,
cos
2
θ
0%
sin
2
θ
,
cos
θ
When the determinant
|
cos
2
x
sin
2
x
cos
4
x
sin
2
x
cos
2
x
cos
2
x
cos
4
x
cos
2
x
cos
2
x
|
is expanded in powers of
sin
x
, then the constant term in that expression is
Report Question
0%
1
0%
0
0%
−
1
0%
2
Explanation
f
(
x
)
=
|
cos
2
x
sin
2
x
cos
4
x
sin
2
x
cos
2
x
cos
2
x
cos
4
x
cos
2
x
cos
2
x
|
the requried constant term is
f
(
0
)
=
|
1
0
1
0
1
1
1
1
1
|
=
|
1
0
0
0
1
1
1
1
0
|
=
1
(
0
−
1
)
=
−
1
If the determinant
|
cos
2
x
sin
2
x
cos
4
x
sin
2
x
cos
2
x
cos
2
x
cos
4
x
cos
2
x
cos
2
x
|
is expanded in powers of
sin
x
then the constant term in the expansion is
Report Question
0%
1
0%
2
0%
−
1
0%
−
2
Explanation
Let
|
cos
2
x
sin
2
x
cos
4
x
sin
2
x
cos
2
x
cos
2
x
cos
4
x
cos
2
x
cos
2
x
|
=
a
1
+
a
2
sin
2
x
+
a
3
sin
3
x
+
.
.
.
.
. . . . . . . . . . .
(
i
)
The constant term in the expansion is clearly the term
a
1
.
Putting
x
=
0
on both sides of
(
i
)
we get-
|
1
0
1
0
1
1
1
1
1
|
=
a
1
⇒
a
1
=
|
1
0
1
0
1
1
0
1
0
|
(
R
3
→
R
3
−
R
1
)
⇒
a
1
=
1
(
0
−
1
)
=
−
1
(Expanding along
C
1
)
Thus, the constant term is
−
1
.
The correct answer is option
C.
If
|
A
|
=
3
and
A
−
1
=
[
3
−
1
−
5
3
2
3
]
then
a
d
j
A
=
?
Report Question
0%
[
9
3
−
5
−
2
]
0%
[
9
−
3
−
5
2
]
0%
[
−
9
3
5
−
2
]
0%
[
9
−
3
5
−
2
]
Explanation
A
−
1
=
1
|
A
|
⋅
a
d
j
A
→
a
d
j
A
=
|
A
|
⋅
A
−
1
=
3
A
−
1
=
[
9
−
3
−
5
2
]
.
If
|
a
b
−
c
c
+
b
a
+
c
b
c
−
a
a
−
b
a
+
b
c
|
=
0
, then the line
a
x
+
b
y
+
c
=
0
passes through the fixed point whcih is
Report Question
0%
(
1
,
2
)
0%
(
1
,
1
)
0%
(
−
2
,
1
)
0%
(
1
,
0
)
Explanation
applying
C
1
→
a
C
!
and then
C
1
→
C
1
+
b
C
2
+
c
C
3
and taking
(
a
2
+
b
2
+
c
2
)
common from
C
1
we get
Δ
=
(
a
2
+
b
2
+
c
2
)
a
|
1
b
−
c
c
+
b
1
b
c
−
a
1
b
+
a
c
|
=
(
a
2
+
b
2
+
c
2
)
a
|
1
b
−
c
c
+
b
0
c
−
a
−
b
0
a
+
c
−
b
|
(
R
2
→
R
2
−
R
1
,
R
3
→
R
3
−
R
1
)
=
(
a
2
+
b
2
+
c
2
)
a
(
−
b
c
+
a
2
+
a
b
+
a
c
+
b
c
)
=
(
a
2
+
b
2
+
c
2
)
(
a
+
b
+
c
)
hence
Δ
=
0
⇒
a
+
b
+
c
=
0
therefore line
a
x
+
b
y
+
c
=
0
passes through the fixed point
(
1
,
1
)
If
A
=
[
2
5
1
3
]
then
a
d
j
A
=
?
Report Question
0%
[
3
−
5
−
1
2
]
0%
[
3
−
1
−
5
2
]
0%
[
−
1
2
3
−
5
]
0%
None of these
If A is singular matrix , then adj A is
Report Question
0%
singular
0%
non-singular
0%
symmetric
0%
not defined
Explanation
A
a
d
j
A
=
|
A
|
I
⇒
|
A
a
d
j
A
|
=
|
A
|
n
[ if A is of order
n
×
n
]
⇒
|
A
|
|
a
d
j
A
|
=
|
A
|
n
⇒
|
a
d
j
A
|
=
|
A
|
n
−
1
Now, A is singular .
∴
|
A
|
=
0
⇒
|
a
d
j
A
|
=
0
Hence adj A is singular.
If
A
=
[
1
−
1
2
0
3
1
0
0
−
1
/
3
]
, then
Report Question
0%
|
A
|
=
−
1
0%
a
d
j
A
=
[
−
1
1
−
2
0
−
3
−
1
0
0
−
1
/
3
]
0%
A
=
[
1
1
/
3
7
0
1
/
3
1
0
0
−
3
]
0%
A
=
[
1
−
1
/
3
−
7
0
−
3
0
0
0
1
]
Explanation
|
A
−
1
|
=
−
1
⇒
|
A
|
=
−
1
Now, use
a
d
j
A
=
|
A
|
A
−
1
a
n
d
A
=
(
A
−
1
)
−
1
There are two values of a which makes determinant
Δ
=
|
1
−
2
5
2
a
−
1
0
4
2
a
|
=
86
then sum of these number is
Report Question
0%
4
0%
5
0%
−
4
0%
9
Explanation
we have
Δ
=
|
1
−
2
5
2
a
−
1
0
4
2
a
|
=
86
⇒
1
(
2
a
2
+
4
)
−
2
(
−
4
a
−
20
)
+
0
=
86
[expanding along first column ]
⇒
2
a
2
+
4
+
8
a
+
40
=
86
⇒
2
a
2
+
8
a
+
44
−
86
=
0
⇒
a
2
+
4
a
−
21
=
0
⇒
a
2
+
7
a
−
3
a
−
21
=
0
⇒
(
a
+
7
)
(
a
−
3
)
=
0
a
=
−
7
and
3
∴
Required sum =
−
7
+
3
=
−
4
If
|
2
x
5
8
x
|
=
|
6
−
2
7
3
|
then value of
x
is
Report Question
0%
3
0%
±
3
0%
±
6
0%
6
Explanation
∵
|
2
x
5
8
x
|
=
|
6
−
2
7
3
|
⇒
2
x
2
−
40
=
18
+
14
⇒
2
x
2
=
32
+
40
⇒
x
2
=
72
2
=
36
∴
x
=
±
6
Choose the correct answer from the given alternatives in the following question:
If
A
=
[
1
2
3
4
]
,
adj
A
=
[
4
a
−
3
b
]
, then the values of
a
and
b
are
Report Question
0%
a
=
−
2
,
b
=
1
0%
a
=
2
,
b
=
4
0%
a
=
2
,
b
=
−
1
0%
a
=
1
,
b
=
−
2
Choose the correct answer from the given alternatives in the following question:
If
A
=
[
2
−
4
3
1
]
, then the adjoint of matrix
A
is
Report Question
0%
[
−
1
3
−
4
1
]
0%
[
1
4
−
3
2
]
0%
[
1
3
4
−
2
]
0%
[
−
1
−
3
−
4
2
]
State whether true or false:
If the value of a third order determinant is
12
, then the value of determinant formed by replacing each element by its co-factor will be
144
Report Question
0%
True
0%
False
Explanation
Let A is determinant
∴
|
A
|
=
12
Also we know that if
A
is a square matrix of order
n
then
|
a
d
j
A
|
=
|
A
|
n
−
1
For
n
=
3
,
|
A
d
j
A
|
=
|
A
|
3
−
1
=
|
A
|
2
=
(
12
)
2
=
144
Choose the correct answer from the given alternatives in the following question:
If
A
=
[
1
2
2
1
]
and
A
(
adj
A
)
=
K
I
, then the value of
k
is
Report Question
0%
1
0%
−
1
0%
0
0%
−
3
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
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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
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Incorrect : 0
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Practice Class 12 Commerce Maths Quiz Questions and Answers
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