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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
]
∴
Hence,
\text{adj}(3A^2+12A)=\begin{bmatrix} 51 & 84 \\ 63 & 72\end{bmatrix}^T=\begin{bmatrix}51&63\\84&72\end{bmatrix}
Let
P = [a_{ij}]
be a 3
\times
3 matrix and let
Q = [b_{ij}]
, where
b_{ij} = 2^{i + j} a_{ij}
for
1 \leq i, j \leq 3
. If the determinant of P is 2, then the determinant of the matrix Q is
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0%
2^{10}
0%
2^{11}
0%
2^{12}
0%
2^{13}
Explanation
Given,
[P]=a_{ij}
is a order 3 matrix.
Let
Q=\begin{bmatrix} { 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 } \end{bmatrix}
|Q|=\begin{vmatrix} { 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 } \end{vmatrix}
\Rightarrow Q={ 2 }^{ 2 }{ 2 }^{ 3 }{ 2 }^{ 4 }\begin{bmatrix} { 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 } \end{bmatrix}
\Rightarrow |Q|={ 2 }^{ 9 }{ \times 2 }\times { 2 }^{ 2 }\begin{vmatrix} { a }_{ 11 } & { a }_{ 12 } & { a }_{ 13 } \\ { a }_{ 21 } & { a }_{ 22 } & { a }_{ 23 } \\ { a }_{ 31 } & { a }_{ 32 } & { a }_{ 33 } \end{vmatrix}
\Rightarrow |Q|=2^{12}|P|
\Rightarrow |Q|=2^{13}
Consider three points
{P}=(-\sin(\beta-\alpha), -\cos\beta) , {Q}=(\cos(\beta-\alpha), \sin\beta)
and
{R}=(\cos(\beta-\alpha +\theta), \sin(\beta-\theta))
, where
0< \alpha,\ \beta,\ \theta <\displaystyle \frac{\pi}{4}
. Then
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0%
P
lies on the line segment
RQ
0%
Q
lies on the line segment
PR
0%
R
lies on the line segment
QP
0%
P, Q, R
are non-collinear
Explanation
\mathrm{P}\equiv(-\sin(\beta-\alpha), -\cos\beta)\equiv(\mathrm{x}_{1}, \mathrm{y}_{1})
\mathrm{Q}\equiv(\cos(\beta-\alpha), \sin\beta)\equiv(\mathrm{x}_{2}, \mathrm{y}_{2})
and
\mathrm{R}\equiv(\mathrm{x}_{2}\cos\theta+\mathrm{x}_{1}\sin\theta, \mathrm{y}_{2}\cos\theta+\mathrm{y}_{1}\sin\theta)
We see that
\displaystyle \mathrm{T}\equiv(\frac{\mathrm{x}_{2}\cos\theta+\mathrm{x}_{1}\sin\theta}{\cos\theta+\sin\theta}, \frac{\mathrm{y}_{2}\cos\theta+\mathrm{y}_{1}\sin\theta}{\cos\theta+\sin\theta})
and
\mathrm{P},\ \mathrm{Q},\ \mathrm{T}
are collinear
\Rightarrow \mathrm{P},\ \mathrm{Q},\ \mathrm{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
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0%
(p - 1)^2
0%
2(p - 1)
0%
(p - 1)^2 + 1
0%
2 p - 1
Explanation
Given
A=\begin{bmatrix} a\quad & b \\ c & a \end{bmatrix},a,b,c\in \left\{ 0,1,2,...,p-1 \right\}
If A is skew -symmetric matrix, then a =0, b=-c
\therefore \left| A \right| =-{ \left| b \right| }^{ 2 }
Thus, p divides
\left| A \right|
only when
b=0
...(1)
Again, if A is symmetric matrix, then
v=c
and
\left| A \right| ={ a }^{ 2 }-{ b }^{ 2 }
Thus p divides
\left| A \right|
is either p divides
(a-b)
or o divides
(a+b)
p divides
(a-b)
only when
a=b
i.e
a=b\in \left\{ 0,1,2,...,\left( p-1 \right) \right\}
i.e p choice
p divides
(a+b)
\Rightarrow
p choice including
a=b=0
in (1)
Therefore total number of choices are
\left( p+p+1 \right) =2p-1
Let k be a positive real number and let
A = \begin{bmatrix}2k-1 & 2\sqrt{k} & 2\sqrt{k}\\ 2\sqrt{k} & 1 & -2k\\ -2\sqrt{k} & 2k & -1\end{bmatrix}
and
B=\begin{bmatrix}0 & 2k-1 & \sqrt{k}\\ 1-2k & 0 & 2\sqrt{k}\\ -\sqrt{k} & -2\sqrt{k} & 0\end{bmatrix}
.
If det
(adj A) + det (adj B) = 10^{6}, then [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].
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0%
3
0%
4
0%
5
0%
6
Explanation
|\mathrm{A}|=(2\mathrm{k}+1)^{3},\ |\mathrm{B}|=0
(Since
\mathrm{B}
is a skew-symmetric matrix of order 3)
\Rightarrow \det(adj \mathrm{A}) =|\mathrm{A}|^{\mathrm{n}-1}=((2\mathrm{k}+1)^{3})^{2}=10^6\Rightarrow 2\mathrm{k}+1=10\Rightarrow 2\mathrm{k}=9
[\mathrm{k}]=4
.
The value of
|\mathrm{U}|
is
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0%
3
0%
-3
0%
3/2
0%
2
Explanation
A=\begin{bmatrix} 1 & 0 & 0 \\ 2 & 1 & 0 \\ 3 & 2 & 1 \end{bmatrix}
|A|=1
Adj\: A=\begin{bmatrix} 1 & 0 & 0 \\ -2 & 1 & 0 \\ 1 & -2 & 1 \end{bmatrix}
\Rightarrow A^{-1}=\begin{bmatrix} 1 & 0 & 0 \\ -2 & 1 & 0 \\ 1 & -2 & 1 \end{bmatrix}
AU_1=\begin{bmatrix} 1 \\ 0 \\ 0 \end{bmatrix}
\Rightarrow U_1=A^{-1}\begin{bmatrix} 1 \\ 0 \\ 0 \end{bmatrix}=\begin{bmatrix} 1 & 0 & 0 \\ -2 & 1 & 0 \\ 1 & -2 & 1 \end{bmatrix}\begin{bmatrix} 1 \\ 0 \\ 0 \end{bmatrix}=\begin{bmatrix} 1 \\ -2 \\ 1 \end{bmatrix}
AU_2=\begin{bmatrix} 2 \\ 3 \\ 0 \end{bmatrix}
\Rightarrow U_2=A^{-1}\begin{bmatrix} 2 \\ 3 \\ 0 \end{bmatrix}=\begin{bmatrix} 1 & 0 & 0 \\ -2 & 1 & 0 \\ 1 & -2 & 1 \end{bmatrix}\begin{bmatrix} 2 \\ 3 \\ 0 \end{bmatrix}=\begin{bmatrix} 2 \\ -1 \\ -4 \end{bmatrix}
AU_3=\begin{bmatrix} 2 \\ 3 \\ 1 \end{bmatrix}
\Rightarrow U_3=A^{-1}\begin{bmatrix} 2 \\ 3 \\ 1 \end{bmatrix}=\begin{bmatrix} 1 & 0 & 0 \\ -2 & 1 & 0 \\ 1 & -2 & 1 \end{bmatrix}\begin{bmatrix} 2 \\ 3 \\ 1 \end{bmatrix}=\begin{bmatrix} 2 \\ -1 \\ -3 \end{bmatrix}
\therefore U=\begin{bmatrix} 1 & 2 & 2 \\ -2 & -1 & -1 \\ 1 & -4 & -3 \end{bmatrix}
U^{-1}=\displaystyle\frac{1}{3}\begin{bmatrix} -1 & -2 & 0 \\ -7 & -5 & -3 \\ 9 & 6 & 3 \end{bmatrix}
\therefore
Sum of elements of
U^{-1}
is
0
.
Hence, option B.
Hence
U=\begin{bmatrix} 1 & 2 & 2 \\ -2 & -1 & -1 \\ 1 & -4 & -3 \end{bmatrix}\Rightarrow \left| U \right| =3
A = \begin{bmatrix}1&2\\3&4\end{bmatrix}, B = \begin{bmatrix}2&1\\3&4\end{bmatrix}
then
\left|(B^TA^T)^{-1}\right|
is equal to
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0%
10
0%
\dfrac{1}{10}
0%
1
0%
- 1
Explanation
A = \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}
B = \begin{bmatrix} 2 & 1 \\ 3 & 4 \end{bmatrix}
B^{T}A^{T} = \begin{bmatrix} 2 & 3 \\ 1 & 4 \end{bmatrix} \begin{bmatrix} 1 & 3 \\ 2 & 4 \end{bmatrix} = \begin{bmatrix} 2 + 6 & 6 + 12 \\ 1 + 8 & 3 + 16 \end{bmatrix} = \begin{bmatrix} 8 & 18 \\ 9 & 19 \end{bmatrix}
det (B^{T}A^{T}) = 8 \times 19 - 9 \times 18 = 152 - 162 = -10
(B^TA^T)^{-1} = \dfrac1{det(B^TA^T)} adj(B^TA^T) = \dfrac{1}{-10}\begin{bmatrix}19&-18\\-9&8\end{bmatrix}
det( (B^{T}A^{T})^{-1}) =\dfrac{19\times8 - 18\times9}{-10} = 1
If three points
(k, 2k), (2k, 3k), (3, 1)
are collinear, then
k
is equal to:
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0%
-2
0%
1
0%
\displaystyle\frac{1}{2}
0%
-\displaystyle\frac{1}{2}
Explanation
If given points are collinear, then
\begin{vmatrix} k&2k&1\\2k&3k&1\\3&1&1\end{vmatrix}=0
\Rightarrow \begin{vmatrix} k&2k&1\\k&k&0\\3&1&1\end{vmatrix}=0, R_2\rightarrow R_2-R_1
\Rightarrow 1(k-3k)+1(k^2-2k^2)=0
\Rightarrow k^2+2k=0
\Rightarrow k=0,-2
\begin{vmatrix} 4\sin^{2}\theta & \cos^{2}\theta \\ 3\sec^{2}\theta & \text{cosec}^{2}\theta \end{vmatrix}=
Report Question
0%
8\sin^{2}\theta \cos^{2}\theta
0%
4\sin 2\theta \cos 2\theta
0%
1
0%
4\cos^{3}\theta -3\cos \theta
Explanation
The value of
\begin{vmatrix} 4\sin^{2}\theta & \cos^{2}\theta \\ 3\sec^{2}\theta & \text{cosec}^{2}\theta \end{vmatrix}
is
=4\sin^{2}\theta .\text{cosec}^{2}\theta -3\sec^{2}\theta .\cos^{2}\theta
=4-3=1
\left|\begin{array}{lllll} 0 & & \mathrm{c}\mathrm{o}\mathrm{s}\alpha & \mathrm{c}\mathrm{o}\mathrm{s} & \beta\\ \mathrm{c}\mathrm{o}\mathrm{s} & \alpha & 0 & \mathrm{c}\mathrm{o}\mathrm{s} & \gamma\\ \mathrm{c}\mathrm{o}\mathrm{s} & \beta & \mathrm{c}\mathrm{o}\mathrm{s}\gamma & 0 & \end{array}\right|=
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0%
\cos\alpha+\cos\beta+\cos\gamma
0%
\cos\alpha\cos\beta\cos\gamma
0%
2\cos\alpha\cos\beta\cos\gamma
0%
2 \displaystyle \sum \cos\alpha\cos\beta
Explanation
Let
A=\begin{vmatrix} 0 & cos \alpha & \cos \beta\\ \cos \alpha & 0 & cos \gamma\\ cos \beta & \cos \gamma & 0 \end{vmatrix}
Expanding the determinant,
A=-cos \alpha[-cos \beta cos \gamma]+cos \beta[cos \alpha \cos \gamma]
=2 \cos \alpha \cos \beta \cos \gamma
A=\left\{\begin{array}{ll} 8 & 9\\ 10 & 11 \end{array}\right\}
, then cofactor of
\mathrm{a}_{12}
is:
Report Question
0%
11
0%
10
0%
-11
0%
-10
Explanation
By property of cofactor (2),
A=\begin{bmatrix} 8 & 9\\ 10 & 11 \end{bmatrix}
minor of
a_{12}=M_{12}=10
So, cofactor of
a_{12}=M_{12}(-1)^{1+2}=-M_{12}=-10
If P =
\begin{bmatrix} 1 & 4\\ 2 & 6 \end{bmatrix}
,then adj (P)
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0%
\begin{bmatrix}1 & 4\\ 2 & 6\end{bmatrix}
0%
\begin{bmatrix}6& -4\\ -2 & 1\end{bmatrix}
0%
\begin{bmatrix}6& -2\\ -4 & 1\end{bmatrix}
0%
\begin{bmatrix}2& 1\\ 6 & 4\end{bmatrix}
Explanation
Cofactor of the martix is
\begin{bmatrix} M11 & -M12 \\ -M21 & M22 \end{bmatrix}\\
Where
M11 = 6
M12 = 2
M21 = 4
M22 = 1
Substituting all value we get
Cofactor of the matrix as
\begin{bmatrix} 6 & -2 \\ -4 & 1 \end{bmatrix}\\
The adjoint of the matrix is transpose of cofactor of the given matrix
\therefore
Adjoint of the matrix is
\begin{bmatrix} 6 & -4 \\ -2 & 1 \end{bmatrix}
Hence the answer is option B.
\begin{vmatrix} x^{2}+3 &x-1 &x+3 \\ x+3 & -2x &x-4 \\ x-3& x+4 & 3x \end{vmatrix}
=px^{4}+qx^3+rx^{2}+sx+t,
then
t =
Report Question
0%
72
0%
0
0%
24
0%
-48
Explanation
On expanding the determinant we get,
(x^2+3)(-6x^2-x^2+16)-(x-1)(3x^2+9x-x^2+7x-12)+(x+3)(x^2+7x+12-2x^2+6x)
Let us consider only the constant terms,
since
t
is constant term
\therefore (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=\begin{bmatrix} a & b\\ c & d \end{bmatrix}
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 \epsilon {0,0}
So, Max value of det
A=\begin{bmatrix} 2 & 0\\ 0 & 2 \end{bmatrix}=4
If
\begin{vmatrix} cos(A+B) & -sin(A+B) &cos2B \\ sin A& cos A &sin B \\ -cos A& sin A & cos B \end{vmatrix}
=0 then B=
Report Question
0%
(2n+1)
\frac{\pi }{2}
0%
n
\pi
0%
(2n+1)
\pi
0%
2n
\pi
Explanation
\Rightarrow cos(A+B)(cosAcos B-sin A sin B)+sin^2(A+B)+cos2B+0
\Rightarrow cos^2(A+B)+sin^2(A+B)+cos2B=0
\Rightarrow cos (2B)=-1
2B=(2n+1)\pi
B=(2n+1)\pi/2
so
(2n+1)\pi/2
If A
=\begin{bmatrix} 0 & c &-b \\ -c& 0& a\\ b & -a & 0 \end{bmatrix}
then
\left ( a^{2}+b^{2}-c^{2} \right )\left | A \right |=
Report Question
0%
abc
0%
a + b + c
0%
\left ( a^{3}+b^{3}+c^{3} \right )
0%
0
Explanation
|A|=0\times(a^2)-c(-ab)-b(ac)
=0+abc-abc
=0
Find x if it is given that:
\det \left[\begin{array}{lll} 2 & 0 & 0\\ 4 & 3 & 0\\ 4 & 6 & x \end{array}\right]=42
Report Question
0%
8
0%
7
0%
6
0%
21/4
Explanation
Given
det\begin{pmatrix} 2 & 0 & 0\\ 4 & 3 & 0\\ 4 & 6 & x \end{pmatrix}=42
By operation of matrix (5)
x(6)=42
\Rightarrow x=7
\mathrm{If}
\left|\begin{array}{lll} 1 & 0 & 0\\ 2 & 3 & 4\\ 5 & -6 & x \end{array}\right|
= 45
\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{n}
\mathrm{x}=
Report Question
0%
4
0%
7
0%
- 5
0%
-7
Explanation
Given
\begin{vmatrix} 1 & 0 & 0\\ 2 & 3 & 4\\ 5 & -6 & x \end{vmatrix}=45
By operation of matrix (5),
1(3x+24)=45
3x=21
\Rightarrow x=7
If A is a singular matrix, then A (adj A) is a
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0%
scalar matrix
0%
zero matrix
0%
identity matrix
0%
orthogonal matrix
Explanation
Given
A
is a singular matrix.
\Rightarrow |A|=0
A(adj\:A)=|A|I=0I=O
\therefore A(adj\:A)
is a zero matrix.
Hence, option B.
If A is a singular matrix, then adj A is
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0%
\displaystyle non-singular
0%
singular
0%
symmetric
0%
not defined
Explanation
Given
|A|=0
We know
|adj A|=|A|^{n-1}
\therefore |adj A|=0
Hence, adj A is singular
If the value of the determinant
\begin{vmatrix}m & 2\\ -5 & 7\end{vmatrix}
is
31
, find
m
.
Report Question
0%
3
0%
8
0%
6
0%
2
Explanation
Given,
\begin{vmatrix}m & 2\\ -5 & 7\end{vmatrix}
\Rightarrow 7m- (-10)=31
\Rightarrow 7m+ 10= 31
\Rightarrow 7m= 21
\Rightarrow m= 3
If
A
is a
3\times 3
matrix and
\text{det} (3A)=k(\text{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.
|kA|=kn|A|
⟹
|3A|=3^3|A|=27|A|
k=27
If A and B are similar matrices such that
det (AB)=0,
then
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0%
det (A)=0
and
det (B)=0
0%
det (A)=0
and
det (B) \neq 0
0%
A=0
and
B=0
0%
A=0
or
B=0
Explanation
Given. two n
\times
n square matrices A and B
Given,
B=P^{-1}AP
det\left( B \right) =det\left( { P }^{ -1 }AP \right)
det\left( B \right) =det\left( { P }^{ -1 }AP \right)
det\left( B \right) =\left( det\left( { P } \right) \right) ^{ -1 }\left( det\left( A \right) \right) \left( det\left( P \right) \right)
det\left( B \right) =\left( det\left( { P } \right) \right) ^{ -1 }\left( det\left( A \right) \right) \left( det\left( P \right) \right)
det\left( B \right) =det\left( A \right)
det\left( AB \right) =\left( det\left( A \right) \right) \left( det\left( B \right) \right)
If
\left( det\left( A \right) \right) =0
then
\left( det\left( B \right) \right)
also zero.
Let a, b, c be three complex numbers, and let
z=\begin{vmatrix} 0 & -b & -c\\ b & 0 & -a\\ c & a & 0 \end{vmatrix}
then z equal
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0%
0
0%
purely imaginary
0%
abc
0%
none of these
Explanation
z=\begin{vmatrix} 0 & -b & -c \\ b & 0 & -a \\ c & a & 0 \end{vmatrix}=0\left( 0+{ a }^{ 2 } \right) +b\left( 0+ac \right) -c\left( ab-0 \right) =0
Two
n\times 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
det (A)=1
, then
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det (B)=1
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det (A)=-det (B)
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det (B)=-1
0%
none of these
Explanation
Given, two n
\times
n square matrices A and B
Given,
B=P^{-1}AP
det\left( B \right) =det\left( { P }^{ -1 }AP \right)
det\left( B \right) =\left( det\left( { P }^{ -1 } \right) \right) \left( det\left( A \right) \right) \left( det\left( P \right) \right)
det\left( B \right) =\left( det\left( { P } \right) \right) ^{ -1 }\left( det\left( A \right) \right) \left( det\left( P \right) \right)
[\because \quad|A||A^{-1}|=|AA^{-1}|=|I|=1]
det\left( B \right) =det\left( A \right) =1
If A is a square matrix of order
n
then adj
\left ( adj A \right )
is equal to
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\displaystyle \left | A \right |^{n}A
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\displaystyle \left | A \right |^{n-1}A
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\displaystyle \left | A \right |^{n-2}A
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\displaystyle \left | A \right |^{n-3}A
Explanation
{ A }^{ -1 }=\cfrac { AdjA }{ detA } =\cfrac { I }{ A } =\cfrac { AdjA }{ \left| A \right| }
A(adjA)=\left| A \right|
Replacing
A=adjA
adjA(AdjAdjA)=\left| AdjA \right|
\Rightarrow Adj\left( AdjA \right) =\cfrac { \left| Adj(A) \right| }{ \left| A \right| } \times A
=\cfrac { { \left| A \right| }^{ n-1 } }{ { \left| A \right| } } \times A
={ \left| A \right| }^{ n-2 }\times A
Adj(AdjA)={ \left| A \right| }^{ n-2 }\times 2
n\rightarrow
order
Option C is correct
If
\displaystyle \left | \begin{matrix}-12 &0 &\lambda \\ 0& 2& -1\\ 2& 1 &15 \end{matrix} \right |=-360
, then the value of
\lambda
,is
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-1
0%
-2
0%
-3
0%
4
Explanation
\left| \begin{matrix} -12 & 0 & \lambda \\ 0 & 2 & -1 \\ 2 & 1 & 15 \end{matrix} \right| =-360
Expanding along first column
\Rightarrow (-12)(31)-4\lambda =-360
\Rightarrow \lambda =-3
If
A
is any skew-symmetric matrix of odd order then
\left| A \right|
equals
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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|
\Rightarrow 2|A|=0
\Rightarrow|A|=0
Ans: B
If
\begin{vmatrix} x & y\\ 4 & 2 \end{vmatrix}=7
and
\begin{vmatrix} 2 & 3\\ y & x \end{vmatrix}=4
then
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x=-3, y=-\dfrac {5}{2}
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x=-\dfrac {5}{2}, y=-3
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x=-3, y=\dfrac {5}{2}
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x=-\dfrac {5}{2}, y=3
Explanation
\begin{vmatrix} x & y\\ 4 & 2 \end{vmatrix}=7
\Rightarrow 2x-4y=7
\begin{vmatrix} 2 & 3\\ y & x \end{vmatrix}=4
\Rightarrow 2x-3y=4
Therefore,
x=-5/2,y=-3
Ans: B
If
\begin{vmatrix}a & -b & -c\\-a & b & -c \\ -a & -b & -c\end{vmatrix}+\lambda abc=0
, then
\lambda
is equal to
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0%
2
0%
4
0%
-2
0%
-4
Explanation
\begin{vmatrix} a & -b & -c \\ -a & b & -c \\ -a & -b & -c \end{vmatrix}+\lambda abc=0
Substitute
a=b=c=1
\begin{vmatrix} 1 & -1 & -1 \\ -1 & 1 & -1 \\ -1 & -1 & -1 \end{vmatrix}+\lambda =0
(-2-2)+\lambda=0
\Rightarrow \lambda =4
The cofactors of elements in second row of the determinant
\begin{vmatrix} 2 & -1 & 4 \\ 4 & 2 & -3 \\1 & 1 & 2 \end{vmatrix}
are
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5, 6, 4
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6, 0, -3
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5, 1, 8
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6, 0, 3
Explanation
\begin{vmatrix} 2 & -1 & 4 \\ 4 & 2 & -3 \\ 1 & 1 & 2 \end{vmatrix}\\ { C }_{ 21 }=-\begin{vmatrix} -1 & 4 \\ 1 & 2 \end{vmatrix}=6\\ { C }_{ 22 }=\begin{vmatrix} 2 & 4 \\ 1 & 2 \end{vmatrix}=0\\ { C }_{ 23 }=-\begin{vmatrix} 2 & -1 \\ 1 & 1 \end{vmatrix}=-3
A set of points which do not lie on the same line are called as
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collinear
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non-collinear
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concurrent
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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
AB=10
,
BC=5
, then
AC
is equal to:
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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
\therefore
In case (i)
AC=AB-BC=10-5=5
In case (ii)
AC=AB+BC=10+5=15
If
A=\begin{bmatrix} 3 & -5 \\ -1 & 0 \end{bmatrix}
, then adj.
A
is equal to
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\begin{bmatrix} 3 & 5 \\ 1 & 0 \end{bmatrix}
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\begin{bmatrix} 0 & -5 \\ -1 & 3 \end{bmatrix}
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\begin{bmatrix} 0 & 5 \\ 1 & 3 \end{bmatrix}
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\begin{bmatrix} 0 & -1 \\ -5 & 3 \end{bmatrix}
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
adjA=\begin{bmatrix} 0 & 5 \\ 1 & 3 \end{bmatrix}
Ans: C
The value of the determinant
\begin{vmatrix}a & b & 0\\ 0 & a & b\\ b & 0 &a \end{vmatrix}
is equal to
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a^3-b^3
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a^3+b^3
0%
0
0%
none of these
Explanation
\begin{vmatrix} a & b & 0 \\ 0 & a & b \\ b & 0 & a \end{vmatrix}=a\left( { a }^{ 2 }-0 \right) -b\left( 0-{ b }^{ 2 } \right) +0={ a }^{ 3 }+{ b }^{ 3 }
The value of the determinant
\begin{vmatrix} 1 & 2 & 3\\ 3 & 5 & 7\\ 8 & 14 & 20\end{vmatrix}
is equal to
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0%
20
0%
10
0%
0
0%
45
Explanation
The value of the determinant is
\begin{vmatrix} 1 & 2 & 3 \\ 3 & 5 & 7 \\ 8 & 14 & 20 \end{vmatrix}\\ =1\left( 100-98 \right) -2\left( 60-56 \right) +3\left( 42-40 \right)
=2-8+6=0
If
\begin{vmatrix} 6i & -3i & 1\\ 4 & 3i & -1 \\ 20 & 3 & i\end{vmatrix}=x+iy
, then
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x = 3, y = 1
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x = 1, y = 3
0%
x = 0, y = 3
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x = 0, y = 0
Explanation
\begin{vmatrix} 6i & -3i & 1 \\ 4 & 3i & -1 \\ 20 & 3 & i \end{vmatrix}=x+iy
\Rightarrow 6i\left( 3{ i }^{ 2 }+3 \right) +3i\left( 4i+20 \right) +1\left( 12-60i \right) =x+iy
\Rightarrow 0=x+iy
Therefore,
x=y=0
Ans: D
The points which are not collinear are:
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(0, 1),\ (8, 3)\ and\ (6, 7)
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(4, 3),\ (5, 1)\ and\ (1, 9)
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(2, 5),\ (-1, 2)\ and\ (4, 7)
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(-3, 2)\, (1, -2)\ and\ (9, -10)
Explanation
Let\quad us\quad take\quad three\quad points\quad A,\quad B\quad \& \quad C.\\ Three\quad points\quad A,\quad B\quad \& \quad C\quad will\quad be\quad collinear\\ If\quad AB+BC=AC.\\ Three\quad points\quad A,\quad B\quad \& \quad C\quad will\quad not\quad be\quad collinear\\ If\quad AB+BC=AC.\\ Let\quad us\quad investigate\quad thr\quad given\quad options.\\ Option\quad A\longrightarrow The\quad points\quad are\\ A(0,1)\quad B(8,3)\quad \& \quad C(6,7).\\ \therefore \quad AB=\sqrt { { \left( 0-8 \right) }^{ 2 }+{ \left( 1-3 \right) }^{ 2 } } units=\sqrt { 68 } =8.25units,\\ \quad \quad \quad BC=\sqrt { { \left( 8-6 \right) }^{ 2 }+{ \left( 3-7 \right) }^{ 2 } } units=\sqrt { 28 } =5.29units,\\ and\quad AC=\sqrt { { \left( 0-6 \right) }^{ 2 }+{ \left( 1-7 \right) }^{ 2 } } units=\sqrt { 72 } =8.49units.\\ \therefore \quad AB+BC=\left( 8.25+5.29 \right) units.=13.54units.\\ So\quad AB+BC\neq AC.\\ \therefore \quad A,\quad B\quad \& \quad C\quad are\quad not\quad collinear.\\ Option\quad B\longrightarrow The\quad points\quad are\\ A(4,3)\quad B(5,1)\quad \& \quad C(1,9).\\ \therefore \quad AB=\sqrt { { \left( 4-5 \right) }^{ 2 }+{ \left( 3-1 \right) }^{ 2 } } units=\sqrt { 5 } =2.24units,\\ \quad \quad \quad BC=\sqrt { { \left( 5-1 \right) }^{ 2 }+{ \left( 1-9 \right) }^{ 2 } } units=\sqrt { 80 } =8.94units,\\ and\quad AC=\sqrt { { \left( 4-1 \right) }^{ 2 }+{ \left( 3-9 \right) }^{ 2 } } units=\sqrt { 45 } =6.71units.\\ \therefore \quad AB+AC=\left( 2.24+6.71 \right) units=8.95units.\\ So\quad AB+AC=BC.\\ \therefore \quad A,\quad B\quad \& \quad C\quad are\quad \quad collinear.\\ Option\quad C\longrightarrow The\quad points\quad are\\ A(2,5)\quad B(-1,2)\quad \& \quad C(4,7).\\ \therefore \quad AB=\sqrt { { \left( 2+1 \right) }^{ 2 }+{ \left( 5-2 \right) }^{ 2 } } units=\sqrt { 18 } units=4.24units,\\ \quad \quad \quad BC=\sqrt { { \left( -1-4 \right) }^{ 2 }+{ \left( 2-7 \right) }^{ 2 } } units=\sqrt { 50 } units=7.07units,\\ and\quad AC=\sqrt { { \left( 2-4 \right) }^{ 2 }+{ \left( 5-7 \right) }^{ 2 } } units=\sqrt { 8 } units=2.83units.\\ \therefore \quad AB+AC=\left( 4.24+2.83 \right) units=7.07units.\\ So\quad AB+AC=BC.\\ \therefore \quad A,\quad B\quad \& \quad C\quad are\quad \quad collinear.\\ Option\quad C\longrightarrow The\quad points\quad are\\ A(-3,2)\quad B(1,-2)\quad \& \quad C(9,-10).\\ \therefore \quad AB=\sqrt { { \left( -3-1 \right) }^{ 2 }+{ \left( 2+2 \right) }^{ 2 } } units=\sqrt { 32 } units=5.66units,\\ \quad \quad \quad BC=\sqrt { { \left( 1-9 \right) }^{ 2 }+{ \left( -2+10 \right) }^{ 2 } } units=\sqrt { 128 } units=11.31units,\\ and\quad AC=\sqrt { { \left( -3-9 \right) }^{ 2 }+{ \left( 2+10 \right) }^{ 2 } } units=\sqrt { 288 } units=16.97units.\\ \therefore \quad AB+BC=\left( 5.66+11.31 \right) units=16.97units.\\ So\quad AB+BC=AC.\\ \therefore \quad A,\quad B\quad \& \quad C\quad are\quad \quad collinear.\\ Ans-\quad Option\quad A.\\ \\ \\
If
A=\begin{bmatrix} a & b \\ b & a \end{bmatrix}
, then
\left| A+{ A }^{ T } \right|
equals
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4({a}^{2}-{b}^{2})
0%
2({a}^{2}-{b}^{2}
0%
{a}^{2}-{b}^{2})
0%
4ab
Explanation
A+{A}^{T}=\begin{bmatrix} a & b \\ b & a \end{bmatrix}\begin{bmatrix} a & b \\ b & a \end{bmatrix}=\begin{bmatrix} 2a & 2b \\ 2b & 2a \end{bmatrix}
\left| A+{ A }^{ T } \right| =4{ a }^{ 2 }-4{ b }^{ 2 }\quad
Ans: A
If
\Delta_1=\begin{vmatrix} 1 & 0\\ a & b\end{vmatrix}
and
\Delta_2=\begin{vmatrix} 1 & 0\\ c & d\end{vmatrix}
then
\Delta_2 \Delta_1
is equal to
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ac
0%
bd
0%
(b-a)(d-c)
0%
none of these
Explanation
\Delta_1=\begin{vmatrix} 1 & 0\\ a & b\end{vmatrix}=1.b-0.a=b
\Delta_2=\begin{vmatrix} 1 & 0\\ c & d\end{vmatrix}=1.d-0.c=d
then
\Delta_2 \Delta_1=bd
Ans: B
If
A
and
B
are two matrices of same order
3\times 3
, where
A=\begin{bmatrix} 1 & 2 & 3 \\ 2 & 3 & 4 \\ 5 & 6 & 8 \end{bmatrix}
and
B=\begin{bmatrix} 3 & 2 & 5 \\ 2 & 3 & 8 \\ 7 & 2 & 9 \end{bmatrix}
The value of
\text{Adj} (\text{Adj}\, A)
is equal to
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-A
0%
A
0%
8A
0%
16A
Explanation
\left| A \right| =1\left( 3\times8-4\times6 \right) -2\left( 2\times8-5\times4 \right) +3\left( 2\times6-5\times3 \right) =-1
Use following results
\text{Adj} (\text{Adj}\, A)={ \left| A \right| }^{ n-2 }A \Rightarrow \text{Adj}(\text{Adj}\,A)={ \left| A \right| }^{ n-2 }A={ \left( -1 \right) }^{ 1 }A=-A
Ans:
A
If
\omega
is a cube root of unity and
\Delta=\begin{vmatrix}1 & 2\omega \\ \omega & \omega^2\end{vmatrix}
, then
\Delta^2
is equal to
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-\omega
0%
\omega
0%
1
0%
\omega^2
Explanation
\Delta=\omega^2-2\omega^2=-\omega^2
\Delta^2=\omega^4=\omega
Ans: B
If
\begin{vmatrix} x & 2 \\ 18 & x \end{vmatrix}=\begin{vmatrix} 6 & 2 \\ 3x & 6 \end{vmatrix}
, then
x
is equal to
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6
0%
\pm 6
0%
-6
0%
0
Explanation
\begin{vmatrix} x & 2 \\ 18 & x \end{vmatrix}=\begin{vmatrix} 6 & 2 \\ 3x & 6 \end{vmatrix}
\Rightarrow x^2-36=36-6x
\Rightarrow x^2+6x-72=0
\Rightarrow \displaystyle x=\frac{-6\pm\sqrt{6^2+4\times 72}}{2}
\quad \displaystyle =\frac{-6\pm \sqrt{36(1+8)}}{2}
\quad =-3\pm 9=6\ or\ -12
Which of the following is correct?
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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| \displaystyle \neq 0
, then
A
is
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zero matrix
0%
singular matrix
0%
non - singular matrix
0%
diagonal matrix
Explanation
If
|A| \neq 0
then by definition,
A
is non-singular matrix
A=\begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}
and
A(\text{adj}\,A)=KI
, then the value of
K
is:
Report Question
0%
2
0%
-2
0%
10
0%
-10
Explanation
Given,
A=\begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}\quad
Adjoint of
A=\begin{bmatrix} 4 & -2 \\ -3 & 1 \end{bmatrix}\quad
\therefore
A(\text{adj}\,A)=KI
\begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}\begin{bmatrix} 4 & 2 \\ -3 & 1 \end{bmatrix}=K\begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}
\begin{bmatrix} -2 & 0 \\ 0 & -2 \end{bmatrix}=\begin{bmatrix} K & 0 \\ 0 & K \end{bmatrix}
\therefore
K=-2
What is the determinant of the matrix
\left [\begin{matrix} 3& 6\\ -1 & 2\end {matrix} \right]
?
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0%
0
0%
12
0%
|0|
0%
|6|
Explanation
Given,
\begin{vmatrix} 3 & 6 \\ -1 & 2 \end{vmatrix}
Let determinent be
\left| d \right|
Value of
\left| d \right|
will be
\left| d \right|
=
3\times 2-\left( 6\times -1 \right)
=6+6=12
If
A = \begin{bmatrix} 1& \log_{b}a\\ \log_{a}b & 1\end{bmatrix}
then
|A|
is equal to
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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
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Practice Class 12 Commerce Maths Quiz Questions and Answers
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