Processing math: 4%
MCQExams
0:0:1
CBSE
JEE
NTSE
NEET
Practice
Homework
×
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
3x-2y=4
can be solved using the above method
x=\dfrac { \left| \underset { 4 }{ 3 } \quad \underset { -2 }{ 1 } \right| }{ \left| \underset { 3 }{ 1 } \quad \underset { -2 }{ 1 } \right| } \quad ;\quad y=\dfrac { \left| \underset { 3 }{ 1 } \quad \underset { 4 }{ 3 } \right| }{ \left| \underset { 3 }{ 1 } \quad \underset { -2 }{ 1 } \right| }
x=\dfrac { -6-4 }{ -2-3 } \quad ;\quad y=\dfrac { 4-9 }{ -5 }
x=\dfrac { 10 }{ -5 } \quad ;\quad y=\dfrac { -5 }{ -5 }
x=2\quad ;\quad y=1
now the quantity
\left| \underset { 3 }{ 1 } \quad \underset { -2 }{ 1 } \right| =D
(determinant)
D=\left| \underset { 3 }{ 1 } \quad \underset { -2 }{ 1 } \right| =-5
So, answer is option C.
The number of distinct real roots of the equation,
\left| \begin{matrix} cosx & sinx & sinx \\ sinx & cosx & sinx \\ sinx & sinx & cosx \end{matrix} \right| =0
In t interval
\left[ -\dfrac { \pi }{ 4 } \dfrac { \pi }{ 4 } \right]
is/are:
Report Question
0%
3
0%
2
0%
1
0%
4
If the point
\left(\lambda+1,1\right),\left(2\lambda+1,3\right)
and
\left(2\lambda+2,2\lambda\right)
are collinear then the possible value of
\lambda
is
Report Question
0%
2
0%
1/2
0%
3
0%
1/3
Solve
\begin{vmatrix} 1 & 1 & 1 \\ 1 & 1+x & 1 \\ 1 & 1 & 1+y \end{vmatrix}
Report Question
0%
1
0%
0
0%
x
0%
xy
Explanation
Let
\begin{array}{l}A= \left| { \begin{array} { *{ 20 }{ c } }1 & 1 & 1 \\ 1 & { 1+x } & 1 \\ 1 & 1 & { 1+y } \end{array} } \right| \\ { R_{ 1 } }\to { R_{ 1 } }-{ R_{ 3 } }\, \, \, \, \, \, \, \, \, \, \, { R_{ 2 } }\to { R_{ 2 } }-{ R_{ 3 } } \\ A=\left| { \begin{array} { *{ 20 }{ c } }0 & 0 & { -y } \\ 0 & x & { -y } \\ 1 & 1 & { 1+y } \end{array} } \right| \\ =-y\left( { 0-x } \right) \\ =xy \end{array}
If
(k,2-2k),(-k+1,2k),(-4-k,6-2k)
are collinear, then
k=
Report Question
0%
+1
0%
-1
0%
-2
0%
2
Explanation
\begin{array}{l} where\, \, more\, than\, \, three\, \, po{ { int } }s\, are\, collinear\, slope\, of\, the\, line\, are\, same \\ So, \\ slope:\, \, \left[ { \left( { k,2-2k } \right) ,\left( { -k+1,2k } \right) } \right] =\left[ { \left( { -k+1,\, 2k } \right) ,\left( { -4-k,6-2k } \right) } \right] \\ \Rightarrow \dfrac { { 2-2k-2k } }{ { k+k-1 } } =\dfrac { { 2k-6+2k } }{ { -k+1+4+k } } \\ \Rightarrow \dfrac { { 2-4k } }{ { 2k-1 } } =\dfrac { { 4x-6 } }{ 5 } \\ \Rightarrow \left( { 1-2k } \right) 5=\left( { 2k-3 } \right) \left( { 2k-1 } \right) \end{array}
5-10k=4k^2-2k-6k+3
4k^2+2k-2=0
2k^2+k-1=0
2k^2+2k-k-1=0
2k(k+1)-1(k+1)=0
(2k-1)(k+1)=0
k=-1\, or\, \dfrac{1}{2}
Hence,option
B
is the correct answer.
If
{ D }_{ P }=\left| \begin{matrix} P & 15 & 8 \\ { P }^{ 2 } & 35 & 9 \\ { P }^{ 3 } & 25 & 10 \end{matrix} \right| ,
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=\begin{vmatrix} p & 15 & 8\\ p^2 & 35 & 9\\ p^3 & 25 & 10\end{vmatrix}
=p(350-225)-15(10p^2-9p^3)+8(25p^2-35p^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
.
\therefore D_1+D_2+D_3+D_4+D_5=30-710-3090-8980-16250
=-29000
.
The value of determinant
\left| \begin{matrix} { bc-a }^{ 2 } & { ac-b }^{ 2 } & ab-c^{ 2 } \\ { ac-b }^{ 2 } & { ab-c }^{ 2 } & { bc-a }^{ 2 } \\ { ab-c }^{ 2 } & { bc-a }^{ 2 } & ac-b^{ 2 } \end{matrix} \right|
is
Report Question
0%
always non-negative
0%
always non-positive
0%
always zero
0%
can't say anything
If
A=\begin{bmatrix} 2 & -3 \\ -4 & 1 \end{bmatrix}
, then
\text{adj}
(3{ A }^{ 2 }+12A)
is equal to
Report Question
0%
\begin{bmatrix} 51 & 63 \\ 84 & 72 \end{bmatrix}
0%
\begin{bmatrix} 51 & 84 \\ 63 & 72 \end{bmatrix}
0%
\begin{bmatrix} 72 & -63 \\ -84 & 51 \end{bmatrix}
0%
\begin{bmatrix} 72 & -84 \\ -63 & 51 \end{bmatrix}
Explanation
A = \begin{bmatrix} 2 & -3 \\ -4 & 1 \end{bmatrix}
then
A^2 = \begin{bmatrix} 2 & -3 \\ -4 & 1 \end{bmatrix} \, \begin{bmatrix} 2 & -3\\ -4 &1 \end{bmatrix}
A^2 = \begin{bmatrix} 16 & -9 \\ -12 & 13 \end{bmatrix}
3A^2 = \begin{bmatrix} 48 & -27 \\ -36 & 39 \end{bmatrix}
3A^2 + 12 A = \begin{bmatrix} 48 & -27 \\ -36 & 39 \end{bmatrix} + \begin{bmatrix} 24 & -36 \\ -48 & 12 \end{bmatrix} = \begin{bmatrix} 72 & -63 \\ -84 & 51 \end{bmatrix}
adj (3A^2 + 12 A) = \begin{bmatrix} 51 & 63 \\ 84 & 72 \end{bmatrix}
If a,b,c are distinct and
\left| \begin{matrix} a & { a }^{ 2 } & { a }^{ 3 }-1 \\ b & { b }^{ 2 } & { b }^{ 3 }-1 \\ c & { c }^{ 2 } & { c }^{ 3 }-1 \end{matrix} \right| =0
then
Report Question
0%
a+b+c=1
0%
ab+bc+ca=0
0%
a+b+c=0
0%
abc=1
Explanation
\begin{vmatrix}a & a^2 & a^3-1\\ b & b^2 & b^3-1\\ c & c^2 & c^3-1\end{vmatrix} =0
\Rightarrow a(b^2c^3-b^2-c^2b^3+c^2)-a^2(bc^3-b-cb^3+c)+(a^3-1)(bc^2-cb^2)=0
\Rightarrow a (b^2c^2(c-b)+c^2-b^2)-a^2(bc(c^2-b^2)+c-b)+(a^3-1)bc(c-b)=0
\Rightarrow a(b^2c^2(c-b)+(c-b)(c+b))-a^2(bc(c-b)(c+b)+(c-b))+bc(a^3-1)(c-b)=0
\Rightarrow a(c-b)(b^2c^2+c+b)-a^2(c-b)(bc(c+b)+1)+bc(a^3-1)(c-b)=0
\Rightarrow (c-b)\left \{ a(b^2c^2+c+b)-a^2(bc(c+b)+1)+bc(a^3-1) \right \}=0
\Rightarrow(c-b)\left \{ ab^2c^2+ac+ab-a^2bc^2-a^2b^2c-a^2+a^3bc-bc \right \}=0
\Rightarrow (c-b)\left \{ abc (bc-ac-ab+b^2)-(bc-ac-ab+a^2) \right \}=0
\Rightarrow(c-b)(bc-ac-ab+a^2)(abc-1)=0
\Rightarrow (c-b)(b(c-a)-a(c-a))(abc-1)=0
\Rightarrow (c-b)(c-a)(b-a)(abc-1)=0
\Rightarrow c-b=0
or
c-a =0
or
b-a =0
\Rightarrow c=b
c=a
b=a
which is not possible since
a, b, c
are distinct.
\therefore abc -1=0
\Rightarrow abc=1
If
\begin{bmatrix} 1 & \alpha & 3 \\ 1 & 3 & 3 \\ 2 & 4 & 4 \end{bmatrix}
is the adjoint of a
3\times 3
matrix
A
and
|A|=4
, then
\alpha
is equal to:
Report Question
0%
4
0%
11
0%
5
0%
0
Explanation
\begin{vmatrix} 1 & \alpha & 3 \\ 1 & 3 & 3 \\ 2 & 4 & 4 \end{vmatrix} = 1 \left( 3 \times 4 - 3 \times 4 \right) - \alpha \left( 1 \times 4 - 3 \times 2 \right) + 3 \left( 1 \times 4 - 3 \times 2 \right) = 2 \alpha - 6
\begin{vmatrix} 1 & \alpha & 3 \\ 1 & 3 & 3 \\ 2 & 4 & 4 \end{vmatrix} = \begin{vmatrix} adj. \; A \end{vmatrix}
\Rightarrow \left( 2 \alpha - 6 \right) = \begin{vmatrix} adj. \; A \end{vmatrix}
Now,
\begin{vmatrix} adj. \; A \end{vmatrix} = {\left| A \right|}^{3 - 1} = {4}^{2} = 16
\therefore 2 \alpha - 6 = 16
\Rightarrow 2 \alpha = 16 + 6
\Rightarrow \alpha = \cfrac{22}{2} = 11
If f(x), g(x), h(x) are polynomials in x of degree 2 and F(x)=
\left| \begin{matrix} f & g & h \\ { f' } & g' & h' \\ f" & g" & h" \end{matrix} \right| ,
, then F(x) is equal to
Report Question
0%
1
0%
0
0%
-1
0%
f(x).g(x).h(x)
Explanation
\begin{array}{l} Let, \\ f\left( x \right) =a{ x^{ 2 } }+bx+c \\ f'\left( x \right) =2ax+b \\ f''\left( x \right) =2a \\ g\left( x \right) ={ a_{ 1 } }{ x^{ 2 } }+{ b_{ 1 } }x+{ c_{ 1 } } \\ g'\left( x \right) =2{ a_{ 1 } }x+{ b_{ 1 } } \\ g''\left( x \right) =2{ a_{ 1 } } \\ and, \\ h\left( x \right) ={ a_{ 2 } }{ x^{ 2 } }+{ b_{ 2 } }x+{ c_{ 2 } } \\ h'\left( x \right) =2{ a_{ 2 } }x+{ b_{ 2 } } \\ f\left( x \right) =\left| { \begin{array} { *{ 20 }{ c } }{ a{ x^{ 2 } }+bx+c } & { { a_{ 1 } }{ x^{ 2 } }+{ b_{ 1 } }x+{ c_{ 1 } } } & { { a_{ 2 } }{ x^{ 2 } }+{ b_{ 2 } }x+{ c_{ 2 } } } \\ { 2ax+b } & { 2{ a_{ 1 } }x+{ b_{ 1 } } } & { 2{ a_{ 2 } }x+{ b_{ 2 } } } \\ { 2a } & { 2{ a_{ 1 } } } & { 2{ a_{ 2 } } } \end{array} } \right| \\ f'\left( x \right) =\left| { \begin{array} { *{ 20 }{ c } }{ 2ax+b } & { 2{ a_{ 1 } }x+{ b_{ 1 } } } & { 2{ a_{ 2 } }x+{ b_{ 2 } } } \\ { 2ax+b } & { 2{ a_{ 1 } }x+{ b_{ 1 } } } & { 2{ a_{ 2 } }x+{ b_{ 2 } } } \\ { 2a } & { 2{ a_{ 1 } } } & { 2{ a_{ 2 } } } \end{array} } \right| +0+0 \\ =0 \end{array}
Hence, the option
B
is the correct answer.
If
\left| \begin{array} { c c } { 1 } & { \sin x } & { \sin ^ { 2 } x } \\ { 1 } & { \cos x } & { \cos ^ { 2 } x } \\ { 1 } & { \tan x } & { \tan ^ { 2 } x } \end{array} \right| = 0 , x \in [ 0,2 \pi ] ,
then number of possible values of
x
is.
Report Question
0%
4
0%
5
0%
6
0%
None of these
Explanation
\left| \overset { 1 }{ \underset { 1 }{ 1 } } \quad \overset { sinx }{ \underset { tanx }{ cosx } } \quad \overset { { \sin }^{ 2 }x }{ \underset { { \tan }^{ 2 }x }{ { \cos }^{ 2 }x } } \right| =0
\left( \underset { { R }_{ 2 }\rightarrow { R }_{ 2 }-{ R }_{ 3 } }{ { R }_{ 1 }\rightarrow { R }_{ 1 }-{ R }_{ 2 } } \right)
\Rightarrow \left| \overset { 0 }{ \underset { 1 }{ 0 } } \quad \quad \overset { sinx-cosx }{ \underset { tanx }{ cosx-tanx } } \quad \quad \overset { { \sin }^{ 2 }x-{ \cos }^{ 2 }x }{ \underset { { \tan }^{ 2 }x }{ { \cos }^{ 2 }x-{ \tan }^{ 2 }x } } \right| =0
\Rightarrow \underset { \left( cosx-tanx \right) }{ \left( sinx-cosx \right) } \left| \overset { 0 }{ \underset { 1 }{ 0 } } \quad \quad \overset { 1 }{ \underset { tanx }{ 1 } } \quad \quad \overset { sinx+cosx }{ \underset { { \tan }^{ 2 }x }{ cosx+tanx } } \right| =0
Expanding along colomn I we get
\Rightarrow \left( cosx-tanx \right) \left( sinx-cosx \right) \left( cosx+tanx-sinx-cosx \right) =0
\Rightarrow \left( cosx-tanx \right) \left( sinx-cosx \right) \left( tanx-sinx \right) =0
\therefore
cosx-tanx=0
or
sinx=cosx
or
tanx=sinx
or
cosx=tanx
;
sinx=cosx
;
tanx=sinx
for
x\epsilon \left[ 0,2\pi \right]
csox=tanx
from graph we see there are
2
solutions.
for
x\epsilon \left[ 0,2\pi \right]
sinx=cosx
from graph we have two solutions
for
x\epsilon \left[ 0,2\pi \right]
from graph we have solutions
So a total of
2+2+3=7
solutions
i.e Answer : D.
If
\begin{vmatrix} x-4 & 2x & 2x \\ 2x & x-4 & 2x \\ 2x & 2x & x-4 \end{vmatrix}
=
(A+Bx)(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
\left|\begin{matrix} x-4 & 2x & 2x \\ 2x & x-4 & 2x \\ 2x & 2x & x-4 \end{matrix}\right|=
(x-4)[(x-4)^2-4x^2]-2x[2x(x-4)-4x^2]+2x[4x^2-2x(x-4)]
=(x-4)[x^2+16-8x-4x^2]-2x[2x^2-8x-4x^2]+2x[4x^2-2x^2+8x]
=(x-4)[16-8x-3x^2]-2x[-8x-2x^2]+2x[2x^2+8x]
=(x-4)(16-8x-3x^2)+4x[2x^2+8x]
=16x-8x^2-3x^3-64+32x+12x^2+8x^3+32x^2
=5x^3+36x^2+48x-64
(A+Bx)(x-A)^2=(A+Bx)(x^2+A^2-2xA)
=Ax^2+A^3-2xA^2+Bx^3+BA^2x-2ABx^2
=Bx^3+(A-2AB)x^2+(BA^2-2x)x+A^3
B=5
A-2AB=36
A(1-2B)=36
A(1-10)=36
A=-4
\therefore (A,B)=(-4,5)
.
If
A=\begin{vmatrix} 10 & 2 \\ 30 & 6 \end{vmatrix}\\
then
\left|A\right|=
Report Question
0%
0
0%
10
0%
12
0%
60
Explanation
A=\begin{vmatrix} 10 & 2 \\ 30 & 6 \end{vmatrix}\\
=10\times 6-30\times 2
=60-60=0
If
\left|( {adj\,A}) \right| = 81,
for
3 \times 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
.
|adjA|=|A|^{n-1}
, where
n
is the order of matrix.
We have,
|adj(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
\therefore \dfrac{1}{2}\left|x_1(y_2-y_3)+x_2(y_3-y_1)+x_3(y_1-y_2)\right|=0
\dfrac{1}{2}\left|-1(2-4)+-3(4-x)+-4(x-2)\right|=0
\dfrac{1}{2}\left|-1(-2)-3(4-x)-4(x-2)\right|=0
\dfrac{1}{2}\left|2-3\times 4+3x-4x+4\times 2\right|=0
\dfrac{1}{2}\left|2-12+3x-4x+8\right|=0
\dfrac{1}{2}\left|-2-x\right|=0
-2-x=0
x=-2
.
\left| \begin{matrix} \frac { 1 }{ a } & { a }^{ 2 } & bc \\ \frac { 1 }{ b } & { b }^{ 2 } & ca \\ \frac { 1 }{ c } & { c }^{ 2 } & ab \end{matrix} \right| =
Report Question
0%
abc
0%
a+b+c
0%
0
0%
4abc
Explanation
Given,
\begin{vmatrix}\frac{1}{a}&a^2&bc\\ \frac{1}{b}&b^2&ca\\ \frac{1}{c}&c^2&ab\end{vmatrix}
=\dfrac{1}{a}\det \begin{pmatrix}b^2&ca\\ c^2&ab\end{pmatrix}-a^2\det \begin{pmatrix}\frac{1}{b}&ca\\ \frac{1}{c}&ab\end{pmatrix}+bc\det \begin{pmatrix}\frac{1}{b}&b^2\\ \frac{1}{c}&c^2\end{pmatrix}
=\dfrac{1}{a}\left(ab^3-ac^3\right)-a^2\cdot \:0+bc\left(\dfrac{c^2}{b}-\dfrac{b^2}{c}\right)
=b^3-c^3-0+bc\left(\dfrac{c^2}{b}-\dfrac{b^2}{c}\right)
=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
\angle ABC=90^{\circ}
and
\angle ADC=90^{\circ}
\therefore ABCD
are concyclic.
So,
\text{B}
is the correct option.
Find the value of
\begin{vmatrix} 5 & 3 \\ -7 & -4 \end{vmatrix}
Report Question
0%
-1
0%
-41
0%
41
0%
1
Explanation
\begin{vmatrix} 5 & 3 \\ -7 & -4 \end{vmatrix}
= 5 \times (-4) - 3 \times (-7) = -20 + 21 = 1
Hence, the correct answer is 1.
If
a,b,c
are
pth
,
qth
and
rth
terms of a GP, then
\begin{vmatrix} \log { a } & p & 1 \\ \log { b } & q & 1 \\ \log { c } & r & 1 \end{vmatrix}
is equal to
Report Question
0%
0
0%
1
0%
\log { abc }
0%
none of these
Explanation
It is given that the given terms are in G.P
\therefore {p}^{th}
term
=a,{q}^{th}
term
=b
and
{r}^{th}
term
=c
\Rightarrow a=A{R}^{p-1}\Rightarrow \log{a}=\log{A{\left(R\right)}^{p-1}}=\log{A}+\left(p-1\right)\log{R}
\Rightarrow b=A{R}^{q-1}\Rightarrow \log{b}=\log{A{\left(R\right)}^{q-1}}=\log{A}+\left(q-1\right)\log{R}
\Rightarrow c=A{R}^{q-1}\Rightarrow \log{c}=\log{A{\left(R\right)}^{r-1}}=\log{A}+\left(r-1\right)\log{R}
Now,
\Delta=\begin{vmatrix} \log{a} & p & 1 \\ \log{b} & q & 1 \\\log{c} & r & 1 \end{vmatrix}
=\begin{vmatrix} \log{A}+\left(p-1\right)\log{R} & p & 1 \\ \log{A}+\left(q-1\right)\log{R} & q & 1\\\log{A}+\left(r-1\right)\log{R} & r & 1 \end{vmatrix}
{C}_{2}\rightarrow {C}_{2}-{C}_{3}
=\begin{vmatrix} \log{A}+\left(p-1\right)\log{R} & p-1 & 1 \\ \log{A}+\left(q-1\right)\log{R} & q-1 & 1\\\log{A}+\left(r-1\right)\log{R} & r-1 & 1 \end{vmatrix}
{C}_{1}\rightarrow {C}_{1}-\left(\log{A}\right){C}_{3}-\left(\log{R}\right){C}_{2}
=\begin{vmatrix} 0 & p-1 & 1 \\0 & q-1 & 1 \\0 & r-1 & 1\end{vmatrix}
=0
|A|=6
and
A
is
3\times 3
matrix then
det(2\ adj(2(A^{-1})^{T}))=
Report Question
0%
162
0%
36
0%
216
0%
512
If
\Delta{(x)} = \begin{vmatrix} e^x & \sin 2x & \tan x^2\\ ln(1 + x) & \cos x & \sin x\\ \cos x^2 & e^x - 1 & \sin x^2 \end{vmatrix} = A + Bx + Cx^2 + .....
then B is equal to
Report Question
0%
0
0%
1
0%
2
0%
4
If
A=\begin{bmatrix} 5 & 5x & x\\ 0 & x & 5x\\ 0 & 0 & 5\end{bmatrix}
and
|A^2|=25
, then
|x|
is equal to?
Report Question
0%
\dfrac{1}{5}
0%
5
0%
5^2
0%
1
Explanation
Given
A=\begin{bmatrix} 5 & 5x & x\\ 0 & x & 5x\\ 0 & 0 & 5\end{bmatrix}
and
|A^2|=25
|A^2|=|A|^2=25
|A|=5[5x-0]+0=25x
\Rightarrow (25x)^2=25
\Rightarrow x^2 =\dfrac {1}{25}
\Rightarrow |x|=\dfrac{1}{5}
.
If a determinant of order
3\times 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
\alpha, \beta
are the roots of
x^2+x+1=0
then
\begin{vmatrix} y+1 & \beta & \alpha\\ \beta & y+\alpha & 1\\ \alpha & 1 & y+\beta\end{vmatrix}=?
Report Question
0%
y^2-1
0%
y(y^2-1)
0%
y^2-y
0%
y^3
\begin{vmatrix} \sin ^{ 2 }{ \theta } & \cos ^{ 2 }{ \theta } \\ -\cos ^{ 2 }{ \theta } & \sin ^{ 2 }{ \theta } \end{vmatrix}=
Report Question
0%
\cos { 2\theta }
0%
\cfrac { 1 }{ 2 } \left( 1+\cos ^{ 2 }{ 2\theta } \right)
0%
\cfrac { 1 }{ 2 } \left( 1-\sin ^{ 2 }{ 2\theta } \right)
0%
\cfrac { 1 }{ 2 } \sin ^{ 2 }{ 2\theta }
Explanation
we know that
(sin^2x+cos^2x)^2=sin^4x+cos^4x+2sin^2xcos^2x
\begin{vmatrix} \sin ^{ 2 }{ \theta } & \cos ^{ 2 }{ \theta } \\ -\cos ^{ 2 }{ \theta } & \sin ^{ 2 }{ \theta } \end{vmatrix}=
\sin ^{ 4 }{ \theta } +\cos ^{ 4 }{ \theta } =1-2\sin ^{ 2 }{ \theta } \cos ^{ 2 }{ \theta }
=1-\cfrac { 1 }{ 2 } \sin ^{ 2 }{ 2\theta }
=1-\cfrac { 1 }{ 2 } \left( 1-\cos ^{ 2 }{ 2\theta } \right)
=\cfrac { 1 }{ 2 } \left( 1+\cos ^{ 2 }{ 2\theta } \right)
The sum of the real roots of the equation
\begin{vmatrix} x & -6 & -1 \\ 2 & -3x & x-3 \\ -3 & 2x & x+2 \end{vmatrix}=0
is equal to
Report Question
0%
6
0%
1
0%
0
0%
-4
Explanation
Given
\begin{vmatrix} x & -6 & -1 \\ 2 & -3x & x-3 \\ -3 & 2x & x+2 \end{vmatrix}=0
By expansion, we get
x(-3x^2+6x)-(-6)(2x+4-3x+3)+(-1)(4x+9x)
\Rightarrow -5{ x }^{ 3 }+30x-30+5x=0\Rightarrow -5{ x }^{ 3 }+35x-30=0\Rightarrow { x }^{ 3 }-7x+6=0
\Rightarrow ({x-1})({x-2})({x-3})
\Rightarrow {x}=1,2,-3
Therefore, All roots are real
So, sum of roots
=0
\begin{vmatrix} a+ib & c+id \\ -c+id & a-ib \end{vmatrix}=
?
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%
none\ of\ these
For square matrices
A
and
B
of the same order, we have
adj (AB) = ?
Report Question
0%
(adj\ A)(adj\ B)
0%
(adj\ B)(adj\ A)
0%
|AB|
0%
None of these
\begin{vmatrix} \cos { { 70 }^{ o } } & \sin { { 20 }^{ o } } \\ \sin { { 70 }^{ o } } & \cos { { 20 }^{ o } } \end{vmatrix}=
?
Report Question
0%
1
0%
0
0%
\cos 50^o
0%
\sin 50^o
Explanation
Let us consider
\Delta=\begin{vmatrix} \cos { { 70 }^{ o } } & \sin { { 20 }^{ o } } \\ \sin { { 70 }^{ o } } & \cos { { 20 }^{ o } } \end{vmatrix}
\Delta =\cos 70^0\cos 20^0-\sin 70^0\sin 20^0
=\cos(70^0+20^0)
=\cos 90^0
\Delta =0
Option
B
.
Evaluate :
\begin{vmatrix} \sin { { 23 }^{ o } } & -\sin { { 7 }^{ o } } \\ \cos { { 23 }^{ o } } & \cos { { 7 }^{ o } } \end{vmatrix}
Report Question
0%
\dfrac {\sqrt 3}{2}
0%
\dfrac {1}{2}
0%
\sin 16^o
0%
\cos 16^o
If
A = \begin{bmatrix} a& b\\c & d\end{bmatrix}
then
adj\ A = ?
Report Question
0%
\begin{bmatrix} d& -c\\ -b & a\end{bmatrix}
0%
\begin{bmatrix} -d& b\\ c & -a\end{bmatrix}
0%
\begin{bmatrix} d& -b\\ -c & a\end{bmatrix}
0%
\begin{bmatrix} -d& -b\\ c & a\end{bmatrix}
If
A
is a
3-rowed
square matrix and
|A| = 5
then
|adj\ A| = ?
Report Question
0%
5
0%
25
0%
125
0%
None of these
Explanation
|adj\ A| = |A|^{(n - 1)} = |A|^{2} = 5^{2} = 25
.
\begin{vmatrix} \cos { { 15 }^{ o } } & \sin { { 15 }^{ o } } \\ \sin { { 15 }^{ o } } & \cos { { 15 }^{ o } } \end{vmatrix}=
?
Report Question
0%
1
0%
\dfrac {1}{2}
0%
\dfrac {\sqrt 3}{2}
0%
none\ of\ these
Explanation
Let us consider
\Delta=\begin{vmatrix} \cos { { 15 }^{ o } } & \sin { { 15 }^{ o } } \\ \sin { { 15 }^{ o } } & \cos { { 15 }^{ o } } \end{vmatrix}=
\Delta=\cos^215^0-\sin^2 15^0
=\cos 30^0
.........
(\because \cos 2\theta=\cos^2 \theta-\sin^2 \theta)
=\dfrac{\sqrt{3}}{2}
Hence
\begin{vmatrix} \cos { { 15 }^{ o } } & \sin { { 15 }^{ o } } \\ \sin { { 15 }^{ o } } & \cos { { 15 }^{ o } } \end{vmatrix}=\dfrac{\sqrt{3}}{2}
Option
C
.
The roots of the equation
\begin{vmatrix} 3{ x }^{ 2 } & { x }^{ 2 }+x\cos { \theta } +\cos ^{ 2 }{ \theta } & { x }^{ 2 }+x\sin { \theta } +\sin ^{ 2 }{ \theta } \\ { x }^{ 2 }+x\cos { \theta } +\cos ^{ 2 }{ \theta } & 3\cos ^{ 2 }{ \theta } & 1+\dfrac { \sin { 2\theta } }{ 2 } \\ { x }^{ 2 }+x\sin { \theta } +\sin ^{ 2 }{ \theta } & 1+\dfrac { \cos { 2\theta } }{ 2 } & 3\sin ^{ 2 }{ \theta } \end{vmatrix}=0
are
Report Question
0%
\sin {\theta},\ \cos {\theta}
0%
\sin^2 {\theta},\ \cos^2 {\theta}
0%
\sin {\theta},\ \cos^2 {\theta}
0%
\sin^2 {\theta},\ \cos {\theta}
When the determinant
\begin{vmatrix} \cos { 2x } & \sin ^{ 2 }{ x } & \cos { 4x } \\ \sin ^{ 2 }{ x } & \cos { 2x } & \cos ^{ 2 }{ x } \\ \cos { 4x } & \cos ^{ 2 }{ x } & \cos { 2x } \end{vmatrix}
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)=\begin{vmatrix} \cos { 2x } & \sin ^{ 2 }{ x } & \cos { 4x } \\ \sin ^{ 2 }{ x } & \cos { 2x } & \cos ^{ 2 }{ x } \\ \cos { 4x } & \cos ^{ 2 }{ x } & \cos { 2x } \end{vmatrix}
the requried constant term is
\quad f(0)=\begin{vmatrix} 1 & 0 & 1 \\ 0 & 1 & 1 \\ 1 & 1 & 1 \end{vmatrix}=\quad \begin{vmatrix} 1 & 0 & 0 \\ 0 & 1 & 1 \\ 1 & 1 & 0 \end{vmatrix}=1(0-1)=-1
If the determinant
\begin{vmatrix} \cos 2x & \sin ^{ 2 } x & \cos 4x \\ \sin ^{ 2 } x & \cos 2x & \cos ^{ 2 } x \\ \cos 4x & \cos ^{ 2 } x & \cos 2x \end{vmatrix}
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
\begin{vmatrix}\cos 2x & \sin^2 x &\cos 4x \\ \sin^2 x &\cos 2x &\cos^2x \\\cos 4x &\cos^2x &\cos 2x\end{vmatrix}= 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-
\begin{vmatrix}1 & 0 & 1\\ 0 & 1 &1 \\ 1 & 1 &1\end{vmatrix} = a_{1}
\Rightarrow a_{1}=\begin{vmatrix}1 & 0 & 1\\ 0 & 1 &1 \\ 0 & 1 &0\end{vmatrix}
(
R_{3} \rightarrow R_{3} - R_{1}
)
\Rightarrow 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} = \begin{bmatrix}3 & -1\\ \dfrac {-5}{3} & \dfrac {2}{3}\end{bmatrix}
then
adj\ A = ?
Report Question
0%
\begin{bmatrix} 9 & 3\\ -5 & -2 \end{bmatrix}
0%
\begin{bmatrix} 9 & -3\\ -5 & 2 \end{bmatrix}
0%
\begin{bmatrix} -9 & 3\\ 5 & -2 \end{bmatrix}
0%
\begin{bmatrix} 9 & -3\\ 5 & -2 \end{bmatrix}
Explanation
A^{-1} = \dfrac {1}{|A|}\cdot adj\ A \rightarrow adj\ A = |A| \cdot A^{-1} = 3A^{-1} = \begin{bmatrix} 9& -3\\ -5 & 2\end{bmatrix}
.
If
\begin{vmatrix} a & b-c & c+b \\ a+c & b & c-a \\ a-b & a+b & c \end{vmatrix}=0
, then the line
ax+by+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}\rightarrow a{C}_{!}
and then
{C}_{1}\rightarrow {C}_{1}+b{C}_{2}+c{C}_{3}
and taking
\left( { a }^{ 2 }+{ b }^{ 2 }+{ c }^{ 2 } \right)
common from
{C}_{1}
we get
\Delta =\cfrac { \left( { a }^{ 2 }+{ b }^{ 2 }+{ c }^{ 2 } \right) }{ a } \begin{vmatrix} 1 & b-c & c+b \\ 1 & b & c-a \\ 1 & b+a & c \end{vmatrix}=\cfrac { \left( { a }^{ 2 }+{ b }^{ 2 }+{ c }^{ 2 } \right) }{ a } \begin{vmatrix} 1 & b-c & c+b \\ 0 & c & -a-b \\ 0 & a+c & -b \end{vmatrix}
({R}_{2}\rightarrow {R}_{2}-{R}_{1},{R}_{3}\rightarrow {R}_{3}-{R}_{1})
=\cfrac { \left( { a }^{ 2 }+{ b }^{ 2 }+{ c }^{ 2 } \right) }{ a } (-bc+{ a }^{ 2 }+ab+ac+bc)=\left( { a }^{ 2 }+{ b }^{ 2 }+{ c }^{ 2 } \right) (a+b+c)
hence
\Delta =0\Rightarrow a+b+c=0
therefore line
ax+by+c=0
passes through the fixed point
(1,1)
If
A = \begin{bmatrix}2 & 5\\ 1 & 3\end{bmatrix}
then
adj\ A = ?
Report Question
0%
\begin{bmatrix}3 & -5\\ -1 & 2\end{bmatrix}
0%
\begin{bmatrix}3 & -1\\ -5 & 2\end{bmatrix}
0%
\begin{bmatrix}-1 & 2\\ 3 & -5\end{bmatrix}
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 adj A = |A| I
\Rightarrow |A adj A | = |A|^n
[ if A is of order
n \times n ]
\Rightarrow |A| |adj A| = |A|^n
\Rightarrow |adj A| = |A|^{n-1}
Now, A is singular .
\therefore |A| = 0
\Rightarrow |adj A | = 0
Hence adj A is singular.
If
A = \left[ \begin{matrix} 1 & -1 & 2 \\ 0 & 3 & 1 \\ 0 & 0 & -1/3 \end{matrix} \right]
, then
Report Question
0%
| A | = -1
0%
adj A = \left[ \begin{matrix} -1 & 1 & -2 \\ 0 & -3 & -1 \\ 0 & 0 & -1/3 \end{matrix} \right]
0%
A = \left[ \begin{matrix} 1 & 1/3 & 7 \\ 0 & 1/3 & 1 \\ 0 & 0 & -3 \end{matrix} \right]
0%
A= \left[ \begin{matrix} 1 & -1/3 & -7 \\ 0 & -3 & 0 \\ 0 & 0 & 1 \end{matrix} \right]
Explanation
|A^{-1}| = -1 \Rightarrow |A|= -1
Now, use
adj A=|A|A^{-1} and A=(A^{-1})^{-1}
There are two values of a which makes determinant
\Delta =\left| \begin{matrix} 1\quad & -2 & 5 \\ 2 & a & -1 \\ 0 & 4 & 2a \end{matrix} \right| =86
then sum of these number is
Report Question
0%
4
0%
5
0%
- 4
0%
9
Explanation
we have
\Delta =\left| \begin{matrix} 1\quad & -2 & 5 \\ 2 & a & -1 \\ 0 & 4 & 2a \end{matrix} \right| =86
\Rightarrow 1(2a^2 +4) -2(-4a -20) + 0 = 86
[expanding along first column ]
\Rightarrow 2a^2 +4 +8a +40 = 86
\Rightarrow 2a^2 + 8a +44 -86 = 0
\Rightarrow a^2 +4a -21 = 0
\Rightarrow a^2 +7a -3a-21 =0
\Rightarrow (a +7)(a-3) = 0
a = -7
and
3
\therefore
Required sum =
-7 +3 = -4
If
\begin{vmatrix} 2x & 5 \\ 8 & x \end{vmatrix}=\begin{vmatrix} 6 & -2 \\ 7 & 3 \end{vmatrix}
then value of
x
is
Report Question
0%
3
0%
\pm 3
0%
\pm 6
0%
6
Explanation
\because \begin{vmatrix} 2x & 5 \\ 8 & x \end{vmatrix}=\begin{vmatrix} 6 & -2 \\ 7 & 3 \end{vmatrix}
\Rightarrow 2x^2 -40 = 18 +14
\Rightarrow 2x^2 = 32 +40
\Rightarrow x^2 = \frac {72}{2} = 36
\therefore x = \pm 6
Choose the correct answer from the given alternatives in the following question:
If
A = \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix} , {\text{adj}} A = \begin{bmatrix} 4 & a \\ -3 & b \end{bmatrix}
, 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 = \begin{bmatrix} 2 & -4 \\ 3 & 1 \end{bmatrix}
, then the adjoint of matrix
A
is
Report Question
0%
\begin{bmatrix} -1 & 3 \\ -4 & 1 \end{bmatrix}
0%
\begin{bmatrix} 1 & 4 \\ -3 & 2 \end{bmatrix}
0%
\begin{bmatrix} 1 & 3 \\ 4 & -2 \end{bmatrix}
0%
\begin{bmatrix} -1 & -3 \\ -4 & 2 \end{bmatrix}
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
\therefore |A| = 12
Also we know that if
A
is a square matrix of order
n
then
| adj A| = |A|^{n-1}
For
n = 3 , |Adj A| = |A|^{3-1} =|A|^2
=(12)^2 = 144
Choose the correct answer from the given alternatives in the following question:
If
A = \begin{bmatrix} 1 & 2 \\ 2 & 1 \end{bmatrix}
and
A ({\text{adj}} A) = KI
, 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
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