CBSE Questions for Class 12 Commerce Maths Determinants Quiz 3 - MCQExams.com

Adj $$\begin{bmatrix}
1 & 0 & 2\\
-1 & 1 & -2\\
0& 2 & 1
\end{bmatrix}$$= $$\begin{bmatrix}
5 & a & -2\\
1 & 1 & 0\\
-2& -2 & b
\end{bmatrix}$$ $$\Rightarrow [a\, \, \, \, \, b]=$$ 
  • $$[-4\, \, \, \, \, \, 1]$$
  • $$[-4\, \, \, \, \, \, -1]$$
  • $$[4\, \, \, \, \, \, 1]$$
  • $$[4\, \, \, \, \, \, -1]$$
A determinant of second order is made with the elements $$0$$ and $$1.$$ The number of determinants with non-negative values is:
  • 3
  • 10
  • 11
  • 13
If $$A=\displaystyle \int_{1}^{sin\theta}\frac{t}{1+t^{2}}dt$$ and
$$B=\displaystyle \int_{1}^{cosec\theta}\frac{1}{t(1+t^{2})}dt$$, then the value of determinant  $$\begin{vmatrix}
A & A^{2}& B\\
e^{A+B}& B^{2} &-1 \\
1& A^{2}+B^{2} & -1
\end{vmatrix}$$ is 
  • $$\sin\theta$$
  • $$cosec \theta$$
  • $$0$$
  • $$1$$
I. If A,B,C are angles of angle and $$\begin{vmatrix}
1 & 1 & 1\\
1+sinA& 1+sinB&1+sinC \\
sinA+sin^{2}A & sinB+sin^{2}B & sinC+sin^{2}C
\end{vmatrix}$$ =0  then triangle is isosceles
 II. lf $$a=1+2+4+---$$ upto $$\mathrm{n}$$ terms $$b=1+3+9+---$$ up to $$\mathrm{n}$$ terms $$c=1+5+25+----$$up to $$\mathrm{n}$$ terms  then $$\Delta \begin{vmatrix}
a &2b &4c \\
2& 2& 2\\
2^{n} & 3^{n} & 5^{n}
\end{vmatrix}$$ =0
  • I, II both are true
  • only I is true
  • only II is true
  • neither of them are true
$$1\mathrm{f}\mathrm{A}=\left[\begin{array}{lll}
1 & 5 & -6\\
-8 & 0 & 4\\
3 & -7 & 2
\end{array}\right]$$ then the cofactors of the elements $$3,-7,2$$ are p,q,r respectively their ascending order is 
  • $$\mathrm{p},\ \mathrm{r}, \mathrm{q}$$
  • $$\mathrm{q},\mathrm{r},\ \mathrm{p}$$
  • $$\mathrm{p},\mathrm{q},\mathrm{r}$$
  • $$\mathrm{r},\mathrm{p},\ \mathrm{q}$$
If $$f(x)=$$ $$\begin{vmatrix}
\sin\, \, x &1 &0 \\
1& 2\sin\, \, x& 1\\
0& 1 & 2\sin\, \, x
\end{vmatrix}$$ then $$\displaystyle \int _{-\frac{\pi }{2}}^{\frac{\pi }{2}} f\left ( x \right )$$ equals 
  • $$0$$
  • $$-1$$
  • $$1$$
  • $$\dfrac{3\pi }{2}$$
$$\begin{vmatrix}
1 & cos\alpha & cos\beta \\
cos\alpha & 1 & cos\gamma \\
cos\beta &cos\gamma & 1
\end{vmatrix}$$ = $$\begin{vmatrix}
0 & cos\alpha & cos\beta \\
cos\alpha & 0 & cos\gamma \\
cos\beta &cos\gamma & 0
\end{vmatrix}$$ then 
  • $$cos\alpha +cos\beta +cos\gamma =0$$
  • $$cos\alpha .cos\beta .cos\gamma =0$$
  • $$cos^{2} \, \, \alpha +cos^{2}\, \, \beta + cos^{2}\gamma =1$$
  • $$\sum cos \: \: \alpha \, cos\, \, \beta \, \, =0$$
Match the following elements of $$\begin{vmatrix}
1 & -1 &0 \\
0& 4 & 2\\
3 & -4 & 6
\end{vmatrix}$$ with their cofactors and choose the correct
answer 
Element                                                        Cofactor
I. -1                                                                   a) -2
II. 1                                                                   b) 32
III. 3                                                                  c) 4
IV. 6                                                                  d) 6
                                                                         e) -6
  • $$I-b,II-d,III-a,IV-c$$
  • $$I-b,II-d,III-c,IV-a$$
  • $$I-d,II-b,III-a,IV-c$$
  • $$I-d,II-a,III-b,IV-c$$
If $${A}=\begin{bmatrix}
cos\theta & sin\theta\\
-sin\theta & cos\theta
\end{bmatrix}$$ then $$\displaystyle \lim_{n\rightarrow\infty}\frac{1}{n}|A^{n}|=$$
  • $$1$$
  • $$0$$
  • $$A$$
  • $$\displaystyle \frac{1}{n}A$$
$$\mathrm{D}\mathrm{e}\mathrm{t} \left\{\begin{array}{lll}
-2a & a+b & c+a\\
b+a & -2b & b+c\\
c+a & c+b & -2c
\end{array}\right\}=$$ 
  • $$(\mathrm{a}+\mathrm{b})(\mathrm{b}+\mathrm{c})(\mathrm{c}+\mathrm{a})$$
  • $$(a-b) (b-c) (c-a)$$
  • $$4(\mathrm{a}+\mathrm{b})(\mathrm{b}+\mathrm{c})(\mathrm{c}+\mathrm{a})$$
  • $$4(a-b) (b-c) (c-a)

    $$
The sum of infinite series $$\begin{vmatrix}
1 &2 \\
6 & 4
\end{vmatrix}+\begin{vmatrix}
\frac{1}{2} &2 \\
2& 4
\end{vmatrix}+\begin{vmatrix}
\frac{1}{4} & 2\\
\frac{2}{3}& 4
\end{vmatrix}+$$ ....... is 
  • $$-10$$
  • $$0$$
  • $$10$$
  • $$\infty $$
lf $$f(x)=\left| \begin{matrix} \sec { x }  & \cos { x }  \\ \cos ^{ 2 }{ x }  & \cos ^{ 2 }{ x }  \end{matrix} \right| $$, then $$\displaystyle \int_{0}^{\pi/2}f(x)dx=$$
  • 1/2
  • 1/3
  • 0
  • 1
lf $$a\neq b\neq c$$ Then one value of $$\mathrm{x}$$ which satisfies the equation $$\begin{vmatrix}
0 & x-a &x-b \\
x+a & 0 & x-c\\
x+b& x+c& 0
\end{vmatrix}$$ = 0 is given by 
  • x = a
  • x = b
  • x = c
  • x = 0
.Let $$\begin{vmatrix}
x&2 & x\\
x^{2}&x & 6\\
x & x& 6
\end{vmatrix}$$ =$$ax^{4}+bx^{3}+cx^{2}+dx+e$$ ,then the value of 5a + 4b + 3c + 2d + e is
equal to 
  • 0
  • 16
  • -16
  • -11
$$\begin{vmatrix}
1+i &1-i &1 \\
1-i& i&1+i \\
i & 1+i & 1-i
\end{vmatrix}$$ is a 
  • real number
  • irrational number
  • complex member
  • Purely imaginary
$${A}=\begin{bmatrix}
-1 & -2 & -2\\
2 & 1 & -2\\
2 & -2 & 1
\end{bmatrix}$$ then $$Adj(A)=$$ 
  • $$\mathrm{A}^{\mathrm{T}}$$
  • $$3\mathrm{A}^{\mathrm{T}}$$
  • $$\mathrm{A}^{-1}$$
  • $$-\mathrm{A}^{\mathrm{T}}$$
If $$\Delta $$ = $$\begin{vmatrix} cos\frac{\theta }{2} &1  &1 \\   1&cos\frac{\theta }{2}  &-cos\frac{\theta }{2} \\ -cos\frac{\theta }{2} &1  & -1
\end{vmatrix}$$ the minimum of $$\Delta $$ is $$m_{1}$$ and maximum of $$\Delta $$ is $$m_{2}$$ then $$[m_{1} , m_{2}]$$ is
  • [ 4, 2]
  • [2,4]
  • [4,0]
  • [0,2]
$$f(x)=$$ $$\begin{vmatrix}
\cos x &x &1 \\
2 \sin x & x^{2} &2x \\
\tan x& x & 1
\end{vmatrix}$$ then $$ \displaystyle \lim _{ x\rightarrow 0 }{ f(x) } =$$
  • 0
  • -1
  • -2
  • 2
The value of $$\begin{vmatrix}
1+x&2 & 3\\
1& 2+x&3\\
1 & 2 & 3+x
\end{vmatrix}$$ is 
  • $$(x-6)x^{2}$$
  • $$(x-6)x$$
  • $$x^{2}(x+6)$$
  • $$(x-6)$$
If $$[\mathrm{x}]$$ stands greatest integer $$\leq \mathrm{x}$$ then the value of $$\begin{vmatrix}
\left [ e \right ]& \left [ \pi \right ] &\left [ \pi ^{2}-6 \right ]\\
\left [ \pi \right ] & \pi ^{2}-6 & \left [ e \right ]\\
\left [ \pi ^{2}-6 \right ]&\left [ e \right ] & \left [ \pi \right ]
\end{vmatrix}$$ equals 
  • -8
  • 8
  • -1
  • 1
lf $$adjA=\left\{\begin{array}{lll}
1 & -1 & 0\\
2 & 3 & 1\\
2 & 1 & -1
\end{array}\right\}$$ then adj 2 $$\mathrm{A}=$$ 
  • $$\left\{\begin{array}{lll}

    2 & -2 & 0\\

    4 & 6 & 2\\

    4 & 2 & -2

    \end{array}\right\}$$
  • $$\left\{\begin{array}{ll}

    4&-4 & 0\\

    8&12 & 4\\

    8&4 & -4

    \end{array}\right\}$$
  • $$\left\{\begin{array}{lll}

    8 & -8 & 0\\

    16 & 24 & 8\\

    16 & 8 & -8

    \end{array}\right\}$$
  • $$\left\{\begin{array}{lll}

    1 & -1 & 0\\

    2 & 3 & 1\\

    2 & 1 & -1

    \end{array}\right\}$$
$$A=\begin{bmatrix}4 &-2&5\end{bmatrix}$$, $$ B=$$ $$\begin{bmatrix}
2\\
0\\
3\end{bmatrix}$$, then $$Adj(BA)=$$ 
  • $$\left\{\begin{array}{lll}

    0 & 0 & 0\\

    0 & 0 & 0\\

    0 & 0 & 0

    \end{array}\right\}$$
  • $$\left\{\begin{array}{lll}

    8 & -4 & 10\\

    0 & 0 & 0\\

    12 & -6 & 15

    \end{array}\right\}$$
  • $$\left\{\begin{array}{lll}

    8 & 0 & 12\\

    -4 & 0 & -6\\

    10 & 0 & 5

    \end{array}\right\}$$
  • None of the above
.Let $$\Delta$$ $$(x)=$$$$\begin{vmatrix}
x+a & x+b &x+a-c \\
x+b & x+c &x-1 \\
x+c & x+d &x-b+d
\end{vmatrix}$$ and $$\displaystyle \int_{0}^{2}\Delta(x) dx =-16$$, where $$\mathrm{a}$$, b,c,d are in A.$$\mathrm{P}$$. then the common difference of the A.$$\mathrm{P}$$. is 
  • $$\pm 1$$
  • $$\pm 2$$
  • $$\pm 3$$
  • $$\pm 4$$
A= $$\begin{bmatrix}
 b^{2}c^{2}& bc & b+c\\
 c^{2}a^{2}& ca &c+a \\
 a^{2}b^{2}& ab & a+b
\end{bmatrix}$$ then $$\left | A \right |$$ =?
  • $$abc$$
  • $$abc-1$$
  • $$abc+1$$
  • $$0$$
The straight lines $$\mathrm{x}+2\mathrm{y}-9=0,3\mathrm{x}+5\mathrm{y}-5=0$$ and $$\mathrm{a}\mathrm{x}+\mathrm{b}\mathrm{y}-1=0$$ are concurrent if the straight line $$22\mathrm{x}-35\mathrm{y}-1=0$$ passes through the point 
  • (a, b)
  • (b,a)
  • (-a,b)
  • (-a, -b)
If maximum and minimum values of the determinant $$\begin{vmatrix} 1+\sin ^{ 2 }{ x }  & \cos ^{ 2 }{ x }  & \sin { 2x }  \\ \sin ^{ 2 }{ x }  & 1+\cos ^{ 2 }{ x }  & \sin { 2x }  \\ \sin ^{ 2 }{ x }  & \cos ^{ 2 }{ x }  & 1+\sin { 2x }  \end{vmatrix}$$ are $$\alpha$$ and $$\beta$$, then 
  • $$\alpha +{ \beta  }^{ 99 }=4$$
  • $${ \alpha  }^{ 3 }-{ \beta  }^{ 17 }=26$$
  • $$\left( { \alpha  }^{ 2n }-{ \beta  }^{ 2n } \right) $$ is always an even integer for $$n\in N$$
  • a triangle can be constructed having its sides as $$\alpha -\beta ,\alpha +\beta $$ and $$\alpha+3\beta$$

$$\mathrm{y}=\sin \mathrm{x},\ y_{n}=\displaystyle \frac{d^{n}(\sin x)}{dx^{n}}$$
then $$\begin{vmatrix}
y& y_{1} & y_{2}\\
 y_{3}& y_{4} &y_{5} \\
 y_{6}& y_{7} & y_{8}
\end{vmatrix}$$ =?
  • -sin x
  • 0
  • sin x
  • cos x
If f (x) = tan x and A, B, C are the angles of $$\Delta ABC$$, then $$\begin{vmatrix}
f(A) & f(\pi /4) & f(\pi /4)\\
f(\pi /4) &f(B) & f(\pi /4)\\
f(\pi /4) &f(\pi /4) & f(C)
\end{vmatrix}$$ 
  • 0
  • -2
  • 2
  • 1
If the lines $$2\mathrm{x}-\mathrm{a}\mathrm{y}+1 =0$$,$$\ 3\mathrm{x}-\mathrm{b}\mathrm{y}+1 =0$$,$$\ 4\mathrm{x}-\mathrm{c}\mathrm{y}+1 =0$$ are concurrent then $$a,b,c$$ are in ?

.
  • G.P.
  • A.P.
  • H.P.
  • A.G.P.
If $$\mathrm{a}\neq b\neq \mathrm{c}$$ and if $$ax+by+\mathrm{c}=0\  bx+cy+\mathrm{a}=0$$ and $$cx+ay+b=0$$ are concurrent, 
then find the value of 
$$ 2^{\mathrm{a}^{2}b^{-1}\mathrm{c}^{-1}}2^{b^{2}\mathrm{c}^{-1}\mathrm{a}^{-1}}2^{\mathrm{c}^{2}\mathrm{a}^{-1}b^{-1}}$$
  • 1
  • 4
  • 8
  • 16
lf the lines $$3\mathrm{x}+2\mathrm{y}-5=0,\ 2\mathrm{x}-5\mathrm{y}+3=0,\ 5\mathrm{x}+\mathrm{b}\mathrm{y}+\mathrm{c}=0$$ are concurrent then $$\mathrm{b}+\mathrm{c}=$$
  • 7
  • -5
  • 6
  • 9
If $$\alpha,\ \beta$$ are the roots of  $$\begin{vmatrix}
x & 1 & 2\\
0 & 1& 1\\
1 & x & 2
\end{vmatrix}$$ = 0 then $$\alpha^{n}+\beta^{n}=?$$ 
  • 0
  • 1
  • 2
  • 2n
If the points $$A (1, 2), O (0, 0)$$ and $$C (a, b)$$ are collinear, then
  • $$a=b$$
  • $$a=2b$$ 
  • $$2a=b$$
  • $$a=-b$$
Relation between $$x$$ and $$y$$, if the points $$(x, y), (1, 2)$$ and $$(7, 0)$$ are collinear is _____
  • $$x+3y=7$$
  • $$x+3y=14$$
  • $$-x+3y=7$$
  • $$-x+3y=14$$
Points $$(1, 5), (2, 3)$$ and $$(-2, -11) $$ are ____
  • Non-collinear
  • Collinear
  • Vertices of equilateral triangle
  • Vertices of right angle triangle
P, Q, R are three collinear points. The coordinates of P and R are (3, 4) and (11, 10) respectively and PQ is equal to 2.5 units. Coordinates of Q are-
  • (5, 11/2)
  • (11, 5/2)
  • (5, -11/2)
  • (-5, 11/2)
Find the value of $$m$$ if the points $$(5, 1), (2, 3)$$ and $$(8, 2m )$$ are collinear.
  • $${-1}$$
  • $$\dfrac{1}{2}$$
  • $$-\dfrac{1}{2}$$
  • $${2}$$
If $$a\neq p, b\neq q, c\neq r$$ and $$\begin{vmatrix}p & b & c\\ a & q & c\\ a & b & r\end{vmatrix}=0$$, then the value of $$\dfrac{p}{p-a}+\dfrac{q}{q-b}+\dfrac{r}{r-c}$$ is
  • $$0$$
  • $$1$$
  • $$-1$$
  • $$-2$$
If the points $$(0, 0), (1, 2)$$ and $$(x, y)$$ are collinear, then
  • $$x = y$$
  • $$2x = y$$
  • $$x =2 y$$
  • $$2x = -y$$
$$\left| {\begin{array}{*{20}{c}}
  {{{\sin }^2}x}&{{{\cos }^2}x}&1 \\
  {{{\cos }^2}x}&{{{\sin }^2}x}&1 \\
  { - 10}&{12}&2
\end{array}} \right| = $$
  • $$0$$
  • $$12 cos^2x -10 sin^2x$$
  • $$12 sin^2x -10 cos^2x - 2$$
  • $$10 sin 2x$$
$$\mathrm{If}\mathrm{a}^{2}+\mathrm{b}^{2}+\mathrm{c}^{2}=-2$$ and $$\mathrm{f}(\mathrm{x})=$$ $$\begin{vmatrix}1+a^{2}x &(1+b^{2})x &(1+c^{2})x \\(1+a^{2})x&1+b^{2}x &(1+c^{2})x\\(1+a^{2})x&(1+b^{2})x & 1+c^2x\end{vmatrix}$$  then $$f(x)$$ is a polynomial of degree 
  • $$1$$
  • $$0$$
  • $$3$$
  • $$2$$
Find the values of $$k,$$ if the points $$A (k + 1, 2k), B (3k, 2k + 3)$$ and $$C (5k +1, 5k)$$ are collinear.
  • $$1$$
  • $$ \dfrac{1}{2}$$
  • $$2$$
  • $$2.5$$
If A is a square matrix, then $$adj A^{T} - (adj  A)^T$$ is equal to
  • 2 |A|
  • 2 |A| I
  • null matrix
  • unit matrix
If $$B=A$$, $$|P|=|Q|=1$$, then $$adj({ Q }^{ -1 }{ BP }^{ -1) }$$ is
  • $$P.Q$$
  • $$Q.adj(A).P$$
  • $$P.adj(A).Q$$
  • $$P.{ A }^{ -1 }.Q$$
If $$A = \begin{bmatrix}-5 & 2\\ 1 & -3\end{bmatrix}$$, then adj A is equal to
  • $$\begin{bmatrix}-3 & -2\\ -1 & -5\end{bmatrix}$$
  • $$\begin{bmatrix}3 & -2\\ -1 & 5\end{bmatrix}$$
  • $$\begin{bmatrix}5 & 1\\ 2 & 3\end{bmatrix}$$
  • $$\begin{bmatrix}3 & 2\\ 1 & 5\end{bmatrix}$$
If $$P$$ is a non singular matrix, then value of $$adj({ P }^{ -1 })$$ in terms of $$P$$ is
  • $${ P }/{ |P| }$$
  • $${ P }{ |P| }$$
  • $$P$$
  • none of these
The adjoint of $$\begin{bmatrix}1 & 1 & 1\\ 1 & 2 & -3\\ 2 & -1 & 3\end{bmatrix}$$ is
  • $$\begin{bmatrix}3 & -9 & -5\\ -4 & 1 & 3\\ -5 & 4 & 1\end{bmatrix}$$
  • $$\begin{bmatrix}3 & -4 & -5\\ -9 & 1 & 4\\ -5 & 3 & 1\end{bmatrix}$$
  • $$\begin{bmatrix}-3 & 4 & 5\\ 9 & -1 & -4\\ 5 & -3 & -1\end{bmatrix}$$
  • None of these
If $$(3, 2)$$, $$\left (x, \dfrac {22}{5}\right), (8, 8)$$ lie on a line, then $$x$$ is equal to
  • $$-5$$
  • $$2$$
  • $$4$$
  • $$5$$
Given the matrix $$A=\begin{bmatrix} x & 3 & 2 \\ 1 & y & 4 \\ 2 & 2 & z \end{bmatrix}$$. If $$xyz=60$$ and $$8x+4y+3z=20$$, then $$A(adj\ A)$$ is equal to
  • $$A=\begin{bmatrix} 64 & 0 & 0 \\ 0 & 64 & 0 \\ 0 & 0 & 64 \end{bmatrix}$$
  • $$A=\begin{bmatrix} 88 & 0 & 0 \\ 0 & 88 & 0 \\ 0 & 0 & 88 \end{bmatrix}$$
  • $$A=\begin{bmatrix} 68 & 0 & 0 \\ 0 & 68 & 0 \\ 0 & 0 & 68 \end{bmatrix}$$
  • $$A=\begin{bmatrix} 34 & 0 & 0 \\ 0 & 34 & 0 \\ 0 & 0 & 34 \end{bmatrix}$$
If the points $$(0, 4), (4, 0)$$ and $$(5,  p)$$ are collinear, then value of $$p$$ is
  • $$- 1$$
  • $$7$$
  • $$6$$
  • $$4$$
0:0:1


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