CBSE Questions for Class 12 Commerce Maths Determinants Quiz 10 - MCQExams.com

Let $$A$$ be a square matrix of order $$3\times 3$$ then $$\left| KA \right| $$ is equal to
  • $$K\left| A \right| $$
  • $${K}^{2}\left| A \right| $$
  • $${K}^{3}\left| A \right| $$
  • $$3\left| KA \right| $$
Find the equation of line joining $$(1,2)$$ and $$(3,6)$$ using determinants. Let $$p(x,y)$$ be any point on the line joining $$(1,2)(3,6)$$
  • $$y=x$$
  • $$2y=x$$
  • $$y=2x$$
  • $$y=-2x$$
Find the equation of the line joining $$(3,1)$$ and $$(9,3)$$ using determinants.
  • $$x=-3y$$
  • $$y=3x$$
  • $$y=-3x$$
  • $$3y=x$$
Value of determinant $$\begin{vmatrix} \cos 50^\circ & \sin 10^\circ\\ \sin 50^\circ & \cos 10^\circ \end{vmatrix}$$ is:
  • $$0$$
  • $$1$$
  • $$1/2$$
  • $$-1/2$$
Value of determinant $$\begin{vmatrix} \cos 80^\circ & -\cos 10^\circ\\ \sin 80^\circ & \sin 10^\circ \end{vmatrix}$$ is:
  • $$0$$
  • $$1$$
  • $$-1$$
  • None of these
Co-factors of the first column of determinant
$$\begin{vmatrix} 5 & 20 \\ 3 & -1\end{vmatrix}$$
  • $$-1, 3$$
  • $$-1,- 3$$
  • $$- 1, 20$$
  • $$- 1,- 20$$
If $$a, b, c$$ are the $$p^{th}, q^{th}$$ and $$r^{th}$$ terms of an $${H}.{P}$$, then the lines $$bcx+py+1=0,\ cax+ qy+1=0$$ and $$abx+ry+1=0$$,
  • are concurrent
  • form a triangle
  • are parallel
  • mutually perpendicular lines
If $$x_{1},y_{1}$$ are the roots of $$x^{2}+8x-20=0$$ and  $$x_{2},y_{2}$$ are the roots of $$4x^{2}+32x-57=0$$ and $$x_{3},y_{3}$$ are the roots of $$9x^{2}+72x-112=0$$ such that $$y_{i}<0,$$ then the points $$(x_{1},y_{1}),(x_{2},y_{2})$$ and $$(x_{3},y_{3})$$
  • are collinear
  • form an equilateral triangle
  • form a right angled isosceles triangle
  • are concyclic
If the lines $$\mathrm{x}+\mathrm{p}\mathrm{y}+\mathrm{p}=0,\ \mathrm{q}\mathrm{x}+\mathrm{y}+\mathrm{q}=0$$ and $$\mathrm{r}\mathrm{x}+\mathrm{r}\mathrm{y}+1 =0 (\mathrm{p},\mathrm{q}, \mathrm{r}$$ being distinct and $$ \neq$$ 1) are concurrent, then the value of
$$\displaystyle \frac{p}{p-1}+\frac{q}{q-1}+\frac{r}{r-1}=$$
  • $$1$$
  • $$-1$$
  • $$2$$
  • $$-2$$
The lines $$px+qy+r=0,qx+ry+p=0 \, \,and\,\,\, rx+py+q=0$$ are concurrent then 
  • $$p+q+r=0$$
  • $$p^{3}+q^{3}+r^{3}=3pqr$$
  • $$p^{2}+q^{2}+r^{2}-pq-qr-rp=0$$
  • $$p^{2}+q^{2}+r^{2}=2(pq+qr+rp)$$
The coordinates of the point $$P$$ on the line $$2x+3y+1=0$$ such that $$|PA-PB|$$ is maximum, where $$A(2, 0)$$ and $$B(0, 2)$$ is
  • $$(4, -3)$$
  • $$(7, -5)$$
  • $$(10, -7)$$
  • $$(-8, 5)$$
If the lines $$p_{1}x+q_{1}y=1,p_{2}x+q_{2}y=1 $$ and $$ p_{3}x+q_{3}y=1$$ be concurrent, then the points $$(p_{1},q_{1}),(p_{2},q_{2})$$ and $$(p_{3},q_{3})$$ ,
  • are collinear
  • form an equilateral triangle
  • form a scalene triangle
  • form a right angled triangle
If the lines $${x}+{a}{y}+{a}=0,\ {b}{x}+{y}+{b}=0,\ {c}{x}+{c}{y}+1 =0 ({a}\neq{b}\neq {c}\neq1)\ $$ are concurrent, then the value of $$\displaystyle \frac{{a}}{{a}-1}+\frac{{b}}{{b}-1}+\frac{{c}}{{c}-1}$$, is
  • $$-1$$
  • $$0$$
  • $$1$$
  • $$3$$
If $$x_1, x_2, x_3$$ as well as $$y_1, y_2, y_3$$ are in G.P. with same common ratio, then the points $$P(x_1, y_1), Q (x_2, y_2)$$ and $$R(x_3, y_3)$$
  • lies on a straight line
  • lie on an ellipse
  • lie on a circle
  • are vertices of a triangle
The values of $$|A^{50}|$$ equals
  • 0
  • 1
  • -1
  • 25
The value of $$|\cup|$$ equals
  • $$0$$
  • $$1$$
  • $$2$$
  • $$-1$$
If $$A =\begin{bmatrix}2 & -1 & 3\\ -5 & 3 & 1\\ -3 & 2 & 3\end{bmatrix}$$, then $$A.(Adj A)=$$
  • $$\left( Adj{ .A }^{ T } \right) $$
  • $$(Adj. A) . A$$
  • $$ |A| . A$$.
  • None of these
If A is a square matrix of order 3, then $$|(A - A^T)^{105}|$$ is equal to
  • $$105|A|$$
  • $$105|A|^2$$
  • $$105$$
  • none of these
Let $$ \begin{vmatrix}
 1+x         &                 x        &                x^{2}\\
   x           &               1+x   &                       x^{2} \\
  x^{2}        &                x       &                1+x
\end{vmatrix} =   ax^{5} + bx^{4} + cx^{3} + dx^{2} + \lambda x + \mu $$ be an identity in x, where a,b,c,d,$$ \lambda, \mu$$ are independent of x. Then the value of $$\lambda$$ is
  • $$3$$
  • $$2$$
  • $$4$$
  • $$1$$
The value of |B| is equal to
  • |A|
  • |A|/2
  • 2|A|
  • none of these
Let f (n)= $$\displaystyle \left | \begin{matrix}n &n+1  &n+2 \\^{n}P_{n}  &^{n+1}P_{n+1}  &^{n+2}P_{n+2} \\^{n}C_{n}  &^{n+1}C_{n+1}  &^{n+2}C_{n+2} \end{matrix} \right |,$$ where the symbols have their usual meanings. The $$f(n)$$ is divisible by
  • $$ n^{2}+n+1 $$
  • $$(n+1)!$$
  • $$n!$$
  • none of these
Consider the points $$P=(-\sin (\beta -\alpha ), -\cos \beta )$$, $$Q=(\cos (\beta -\alpha ), \sin \beta )$$ and $$R=(\cos (\beta -\alpha +\theta ), \sin (\beta -\theta ))$$, where $$0< \alpha , \beta < \dfrac{\pi }{4}$$ then 
  • $$P$$ lies on the line segment $$RQ$$
  • $$Q$$ lies on the line segment $$PR$$
  • $$R$$ lies on the line segment $$QP$$
  • $$P, Q, R$$ are non-collinear.
If the points $$(a, 1), (1, b)$$ and $$(a -1, b -1)$$ are collinear, $$\alpha ,\beta $$ are respectively the arithmetic and geometric means of $$a$$ and $$b $$, then $$4\alpha -\beta^{2}$$ is equal to
  • $$-1$$
  • $$0$$
  • $$3$$
  • $$2$$
If the points $$\displaystyle(-2,0),(-1,\dfrac{1}{\sqrt{3}})$$ and $$\displaystyle(\cos\theta,\sin \theta)$$ are collinear, then the number of values of $$\displaystyle \theta \in [0,2\pi]$$ :
  • $$0$$
  • $$1$$
  • $$2$$
  • infinite
If $$A=\begin{bmatrix}a&b  &c \\x &y  &z \\p &q  &r \end{bmatrix}$$, $$B=\begin{bmatrix}q&-b  &y \\-p &a  &-x \\r &-c  &z \end{bmatrix}$$ then
  • $$|A| = |B|$$
  • $$|A| = -|B|$$
  • $$|A| =2|B|$$
  • $$A$$ is invertible if and only if $$B$$ is invertible.
Let $$\begin{vmatrix} x& 2 & x\\ x^2 & x & 6\\ x & x & 6\end{vmatrix}  = \alpha x^4 + \beta x^3 + \gamma x^2 + \delta x + \lambda$$ then the value of $$5 \alpha + 4 \beta + 3\gamma + 2 \delta + \lambda = $$
  • $$-11$$
  • $$0$$
  • $$-16$$
  • $$16$$
If [x] stands greatest integer $$\leq x$$ then the value of
$$\begin{vmatrix}
\left [ e \right ] & \left [ \pi  \right ] & \left [ \pi ^{2}-6 \right ]\\
\left [ \pi  \right ] & \left [ \pi ^{2}-6 \right ] & \left [ e \right ]\\
\left [ \pi ^{2}-6 \right ] & \left [ e \right ] & \left [ \pi  \right ]
\end{vmatrix}$$ equals to=?
  • -8
  • 8
  • -1
  • 1
If $$\Delta =\begin{vmatrix}
x+1 & x+2 & x+a\\
x+2 & x+3 & x+b\\
x+3 & x+4 & x+c
\end{vmatrix}=0$$, then
the family of lines $$ax+by+c=0$$ passes through
  • $$(1, -1)$$
  • $$(1, -2)$$
  • $$(2, -3)$$
  • $$(0, 0)$$
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 two similar matrices, then
  • $$det \: (A) = det \: (B)$$
  • $$det \: (A) + det \: (B)=0$$
  • $$det (AB)\neq 0$$
  • none of these
Let $$0< \theta < \pi /2$$ and
$$\Delta \left ( x, \theta  \right )=\begin{vmatrix}
x & \tan \theta  & \cot \theta \\
-\tan \theta  & -x & 1\\
\cot \theta  & 1 & x
\end{vmatrix}$$
then
  • $$\Delta \left ( 0, \theta \right )=0$$
  • $$\Delta \left ( x, \dfrac {\pi }{4} \right )=x-x^3$$
  • $$Min_{0< \theta < \pi /2}\Delta \left ( 1, \theta \right )=0$$
  • $$\Delta \left ( x, \theta \right )$$ is independent of x
If $$\Delta =\begin{vmatrix} a_{11} & a_{12} & a_{13}\\ a_{21} & a_{22} & a_{23}\\ a_{31} & a_{32} & a_{33} \end{vmatrix}$$ and $$c_{ij}=\left ( -1 \right )^{i+j}$$ (determinant obtained by deleting ith row and jth column), then $$\begin{vmatrix} c_{11} & c_{12} & c_{13}\\ c_{21} & c_{22} & c_{23}\\ c_{31} & c_{32} & c_{33} \end{vmatrix}=\Delta ^{2}$$


If $$\begin{vmatrix} 1 & x & x^{ 2 } \\ x & x^{ 2 } & 1 \\ x^{ 2 } & 1 & x \end{vmatrix}=7$$ and $$\Delta =\begin{vmatrix}
x^{3}-1 & 0 & x-x^{4}\\
0 & x-x^{4} & x^{3}-1\\
x-x^{4} & x^{3}-1 & 0
\end{vmatrix}$$, then
  • $$\Delta =7$$
  • $$\Delta =343$$
  • $$\Delta =-49$$
  • $$\Delta =49$$
The determinant $$\begin{vmatrix}
\sin \alpha  & \cos \alpha  & 1\\
\sin \beta  & \cos \beta  & 1\\
\sin \gamma  & \cos \gamma  & 1
\end{vmatrix}$$ is equal to
  • $$\displaystyle -4\sin \frac{\alpha -\beta }{2}\sin \frac{\alpha -\gamma }{2}\sin \frac{\gamma -\alpha }{2}$$
  • $$\sin \alpha +\sin \beta +\sin \gamma $$
  • $$\sin \left ( \alpha -\beta \right )+\sin \left ( \beta -\gamma \right )+\sin \left ( \gamma -\alpha \right )$$
  • none of these
The number of distinct real roots of $$\begin{vmatrix} \sin\, x&\cos\, x&\cos \,x \\ \cos\, x &\sin\, x&\cos\, x \\ \cos\, x&\cos\, x&\sin \, x\end{vmatrix}=0$$ in the interval $$-\dfrac{\pi}{4} < x \le \dfrac{\pi}{4}$$ is
  • $$0$$
  • $$2$$
  • $$1$$
  • $$> 2$$
Say true or false:
Points $$P(10,\,-6),\,Q(6,\,-4)$$ and $$C(-8,\,3)$$ are collinear. 
  • True
  • False
If the points $$(-1,3), (2,p)$$ and $$(5,-1)$$ are collinear, the value of $$p$$ is
  • $$1$$
  • $$-1$$
  • $$0$$
  • $$\displaystyle \sqrt{2}$$
$$(1,6), (3.-2)$$ and $$(-2,K)$$ are collinear points. What is $$K$$?
  • $$-6$$
  • $$2$$
  • $$8$$
  • $$10$$
  • $$18$$
If $$a_{1}, a_{2}, ...., a_{n}$$, ..... are in G.P. then $$\begin{vmatrix}\log a_{n} & \log a_{n + 1} & \log a_{n + 2}\\ \log a_{n + 3} & \log a_{n + 4} & \log a_{n + 5}\\ \log a_{n + 6} & \log a_{n + 7} & \log a_{n + 8}\end{vmatrix}$$ is
  • $$0$$
  • $$1$$
  • $$-1$$
  • None of these
If the determinant $$\begin{vmatrix} a+p & 1+x & u+f \\ b+q & m+y & v+g \\ c+r & n+z & w+h \end{vmatrix}$$ splits into exactly K determinants of order 3, each element of which contains only one term, then the value of K is
  • 9
  • 8
  • 24
  • 12
The coefficient of $${x}^{2}$$ in the expansion of the determinant
$$\begin{vmatrix} { x }^{ 2 } & { x }^{ 3 }+1 & { x }^{ 5 }+2 \\ { x }^{ 3 }+3 & { x }^{ 2 }+x & { x }^{ 3 }+{ x }^{ 4 } \\ x+4 & { x }^{ 3 }+{ x }^{ 4 } & { 2 }^{ 3 } \end{vmatrix}$$ is
  • $$-10$$
  • $$-8$$
  • $$-2$$
  • $$-6$$
  • $$8$$
The Value of the determinant $$\begin{vmatrix} { b }^{ 2 }-ab & \quad b-c & \quad bc-ac \\ ab-{ a }^{ 2 } & \quad a-b & { \quad b }^{ 2 }-ab \\ bc-ac & \quad c-a & \quad ab-{ a }^{ 2 } \end{vmatrix}$$ =
  • abc
  • a + b + c
  • 0
  • ab + bc + ca
 Points (a, 0), (0, b) and (1, 1)are collinear, if:
  • $$\displaystyle \frac{1}{a} + \frac{1}{b} = 1$$
  • $$\displaystyle \frac{1}{a} - \frac{1}{b} = 1$$
  • $$\displaystyle a+b = 1$$
  • $$\displaystyle a-b = ab$$
The value of $$x$$ satisfying the equation $$\begin{vmatrix}\cos 2x & \sin 2x & \sin 2x\\ \sin 2x & \cos 2x & \sin 2x\\ \sin 2x & \sin 2x & \cos 2x\end{vmatrix} = 0$$ and $$x\epsilon \left [0, \dfrac {\pi}{4}\right ]$$ is
  • $$\dfrac {\pi}{4}$$
  • $$\dfrac {\pi}{2}$$
  • $$\dfrac {\pi}{16}$$
  • $$\dfrac {\pi}{3}$$
  • $$\dfrac {\pi}{8}$$
Consider the following statements:
$$1$$. Determinant is a square matrix.
$$2$$. Determinant is a number associated with a square matrix.
Which of the above statements is/are correct?
  • $$1$$ only
  • $$2$$ only
  • Both $$1$$ and $$2$$
  • Neither $$1$$ nor $$2$$
If $$A$$ is $$3*3$$ show symmetric matrix then $$|A| = ?: - $$
  • $$0$$
  • $$1$$
  • $$-1$$
  • N.O.T.
The value of the determinant $$\begin{vmatrix} b^2-ab & b-c & bc-ac \\ ab -a^2 & a-b & b^2-ab \\ bc-ac & c-a & ab -a^2 \end{vmatrix}$$ =
  • abc
  • a+b+c
  • 0
  • ab+bc+ca
If  $$B$$ is a square matrix of order 4 such that $$ |B|= 24$$ ,then the value of $$|adj B|$$ is equal to
  • $$24$$
  • $${24}^{2}$$
  • $${24}^{3}$$
  • $${24}^{4}$$
if $$a^{2},b^{2}+c^{2}+ab+bc+ca \le 0 \forall a,b,c \epsilon R$$, then value of the determinant $$\left| \begin{matrix} \left( a^{ 2 }+b^{ 2 }+c^{ 2 } \right) ^{ 2 } & a^{ 2 }+b^{ 2 } & 1 \\ 1 & (b+c+2) & b^{ 2 }+c^{ 2 } \\ c^{ 2 }+a^{ 2 } & 1 & (c+a+2)^{ 2 } \end{matrix} \right| $$ equals
  • $$65$$
  • $$a^{2}+b^{2}+c^{2}+31$$
  • $$4(a^{2}+b^{2}+c^{2})$$
  • $$0$$
If $$A$$ is a square matrix of order $$3$$ such that $$A(adjA) =$$ 
$$
\left [
\begin {array}{111}
2 & 0 & 0 \\
0 & 2 & 0 \\
0 & 0 & 2 \\
\end {array}
\right ]
$$ then $$|adj A|=$$
  • $$2$$
  • $$4$$
  • $$8$$
  • $$16$$
Adj $$\begin{bmatrix} 1 & 0 & 2 \\ -1 & 5 & -2 \\ 0 & 2 & 1 \end{bmatrix}=\begin{bmatrix} 9 & a & -2 \\ -1 & 1 & 0 \\ -2 & 2 & b \end{bmatrix}\Rightarrow \left[ \begin{matrix} a & b \end{matrix} \right] =$$
  • $$\left[ \begin{matrix} -4 & 5 \end{matrix} \right]$$
  • $$\left[ \begin{matrix} -4 & -1 \end{matrix} \right]$$
  • $$\left[ \begin{matrix} 4 & 1 \end{matrix} \right]$$
  • $$\left[ \begin{matrix} 4 & -1 \end{matrix} \right]$$
x - 4 is factor of $$\begin{vmatrix}x - 2& 2x -3& 3x -4\\ x -4 &2x-9& 3x -16\\ x-8&2x-27&3x-64\end{vmatrix}$$
  • True
  • False
0:0:1


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