CBSE Questions for Class 12 Commerce Maths Determinants Quiz 1 - MCQExams.com

The number of distinct real roots of the equation, $$\begin{vmatrix} \cos x& \sin x & \sin x\\ \sin x & \cos x & \sin x\\ \sin x & \sin x & \cos x\end{vmatrix} = 0$$ 
in the interval $$\left [-\dfrac {\pi}{4}, \dfrac {\pi}{4}\right ]$$ is/are :
  • $$3$$
  • $$2$$
  • $$1$$
  • $$4$$
The points $$\displaystyle \left( 0, \frac{8}{3} \right), (1, 3)$$ and $$(82, 30)$$ :
  • form an obtuse angled triangle.
  • form a right angled triangle.
  • lie on a straight line.
  • form an acute angled triangle.
If $$A=\begin {bmatrix} 2 & -3\\ -4 & 1\end{bmatrix}$$, then adj $$(3A^2+12A)$$ is  equal to.
  • $$\begin{bmatrix} 72 & -84\\ -63 & 51\end{bmatrix}$$
  • $$\begin{bmatrix} 51 & 63\\ 84 & 72\end{bmatrix}$$
  • $$\begin{bmatrix} 51 & 84 \\ 63 & 72\end{bmatrix}$$
  • $$\begin {bmatrix} 72 & -63\\ -84 & 51\end{bmatrix}$$
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
  • $$2^{10}$$
  • $$2^{11}$$
  • $$2^{12}$$
  • $$2^{13}$$
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
  • $$P$$ 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
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
  • $$(p - 1)^2$$
  • $$2(p - 1)$$
  • $$(p - 1)^2 + 1$$
  • $$2 p - 1$$
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].
  • $$3$$
  • $$4$$
  • $$5$$
  • $$6$$
The value of $$|\mathrm{U}|$$ is
  • 3
  • $$-3$$
  • 3/2
  • 2
$$A = \begin{bmatrix}1&2\\3&4\end{bmatrix}, B = \begin{bmatrix}2&1\\3&4\end{bmatrix}$$ then $$\left|(B^TA^T)^{-1}\right|$$ is equal to
  • $$10$$
  • $$\dfrac{1}{10}$$
  • $$1$$
  • $$- 1$$
If three points $$(k, 2k), (2k, 3k), (3, 1)$$ are collinear, then $$k$$ is equal to:
  • $$-2$$
  • $$1$$
  • $$\displaystyle\frac{1}{2}$$
  • $$-\displaystyle\frac{1}{2}$$
$$\begin{vmatrix} 4\sin^{2}\theta  & \cos^{2}\theta \\ 3\sec^{2}\theta & \text{cosec}^{2}\theta \end{vmatrix}=$$
  • $$8\sin^{2}\theta \cos^{2}\theta $$
  • $$4\sin 2\theta \cos 2\theta $$
  • $$1$$
  • $$4\cos^{3}\theta -3\cos \theta $$
$$\left|\begin{array}{lllll}
0 & & \mathrm{c}\mathrm{o}\mathrm{s}\alpha & \mathrm{c}\mathrm{o}\mathrm{s} & \beta\\
\mathrm{c}\mathrm{o}\mathrm{s} & \alpha & 0 & \mathrm{c}\mathrm{o}\mathrm{s} & \gamma\\
\mathrm{c}\mathrm{o}\mathrm{s} & \beta & \mathrm{c}\mathrm{o}\mathrm{s}\gamma & 0 &
\end{array}\right|=$$
  • $$\cos\alpha+\cos\beta+\cos\gamma$$
  • $$\cos\alpha\cos\beta\cos\gamma$$
  • $$ 2\cos\alpha\cos\beta\cos\gamma$$
  • $$2 \displaystyle \sum \cos\alpha\cos\beta$$
$$A=\left\{\begin{array}{ll}
8 & 9\\
10 & 11
\end{array}\right\}$$, then cofactor of $$\mathrm{a}_{12}$$ is:
  • 11
  • 10
  • -11
  • -10
If P =$$\begin{bmatrix}
1 & 4\\
2 & 6
\end{bmatrix}$$ ,then adj (P) 
  • $$\begin{bmatrix}1 & 4\\ 2 & 6\end{bmatrix}$$
  • $$\begin{bmatrix}6& -4\\ -2 & 1\end{bmatrix}$$
  • $$\begin{bmatrix}6& -2\\ -4 & 1\end{bmatrix}$$
  • $$\begin{bmatrix}2& 1\\ 6 & 4\end{bmatrix}$$
$$\begin{vmatrix}
x^{2}+3 &x-1 &x+3 \\
x+3 & -2x &x-4 \\
x-3& x+4 & 3x
\end{vmatrix}$$ $$=px^{4}+qx^3+rx^{2}+sx+t,$$  then $$t = $$
  • $$72$$
  • $$0$$
  • $$24$$
  • $$-48$$
Maximum value of a second order determinant whose every element is either 0,1 or 2 only is:
  • 0
  • 1
  • 2
  • 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=


  • (2n+1)$$\frac{\pi }{2}$$
  • n$$\pi $$
  • (2n+1)$$\pi $$
  • 2n$$\pi $$
If A $$ =\begin{bmatrix}
0 & c &-b \\
-c& 0& a\\
b & -a & 0
\end{bmatrix}$$ then $$\left ( a^{2}+b^{2}-c^{2} \right )\left | A \right |=$$ 
  • $$abc$$
  • $$a + b + c$$
  • $$\left ( a^{3}+b^{3}+c^{3} \right )$$
  • $$0$$
Find x if it is given that:
$$\det \left[\begin{array}{lll}
2 & 0 & 0\\
4 & 3 & 0\\
4 & 6 & x
\end{array}\right]=42$$
  • $$8$$
  • $$7$$
  • $$6$$
  • $$21/4$$
$$\mathrm{If}$$ $$\left|\begin{array}{lll}
1 & 0 & 0\\
2 & 3 & 4\\
5 & -6 & x
\end{array}\right|$$ $$= 45$$ $$\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{n}$$ $$\mathrm{x}=$$

  • $$4$$
  • $$7$$
  • $$- 5$$
  • $$-7$$
If A is a singular matrix, then A (adj A) is a 
  • scalar matrix
  • zero matrix
  • identity matrix
  • orthogonal matrix
If A is a singular matrix, then adj A is
  • $$\displaystyle non-singular$$
  • singular
  • symmetric
  • not defined
If the value of the determinant $$\begin{vmatrix}m & 2\\ -5 & 7\end{vmatrix}$$ is $$31$$, find $$m$$.
  • $$3$$
  • $$8$$
  • $$6$$
  • $$2$$
If $$A$$ is a $$3\times 3$$ matrix and $$\text{det}  (3A)=k(\text{det}  A)$$, then $$k=$$
  • $$9$$
  • $$6$$
  • $$1$$
  • $$27$$
If A and B are similar matrices such that $$det  (AB)=0, $$ then 
  • $$det (A)=0 $$ and $$ det (B)=0$$
  • $$det (A)=0$$ and $$ det (B) \neq 0$$
  • $$A=0$$ and $$B=0$$
  • $$A=0 $$ or $$B=0$$
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
  • $$0$$
  • purely imaginary
  • $$abc$$
  • none of these
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
  • $$det (B)=1$$
  • $$det (A)=-det (B)$$
  • $$det (B)=-1$$
  • none of these
If A is a square matrix of order $$n$$ then adj $$\left ( adj A \right )$$ is equal to
  • $$\displaystyle \left | A \right |^{n}A$$
  • $$\displaystyle \left | A \right |^{n-1}A$$
  • $$\displaystyle \left | A \right |^{n-2}A$$
  • $$\displaystyle \left | A \right |^{n-3}A$$
If$$\displaystyle \left | \begin{matrix}-12 &0   &\lambda  \\  0&  2& -1\\  2& 1 &15 \end{matrix} \right |=-360$$, then the value of $$\lambda$$,is
  • $$-1$$
  • $$-2$$
  • $$-3$$
  • $$4$$
If $$A$$ is any skew-symmetric matrix of odd order then $$\left| A \right| $$ equals
  • $$-1$$
  • $$0$$
  • $$1$$
  • none of these
If $$\begin{vmatrix} x & y\\ 4 & 2 \end{vmatrix}=7$$ and $$\begin{vmatrix} 2 & 3\\ y & x \end{vmatrix}=4$$ then
  • $$x=-3, y=-\dfrac {5}{2}$$
  • $$x=-\dfrac {5}{2}, y=-3$$
  • $$x=-3, y=\dfrac {5}{2}$$
  • $$x=-\dfrac {5}{2}, y=3$$
If $$\begin{vmatrix}a & -b & -c\\-a & b & -c \\ -a & -b & -c\end{vmatrix}+\lambda abc=0$$, then $$\lambda$$ is equal to
  • $$2$$
  • $$4$$
  • $$-2$$
  • $$-4$$
The cofactors of elements in second row of the determinant $$\begin{vmatrix} 2 & -1 & 4 \\ 4 & 2 & -3 \\1 & 1 & 2 \end{vmatrix}$$ are
  • $$5, 6, 4$$
  • $$6, 0, -3$$
  • $$5, 1, 8$$
  • $$6, 0, 3$$
A set of points which do not lie on the same line are called as
  • collinear
  • non-collinear
  • concurrent
  • square
$$A$$ 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:
  • either $$15$$ or $$5$$
  • necessarily $$5$$
  • necessarily $$16$$
  • none of these
If $$A=\begin{bmatrix} 3 & -5 \\ -1 & 0 \end{bmatrix}$$, then adj. $$A$$ is equal to
  • $$\begin{bmatrix} 3 & 5 \\ 1 & 0 \end{bmatrix}$$
  • $$\begin{bmatrix} 0 & -5 \\ -1 & 3 \end{bmatrix}$$
  • $$\begin{bmatrix} 0 & 5 \\ 1 & 3 \end{bmatrix}$$
  • $$\begin{bmatrix} 0 & -1 \\ -5 & 3 \end{bmatrix}$$
The value of the determinant $$\begin{vmatrix}a & b & 0\\ 0 & a & b\\ b & 0 &a \end{vmatrix}$$ is equal to
  • $$a^3-b^3$$
  • $$a^3+b^3$$
  • 0
  • none of these
The value of the determinant $$\begin{vmatrix} 1 & 2 & 3\\ 3 & 5 & 7\\ 8 & 14 & 20\end{vmatrix}$$ is equal to
  • $$20$$
  • $$10$$
  • $$0$$
  • $$45$$
If $$\begin{vmatrix} 6i & -3i & 1\\ 4 & 3i & -1 \\ 20 & 3 & i\end{vmatrix}=x+iy$$, then
  • $$x = 3, y = 1$$
  • $$x = 1, y = 3$$
  • $$x = 0, y = 3$$
  • $$x = 0, y = 0$$
The points which are not collinear are:
  • $$(0, 1),\  (8, 3)\ and\ (6, 7)$$
  • $$(4, 3),\ (5, 1)\ and\ (1, 9)$$
  • $$(2, 5),\ (-1, 2)\ and\ (4, 7)$$
  • $$(-3, 2)\, (1, -2)\ and\ (9, -10)$$
If $$A=\begin{bmatrix} a & b \\ b & a \end{bmatrix}$$, then $$\left| A+{ A }^{ T } \right| $$ equals
  • $$4({a}^{2}-{b}^{2})$$
  • $$2({a}^{2}-{b}^{2}$$
  • $${a}^{2}-{b}^{2})$$
  • $$4ab$$
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
  • $$ac$$
  • $$bd$$
  • $$(b-a)(d-c)$$
  • none of these
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
  • $$-A$$
  • $$A$$
  • $$8A$$
  • $$16A$$
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
  • $$-\omega$$
  • $$\omega$$
  • $$1$$
  • $$\omega^2$$
If $$\begin{vmatrix} x & 2 \\ 18 & x \end{vmatrix}=\begin{vmatrix} 6 & 2 \\ 3x & 6 \end{vmatrix}$$, then $$x$$ is equal to
  • $$6$$
  • $$\pm 6$$
  • $$-6$$
  • $$0$$
Which of the following is correct?
  • Determinant is a square matrix
  • Determinant is a number associated to a matrix
  • Determinant is a number associated to a square matrix
  • None of these
If $$|A|  \displaystyle \neq    0$$, then $$A$$ is
  • zero matrix
  • singular matrix
  • non - singular matrix
  • diagonal matrix
$$A=\begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}$$ and $$A(\text{adj}\,A)=KI$$, then the value of $$K$$ is:
  • $$2$$
  • $$-2$$
  • $$10$$
  • $$-10$$
What is the determinant of the matrix $$\left [\begin{matrix} 3& 6\\ -1 & 2\end {matrix} \right]$$?
  • $$0$$
  • $$12$$
  • $$|0|$$
  • $$|6|$$
If $$A = \begin{bmatrix} 1& \log_{b}a\\ \log_{a}b & 1\end{bmatrix}$$ then $$|A|$$ is equal to
  • $$0$$
  • $$\log_{a}b$$
  • $$-1$$
  • $$\log_{b}a$$
0:0:1


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