List of top Mathematics Questions on Determinants asked in CBSE CLASS XII, CBSE Compartment XII

If $ \left| \begin{matrix} 2x & 5 \\ 12 & x \end{matrix} \right| = \left| \begin{matrix} 6 & -5 \\ 4 & 3 \end{matrix} \right| $, then the value of x is:

If $\begin{vmatrix} 2x & 3 \\ x & -8 \\ \end{vmatrix} = 0$, then the value of $x$ is:

The value of the determinant \[ \begin{vmatrix} \cos 75^\circ & \sin 75^\circ \\ \sin 15^\circ & \cos 15^\circ \end{vmatrix} \] is:

The area of a triangle with vertices $(3, 0), (0, k), (-3, 0)$ is $9$ sq units. The value of $k$ is:

If $$ \int_{0}^{a} \frac{1}{4 + x^2} \, dx = \frac{\pi}{6}, $$ then the value of $ a $ is:

If $ a_{ij} $ and $ A_{ij} $ represent the $(i,j)^\text{th}$ element and its cofactor of the matrix: \[ \begin{bmatrix} 2 & -3 & 5 \\ 6 & 0 & 4 \\ 1 & 5 & -7 \end{bmatrix}, \] respectively, then the value of $ a_{11}A_{21} + a_{12}A_{22} + a_{13}A_{23} $ is:

For which value of x, are the determinants \[ \begin{vmatrix} 2x & -3 \\ 5 & x \end{vmatrix} \text{and} \begin{vmatrix} 10 & 1 \\ -3 & 2 \end{vmatrix} \text{ equal?} \]

The value of the cofactor of the element of second row and third column in the matrix \[ \begin{bmatrix} 4 & 3 & 2 \\ 2 & -1 & 0 \\ 1 & 2 & 3 \end{bmatrix} \text{ is:} \]

Evaluate $\begin{vmatrix} cos\alpha cos\beta &cos\alpha sin\beta &-sin\alpha \\ -sin\beta&cos\beta &0 \\ sin\alpha cos\beta&sin\alpha\sin\beta &cos\alpha \end{vmatrix}$

Examine the consistency of the system of equations.
x+y+z=12x+3y+2z=2,
ax+ay+2az=4

Using Cofactors of elements of third column,evaluate$△=\begin{bmatrix}1&x&yz\\ 1&y&zx\\ 1&z&xy\end{bmatrix}$

If A=$\begin{bmatrix}2&-1&1\\-1&2&-1\\1&-1&2\end{bmatrix}$verify that A³-6A²+9A-4 I=0 and hence find A^-1

Evaluate the determinants
(i) $\begin{vmatrix}3 & -1 & -2\\ 0 & 0 & -1 \\ 3&-5&0\end{vmatrix}$(iii) $\begin{vmatrix} 3 & -4 & 5\\ 1 & 2 & -2 \\ 2&3&1\end{vmatrix}$
(ii)$\begin{vmatrix} 0 & 1 & 2\\ -1 & 0 & -3 \\ -2&3&0\end{vmatrix}$ (iv)$\begin{vmatrix} 2 & -1 & -2\\ 0 & 2 & -1\\3&-5&0 \end{vmatrix}$

Let A=$\begin{bmatrix}3&7\\2&5\end{bmatrix}$and B=$\begin{bmatrix}6&8\\7&9\end{bmatrix}$,Verify that (AB)^-1=B^-1A^-1.

Solve system of linear equations, using matrix method.
x-y+2z=7
3x+4y-5z=-5
2x-y+3z=12

For the matrix $A=\begin{bmatrix}1&1&1\\ 1&2&-3\\ 2&-1&3\end{bmatrix}$show that $A^3−6A^2+5A+11I=O$.Hence,find A−1.

$ \begin{vmatrix} x + 1 & x - 1 \\ x^2 + x + 1 & x^2 - x + 1 \end{vmatrix} $ is equal to:

Assertion (A): For the matrix \[ A = \begin{bmatrix} 1 & \cos \theta & 1 \\ -\cos \theta & 1 & \cos \theta \\ -1 & -\cos \theta & 1 \end{bmatrix}, \quad \text{where } \theta \in [0, 2\pi], \] $|A| \in [2, 4].$

Reason (R): $\cos \theta \in [-1, 1], \ \forall \ \theta \in [0, 2\pi].$

If $ \begin{vmatrix} -a & b & c \\ a & -b & c \\ a & b & -c \end{vmatrix} = kabc $, then the value of $ k $ is:

If $ F(x) = \begin{bmatrix} \cos x & -\sin x & 0 \\ \sin x & \cos x & 0 \\ 0 & 0 & 1 \end{bmatrix} $ and $ [F(x)]^2 = F(kx) $, then the value of $ k $ is:

(a) If \[ A = \begin{bmatrix} 5 & 0 & 4 \\ 2 & 3 & 2 \\ 1 & 2 & 1 \end{bmatrix}, \quad B^{-1} = \begin{bmatrix} 1 & 3 & 3 \\ 1 & 4 & 3 \\ 1 & 3 & 4 \end{bmatrix}, \] find $ (AB)^{-1} $. Also, find $ |AB|^{-1} $.