List of top Mathematics Questions on Determinants

If \[ f(x) = \begin{vmatrix} x^3 & 2x^2 + 1 & 1 + 3x \\ 3x^2 + 2 & 2x & x^3 + 6 \\ x^3 - x & 4 & x^2 - 2 \end{vmatrix} \] for all $ x \in \mathbb{R} $, then $ 2f(0) + f'(0) $ is equal to

Let $ D = \begin{vmatrix} n & n^2 & n^3 \\ n^2 & n^3 & n^5 \\ 1 & 2 & 3 \end{vmatrix} $. Then $ \lim_{n \to \infty} \frac{M_{11} + C_{33}}{(M_{13})^2} $ is:

The derivative of $\sin^2$$( \cot^{-1} {\sqrt {( \frac{1 + x}{1 - x}} })$ with respect to $ x $ is equal to: (a) 0

If $ f(x) $ is differentiable at $ x = 1 $ and \[ \lim_{h \to 0} \frac{1}{h} f(1 + h) = 5, \] then $ f'(1) $ is equal to:

If $ \sin\left(\frac{y}{x}\right) = \log_e |x| + \frac{\alpha}{2} $ is the solution of the differential equation \[x \cos\left(\frac{y}{x}\right) \frac{dy}{dx} = y \cos\left(\frac{y}{x}\right) + x\]and $ y(1) = \frac{\pi}{3} $, then $ \alpha^2 $ is equal to

Let $ A $ be a $ 3 \times 3 $ matrix and $ \det(A) = 2 $. If $n = \det(\text{adj}(\text{adj}(\ldots(\text{adj}(A))\ldots)))$ with adjoint applied 2024 times, then the remainder when $ n $ is divided by 9 is equal to $\_\_\_\_\_.$

If \[ f(x) = \begin{vmatrix} 2 \cos^4 x & 2 \sin^4 x & 3 + \sin^2 2x \\ 3 + 2 \cos^4 x & 2 \sin^4 x & \sin^2 2x \\ 2 \cos^4 x & 3 + 2 \sin^4 x & \sin^2 2x \end{vmatrix} \] then $ \frac{1}{5} f'(0) $ is equal to

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:

Evaluate the determinant \[ \begin{vmatrix} 1+a & 1 & 1 \\ 1 & 1+b & 1 \\ 1 & 1 & 1+c \end{vmatrix}. \]

Evaluate the determinant: \[ \begin{vmatrix} 1 & 1 & 1 \\ a & b & c \\ a^3 & b^3 & c^3 \end{vmatrix}. \]

Evaluate the determinant of the following matrix: \[ \begin{vmatrix} 3 & 1 & 2 \\ -4 & -2 & 3 \\ 5 & 1 & 1 \end{vmatrix} \]

Evaluate the determinant of the following matrix: \[ \begin{vmatrix} 1 & 2 & 5 \\ 1 & 1 & 4 \\ -2 & -3 & -9 \end{vmatrix} \]

If \[ \left| \begin{pmatrix} 2017 & 2018 \\ 2019 & 2020 \end{pmatrix} \right| + \left| \begin{pmatrix} 2021 & 2022 \\ 2023 & 2024 \end{pmatrix} \right| = 2k \] find $ k^3 $.

If $ A = \begin{bmatrix} 0 & x & 16 \\ x & 5 & 7 \\ 0 & 9 & x \end{bmatrix} $ is a singular matrix, then $ x $ is equal to

The system of linear equations $ x + y + z = 2, \quad 2x + y - 2 = 3, \quad 3x + 2y + kz = 4 \text{ has a unique solution if} $

If $a^2 + b^2 + c^2 = 0$ and $$\begin{vmatrix} b^2+c^2 & ab & ac \\ ab & c^2+a^2 & bc \\ ac & bc & a^2+b^2 \end{vmatrix}=ka^2b^2c^2$$ then k is equal to

Let $A =\left[ a _{i j}\right], a _{i j} \in Z \cap[0,4], 1 \leq i, j \leq 2$ .
The number of matrices $A$ such that the sum of all entries is a prime number $p \in(2,13)$ is _____.

Let $A$ be a $3 \times 3$ matrix such that $|\operatorname{adj}(\operatorname{adj}(\operatorname{adj} A ))|=12^4$ Then $\mid A ^{-1}$ adj $A \mid$ is equal to

Let $x, y, z>1$ and $A=\begin{bmatrix}1 & \log _x y & \log _x z \\ \log _y x & 2 & \log _y z \\ \log _z x & \log _z y & 3\end{bmatrix}$ Then ∣adj(adjA2)∣ is equal to

Let $A= \begin{pmatrix} m & n \\ p & q\end{pmatrix}, d=|A| \neq 0$ and $|A-d(\operatorname{Adj} A)|=0$ Then

If $\frac{1^3+2^3+3^3+\ldots \text { up to } n \text { terms }}{1 \cdot 3+2 \cdot 5+3 \cdot 7+\ldots \text { up to } n \text { terms }}=\frac{9}{5}$, then the value of $n$ is

If the system of equations $ x+2 y+3 z=3$ $ 4 x+3 y-4 z=4 $ $ 8 x+4 y-\lambda z=9+\mu$ has infinitely many solutions, then the ordered pair $(\lambda, \mu)$ is equal to :

Let $A$ be the event that the absolute difference between two randomly choosen real numbers in the sample space $[0,60]$ is less than or equal to a If $P(A)=\frac{11}{36}$, then a is equal to _____

Let A be a $n \times n$ matrix such that $| A |=2$ If the determinant of the matrix$\operatorname{Adj}\left(2 \cdot \operatorname{Adj}\left(2 A ^{-1}\right)\right) \cdot$ is $2^{84}$, then $n$ is equal to ____