JEE MAIN Determinants Previous Year Questions (PYQs) – Page 1 of 6

Let $f(x)=\begin{vmatrix} 1+\sin^{2}x & \cos^{2}x & \sin 2x\\ \sin^{2}x & 1+\cos^{2}x & \sin 2x\\ \sin^{2}x & \cos^{2}x & 1+\sin 2x \end{vmatrix},\ x\in\left[\dfrac{\pi}{6},\dfrac{\pi}{3}\right].$ If $\alpha$ and $\beta$ respectively are the maximum and the minimum values of $f$, then

If $A,B$ and $\big(\operatorname{adj}(A^{-1})+\operatorname{adj}(B^{-1})\big)$ are non-singular matrices of the same order, then the inverse of $A\Big(\operatorname{adj}(A^{-1})+\operatorname{adj}(B^{-1})\Big)^{-1}B$ is equal to:

Let $A = \begin{bmatrix} 2 & b & 1 \\ b & b^2 + 1 & b \\ 1 & b & 2 \end{bmatrix}$ where $b > 0$. Then the minimum value of $\dfrac{\det(A)}{b}$ is –

If $g$ is the inverse of a function $f$ and $f'(x)=\dfrac{1}{1+x^{5}}$, then $g'(x)$ is equal to:

Let $a_1,a_2,\dots,a_{10}$ be in G.P. with $a_i>0$ for $i=1,2,\dots,10$ and $S$ be the set of pairs $(r,k)$, $r,k\in\mathbb{N}$, for which $ \begin{vmatrix} \log_e(a_1^{\,r}a_2^{\,k}) & \log_e(a_2^{\,r}a_3^{\,k}) & \log_e(a_3^{\,r}a_4^{\,k})\\ \log_e(a_4^{\,r}a_5^{\,k}) & \log_e(a_5^{\,r}a_6^{\,k}) & \log_e(a_6^{\,r}a_7^{\,k})\\ \log_e(a_7^{\,r}a_8^{\,k}) & \log_e(a_8^{\,r}a_9^{\,k}) & \log_e(a_9^{\,r}a_{10}^{\,k}) \end{vmatrix} =0. $ Then the number of elements in $S$, is –

Let $A$ and $B$ be two square matrices of order $3$ such that $|A|=3$ and $|B|=2$. Then $\left|,A^{T}A,( \operatorname{adj}(2A))^{-1},(\operatorname{adj}(4B)),(\operatorname{adj}(AB))^{-1},A A^{T}\right|$ is equal to:

Let $A$ be a $3 \times 3$ matrix such that $|\operatorname{adj}(\operatorname{adj}(\operatorname{adj} \mathrm{A}))|=81$.

If $S=\left\{n \in \mathbb{Z}:(|\operatorname{adj}(\operatorname{adj} A)|)^{\frac{(n-1)^2}{2}}=|A|^{\left(3 n^2-5 n-4\right)}\right\}$, then $\sum_\limits{n \in S}\left|A^{\left(n^2+n\right)}\right|$ is equal to :

If the system $11x+y+\lambda z=-5,\quad 2x+3y+5z=3,\quad 8x-19y-39z=\mu$ has infinitely many solutions, then $\lambda^{4}-\mu$ equals: