JEE MAIN Matrices Previous Year Questions (PYQs) – Page 12 of 15

Let $A = [{a_{ij}}]$ be a square matrix of order 3 such that ${a_{ij}} = {2^{j - i}}$, for all i, j = 1, 2, 3. Then, the matrix A² + A³ + ...... + A¹⁰ is equal to :

Let $B=\left[\begin{array}{ll}1 & 3 \\ 1 & 5\end{array}\right]$ and $A$ be a $2 \times 2$ matrix such that $A B^{-1}=A^{-1}$. If $B C B^{-1}=A$ and $C^4+\alpha C^2+\beta I=O$, then $2 \beta-\alpha$ is equal to

If $f(x)=[x]-\left[\dfrac{x}{4}\right],\ x\in\mathbb{R}$, where $[\cdot]$ denotes the greatest integer function, then

Let $A = \left[ {\matrix{ 2 & 3 \cr a & 0 \cr } } \right]$, a$\in$R be written as P + Q where P is a symmetric matrix and Q is skew symmetric matrix. If det(Q) = 9, then the modulus of the sum of all possible values of determinant of P is equal to :

Let $A = [{a_{ij}}]$ be a 3 $\times$ 3 matrix, where ${a_{ij}} = \left\{ {\matrix{ 1 & , & {if\,i = j} \cr { - x} & , & {if\,\left| {i - j} \right| = 1} \cr {2x + 1} & , & {otherwise.} \cr } } \right.$ Let a function f : R $\to$ R be defined as f(x) = det(A). Then the sum of maximum and minimum values of f on R is equal to:

If $P = \begin{bmatrix} \dfrac{\sqrt{3}}{2} & \dfrac{1}{2} \\ -\dfrac{1}{2} & \dfrac{\sqrt{3}}{2} \end{bmatrix}$ and $A = \begin{bmatrix} 1 & 1 \\ 0 & 1 \end{bmatrix}$, $Q = P A P^{T}$, then $P^{T} Q^{2015} P$ is:

If $A = \left[ {\matrix{ {\cos \theta } & {i\sin \theta } \cr {i\sin \theta } & {\cos \theta } \cr } } \right]$, $\left( {\theta = {\pi \over {24}}} \right)$

and ${A^5} = \left[ {\matrix{ a & b \cr c & d \cr } } \right]$, where $i = \sqrt { - 1} $ then which one of the following isnot true?

For $x \in \mathbb{R} - \{0,1\}$, let $f_1(x)=\dfrac{1}{x}$, $f_2(x)=1-x$, and $f_3(x)=\dfrac{1}{1-x}$ be three given functions. If a function $J(x)$ satisfies $(f_2 \circ J \circ f_1)(x)=f_3(x)$, then $J(x)$ is equal to:

Let the system of linear equations \[ \begin{cases} x + y + kz = 2,\\ 2x + 3y - z = 1,\\ 3x + 4y + 2z = k \end{cases} \] have infinitely many solutions. Then the system \[ \begin{cases} (k+1)x + (2k-1)y = 7,\\ (2k+1)x + (k+5)y = 10 \end{cases} \] has: