Let $B=\begin{bmatrix} 1 & 3 \\ 1 & 5 \end{bmatrix}$ and $A$ be a $2 \times 2$ matrix such that $AB^{-1}=A^{-1}$. If $BCB^{-1}=A$ and $C^4+\alpha C^2+\beta I=O$,then $2\beta-\alpha$ is equal to:

Let $A = \begin{bmatrix} p & 13 \\ -13 & p \end{bmatrix}$ and $B = \begin{bmatrix} 4q & 85 \\ -2 & 1 \end{bmatrix}$ where $p, q \in N$. It is given that $|A| = |B|$ and $p, q \in [1, 1000]$. Then the total number of ordered pairs $(p, q)$ is:

The maximum value of $f(x) = \left|\begin{array}{ccc} \sin^{2} x & 1+\cos^{2} x & \cos 2x \\ 1+\sin^{2} x & \cos^{2} x & \cos 2x \\ \sin^{2} x & \cos^{2} x & \sin 2x \end{array}\right|, x \in R$ is:

Let $A = \begin{bmatrix} 1 & 2 \\ -2 & -5 \end{bmatrix}$. Let $\alpha, \beta \in \mathbb{R}$ be such that $\alpha A^{2} + \beta A = 2I$. Then $\alpha + \beta$ is equal to -

Let $a$ and $b$ be non-zero real numbers such that $ab = 5/2$. Given $A = \begin{bmatrix} a & -b \\ b & a \end{bmatrix}$ and $AA^T = 20I$ (where $I$ is the identity matrix),the quadratic equation whose roots are $a$ and $b$ is:

If $\frac{x^2+7}{(x^2+1)(x-2)}=\frac{A}{x-2}+\frac{Bx+C}{x^2+1}$, then the determinant of the matrix $\begin{bmatrix} A & B \\ C & \frac{2}{5} \end{bmatrix}$ is