If $a, b, c$ are non-zero complex numbers satisfying $a^2 + b^2 + c^2 = 0$ and $\left| \begin{array}{ccc} b^2 + c^2 & ab & ac \\ ab & c^2 + a^2 & bc \\ ac & bc & a^2 + b^2 \end{array} \right| = k a^2 b^2 c^2$,then $k$ is equal to

If $A = \begin{bmatrix} 1 & 1 & 2 \\ 0 & 2 & 1 \\ 1 & 0 & 2 \end{bmatrix}$ and $A^3 = (aA - I)(bA - I)$,where $a, b$ are integers and $I$ is a $3 \times 3$ unit matrix,then the value of $(a + b)$ is equal to:

Let $A = \begin{bmatrix} 2 & -1 \\ 0 & 2 \end{bmatrix}$. If $B = I - {}^{3}C_{1}(\operatorname{adj} A) + {}^{3}C_{2}(\operatorname{adj} A)^{2} - {}^{3}C_{3}(\operatorname{adj} A)^{3}$,then the sum of all elements of the matrix $B$ is

Let $A$ be a symmetric matrix such that $|A|=2$ and $\begin{bmatrix} 2 & 1 \\ 3 & \frac{3}{2} \end{bmatrix} A = \begin{bmatrix} 1 & 2 \\ \alpha & \beta \end{bmatrix}$. If the sum of the diagonal elements of $A$ is $s$,then $\frac{\beta s}{\alpha^2}$ is equal to $..........$.

Let $a = \lim_{x \to 1} \left( \frac{x}{\ln x} - \frac{1}{x \ln x} \right)$,$b = \lim_{x \to 0} \frac{x^3 - 16x}{4x + x^2}$,$c = \lim_{x \to 0} \frac{\ln(1 + \sin x)}{x}$,and $d = \lim_{x \to -1} \frac{(x + 1)^3}{3(\sin(x + 1) - (x + 1))}$. Then the matrix $\begin{bmatrix} a & b \\ c & d \end{bmatrix}$ is:

If the polynomial $f(x) = \left|\begin{array}{ccc} (1+x)^{a} & (2+x)^{b} & 1 \\ 1 & (1+x)^{a} & (2+x)^{b} \\ (2+x)^{b} & 1 & (1+x)^{a} \end{array}\right|$,then the constant term of $f(x)$ is ($a$ and $b$ are positive integers).