If $A = \begin{bmatrix} 3 & 7 \\ 1 & 2 \end{bmatrix}$,then $|A^{2011} - 5A^{2010}|$ is equal to

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 $A = (a_{ij})$ be an $n \times n$ matrix defined by $a_{ij} = \begin{cases} k^i, & \forall i=j \\ 0, & \text{otherwise} \end{cases}$. If $m = \text{trace of } A$ and $\lim_{k \rightarrow 1} \frac{n-m}{1-k} = 171$, then the value of $n$ is:

Which of the following is(are) $NOT$ the square of a $3 \times 3$ matrix with real entries?

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 $z = \begin{bmatrix} 1 & 1+2i & -5i \\ 1-2i & -3 & 5+3i \\ 5i & 5-3i & 7 \end{bmatrix}$, then which of the following is true? (where $i = \sqrt{-1}$)