$$\text { If } A=\left[\begin{array}{ccc} 1 & 2 & 3 \\ 3 & -2 & 1 \\ 4 & 2 & 1 \end{array}\right] \text {, then show that } A^3-23 A-40 I=O \text {. }$$

(a) Differentiate $$\quad \sec ^{-1}\left(\frac{1}{\sqrt{1-x^2}}\right) \quad$$ w.r.t. $$\sin ^{-1}\left(2 x \sqrt{1-x^2}\right)$$.

OR

(b) If $$y=\tan x+\sec x$$, then prove that $$\frac{d^2 y}{d x^2}=\frac{\cos x}{(1-\sin x)^2}$$.

(a) Evaluate: $$\int_\limits0^{2 \pi} \frac{1}{1+e^{\sin x}} d x$$

OR

$$\text { (b) Find: } \int \frac{x^4}{(x-1)\left(x^2+1\right)} d x$$

Find the area of the following region using integration:

$$\left\{(x, y): y^2 \leq 2 x \text { and } y \geq x-4\right\}$$