If $x=a \sin ^3 \theta, y=b \cos ^3 \theta$,then find $\frac{d^2 y}{d x^2}$ at $\theta=\frac{\pi}{4}$.

(a) Evaluate: $\int_0^\pi \frac{e^{\cos x}}{e^{\cos x}+e^{-\cos x}} d x$

OR

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

(a) Find the particular solution of the differential equation $\frac{d y}{d x}-2 x y=3^{x^2} e^{x^2} ; y(0)=5$.

OR

(b) Solve the following differential equation: $$ x^2 d y+y(x+y) d x=0 $$

Find a vector of magnitude 4 units perpendicular to each of the vectors $2 \hat{i}-\hat{j}+\hat{k}$ and $\hat{i}+\hat{j}-\hat{k}$ and hence verify your answer.