Q1.
(a) Evaluate the following:  
(i) \(\lim_{x \to 0} (\cos x)^{\csc^2 x}\)        
(ii) \(\lim_{x \to 0} \frac{e^{2x} – e^{-2x}}{\ln(1+x)}\)
(10)

(b) Discuss the differentiability of \(f(x) = |x|\) at \(x = 0\). (10)

Q2.
(a) Find the asymptotes of the curve \(y = \frac{x^3 + 2x – 1}{x^2 – 1}\). (10)

(b) Find the extreme values of the function \(f(x,y) = y^2 + 4xy + 3x^2 + x^3\). (10)

Q3.
(a) Determine the volume of the solid bounded by planes \(z = 0\), \(z = 2 + x\) and the cylinder \(x^2 + y^2 = 1\). (10)

(b) Discuss the convergence of the series:
    (i) \(\sum_{n=1}^{\infty} \frac{n+1}{2n+3}\)                        
(ii) \(\sum_{n=1}^{\infty} \frac{2^n}{n(n+1)}\)
(10)

Q4.
(a) Prove that \(\frac{\pi}{6} < \int_{0}^{\pi/2} \frac{1}{2 + \sin x} dx < \frac{\pi}{4}\). (10)

(b) Evaluate the following by using Beta and Gamma integrals:
    (i) \(\int_{0}^{1} t^4 (1 – t^2)^3 dt\)                
(ii) \(\int_{0}^{\infty} e^{-x^2} dx\)
(10)

Q5.
(a) Simplify \(\left(\frac{\sqrt{3} + i}{1 + \sqrt{3}i}\right)^5\) using De Moivre’s Theorem. (10)

(b) Let \(f(z) = u(x,y) + i v(x,y)\) be an analytic function. If \(u(x,y) = 3x – 2xy\), then find \(v(x,y)\) and express \(f(z)\) in terms of \(z\). (10)

Q6.
(a) Evaluate \(\int_{C} \frac{1 – 2z}{z(z-1)(z-2)} dz\) where \(C\) is the circle \(|z| = 1.5\). (10)

(b) Evaluate by Residue Theorem \(\int_{0}^{2\pi} \frac{\cos 2\theta}{5 + 4\cos\theta} d\theta\). (10)

Q7.
(a) Find a Fourier series to represent \(f(x) = x^2 + x\) from \(-\pi\) to \(\pi\) and show that:
    \(\frac{\pi^2}{6} = 1 + \frac{1}{2^2} + \frac{1}{3^2} + \frac{1}{4^2} + \dots\)
(10)

(b) Find the Fourier transform of the function \(f(x) = \begin{cases} 1, & |x| < a \\ 0, & |x| > a \end{cases}\). (10)