Subject 2018 Question Paper — Government College of Engineering and T…

Instructions

• Attempt five questions in all selecting at least two questions from each section.
• Each question carries 20 marks.

Questions

Section -A

• Q1. (a) Find the limit $\lim_{z \to 0} \frac{\text{Re } z + \text{Im } z}{|z|^2}$. (b) Prove that $f(z) = \cosh z$ is an entire function and find its derivative. (c) Find the bilinear transformation that maps from \{1, i, -1\} to \{i, 0, -1\}. [20 marks]
• Q2. (a) If $f(z)$ is an analytic function with its real part is constant, then prove that it is a constant function. (b) Determine the analytic function whose imaginary part is $\sinh x \cos y$. (c) Find the image of the line $2x - y + 3 = 0$ under the transformation $w = \frac{1}{z}$. [20 marks]
• Q3. (a) Evaluate $\int_{0}^{1+i} (x^2 - iy) dz$ along the line $x = y$. (b) Evaluate $\int_{a}^{b} f(z) dz$ along the curve given by $x = t + 1$, $y = 2t^2 - 1$, where $a = 1 - i, b = 2 + i$, and $f(z) = 2x + iy + 1$. (c) Find the nature of singularities and residue at these points of the function $f(z) = \frac{e^z}{(z^2 + \pi^2)^2}$. [20 marks]
• Q4. (a) Evaluate $\oint_C \frac{(z+1)}{(z-1)(z+2)} dz$ where $C$ is the circle $|z| = 3$, using Cauchy integral formula. (b) Find the Taylor series expansion of $\sin z$ at $z = \frac{\pi}{3}$. (c) Using Cauchy Residue theorem, evaluate $\oint_C \frac{(1+2z)}{(z-1)(z-2)(z+3)} dz$ where $C$ is the circle $|z| = 4$. [20 marks]

Section -B

• Q5. (a) Evaluate $(\Delta + \nabla)^2 (x^2 + x)$. (b) Prove that $\Delta = E\nabla, \nabla = E^{-1}\Delta$. (c) State Lagrange's interpolation formula and apply it to find $f(7)$ from the following data: $x$: 5, 6, 9, 11 $f(x)$: 12, 13, 14, 16 [20 marks]
• Q6. (a) Using Newton's interpolation formula, compute the value of $\sqrt{15.5}$, given that $\sqrt{5} = 2.236$, $\sqrt{10} = 3.162$, $\sqrt{15} = 3.872$, $\sqrt{20} = 4.472$, $\sqrt{25} = 5$. (b) Using Newton's divided difference formula, construct a polynomial which takes the following values: $x$: 3, 7, 9, 10 $f(x)$: 168, 120, 72, 63 Hence or otherwise evaluate $f(6)$. (c) Evaluate $\int_{0}^{1} \sqrt{1 - x^2} dx$ by Trapezoidal and Simpson's $1/3$ rule taking $h = 0.1$. [20 marks]
• Q7. (a) Find a real root of the equation $x^3 - 3x + 4 = 0$, by the method of false position. (b) Using iteration method, find a real root of the equation $x^3 - 9x + 1 = 0$. (c) Using the Newton-Raphson method, find a real root of the equation $x^3 - 5x - 11 = 0$. [20 marks]
• Q8. (a) Given $y' = x - y$ with $y(0) = 1$, compute $y(0.2)$ using Picard's method. (b) Using Euler's method, find an approximate value of $y$ at $x = 0.1$, given that $\frac{dy}{dx} = \frac{y - x}{y + x}$ and $y = 1$ when $x = 0$ taking $h = 0.02$. (c) Use the Runge-Kutta method of fourth order to find $y(1.2)$ with $h = 0.1$ for the initial value problem $\frac{dy}{dx} = x^2 + y^2$, $y(1) = 1.5$. [20 marks]