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

Instructions

• Attempt five questions in all, selecting at least two questions from each section.
• Use of calculator is allowed.

Questions

Section A

• Q1. (a) Show that the function : $u(r, \theta) = r^3 \cos 3\theta - r \sin \theta$ is harmonic and find its conjugate function. (b) Take a rectangular region C with vertices $2 \pm i$ and $-2 \pm i$ and let $g(a) = \oint_C \frac{(3z^2 - 2z + 5)}{z - a} dz$. Then find the values of $g(2)$, $g(-i)$, $g'(i)$ and $g'(2 + 2i)$. (c) Explain four standard transformations with example in each case.
• Q2. (a) Using contour integration, evaluate : $\int_{-\infty}^{\infty} \frac{(x^2 - x + 2)dx}{x^4 + 10x^2 + 9}$ (b) Expand $f(z) = \frac{1}{z^2 \sinh z}$ as a Laurent's series about $z = 0$ and hence evaluate $\oint_C \frac{dz}{z^2 \sinh z}$ around the unit circle C. (c) State and prove necessary conditions for a function $f(z)$ to be analytic in a region.
• Q3. (a) Consider a function $f(z)$ given by $f(z) = \frac{(1+i)x^3 + (1-i)y^3}{|z|^2}$ for $z \neq 0$ and $f(0) = 0$. Examine the continuity of $f(z)$ at $z = 0$. Is it differentiable at the origin ? (b) Evaluate : $\oint_C \frac{(\sin \pi z + \cos \pi z)dz}{(z-1)(z^2 - 4)}$ where C is the circle $|z| = \sqrt{5}$. (c) Evaluate $\int_a^b f(z) dz$ along the straight line joining $a$ to $b$, where $a = 2 - i$, $b = 3 + 2i$ and $f(z) = 2x + 3iy + 1$.
• Q4. (a) State and prove Liouville's theorem. (b) Using calculus of residues, evaluate : $\int_0^\infty \frac{\sin mx}{x} dx$ for $m > 0$. (c) Determine the analytic function $f(z) = u + iv$, where $v = \frac{\sinh 2y}{\cos 2x + \cosh 2y}$.

Section B

• Q5. (a) Define operators : $\Delta, \nabla, E$. Also evaluate $\Delta(e^x \cos 2x)$ and $\nabla(x^2 \sin x)$. (b) The following table gives the values of $f(x) = \tan x$ for $x$ lying within $0.10$ and $0.30$ : | $x$ | $f(x)$ | |---|---| | 0.10 | 0.1003 | | 0.15 | 0.1511 | | 0.20 | 0.2027 | | 0.25 | 0.2553 | | 0.30 | 0.3093 | Find the value of $\tan(0.26)$. (c) Solve the difference equation : $y_{n+2} - 2y_{n+1} + 4y_n = n^2 . 6^n$.
• Q6. (a) Using Lagrange's interpolation formula express the following function as a sum of partial fractions : $\frac{x^2 + 6x - 1}{(x^2 - 1)(x - 4)(x - 6)}$. (b) Let $f(x) = x^2 e^x + 2x$. Then taking $x_0 = 1.5, x_1 = 1.55, x_2 = 1.62$, approximate $f(x)$ at $x = 1.52$ by Newton's divided difference formula. Compare the result with the actual value. (c) Evaluate $\int_0^{\pi/2} \sqrt{\sin \theta} d\theta$ by Simpson's $\frac{1}{3}$ rule taking seven ordinates.
• Q7. (a) Find the first derivative of $y$ at $x = 6$, given that : | $x$ | $y$ | |---|---| | 4.5 | 9.69 | | 5 | 12.9 | | 5.5 | 16.71 | | 6.0 | 21.18 | | 6.5 | 26.37 | | 7.0 | 32.34 | (b) Using Regula falsi method, find a real root of the equation : $x \log_{10} x = 1.2$ (c) Evaluate $y(0.2)$ by Runge-Kutta method of 4th order, given that : $y' = x^2 - y^2$ with $y(0) = 1$.
• Q8. (a) Describe the Newton-Raphson method for finding the cube root of a real number $N$. Apply it to calculate the cube root of $18$, correct to three decimal places. (b) Using Picard's method, find the approximate value of $y$ at $x = 0.2$, given that $y' = x + y + xy$ with $y(0) = 1$. (c) Using Euler's method, evaluate $y(0.8)$, given that $y' = x^2 + y^2$ with $y(0) = 1$. Take $h = 0.2$.