Mathematics-III 2022 Question Paper — Government College of Engineeri…

The 2022 Mathematics-III previous year question paper (Government College of Engineering and Technology, Jammu), with every question and its marks.

Instructions

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

Q1. (a) Show that $f(z) = \sin z$ is an entire function and find its derivative. (b) If $f(z)$ is an analytic function with its real part as constant, then prove that it is a constant function. [20 marks]
Q2. (a) Determine the analytic function whose imaginary part is $\sinh x \cos y$. (b) Find the image of the line $2x - y + 3 = 0$ under the transformation $w = \frac{1}{z}$. [20 marks]
Q3. (a) 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 $f(z) = 2x + iy + 1$. (b) Evaluate $\oint_{C} \frac{(5z+3)}{(z+1)(z-2)} \, dz$ where $C$ is the circle $|z| = 3$, using Cauchy integral formulae. [20 marks]
Q4. (a) Find the nature of singularities and residue at these points of the function $f(z) = \frac{e^z}{(z^2 + \pi^2)}$. (b) State Residue theorem and evaluate the following integral using the Residue theorem $$\oint_{C} \frac{1 - 3z}{(z-1)(z-2)(z+4)} \, dz$$ where $C$ is the circle $|z| = 3$. [20 marks]

Q5. (a) Evaluate $\Delta (e^{2x} \log 3x)$ and $\Delta (x^2 e^x)$. (b) Prove that $\Delta = E\nabla$, $\nabla = E^{-1}\Delta$ and $\Delta\nabla = \nabla\Delta = \delta^2$. [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 Lagrange's interpolation formula fit a polynomial to the data: \begin{tabular}{c c c c c} x: & 0 & 1 & 3 & 4 \\ f(x): & -12 & 0 & 12 & 24 \end{tabular} [20 marks]
Q7. (a) Using the Newton-Raphson method, find a real root of the equation $x^3 - 5x - 11 = 0$. (b) Find the real root of the equation $x^3 + x - 1 = 0$ using Regula falsi method correct to four decimal places. [20 marks]
Q8. (a) Using Picard's method, solve $y' = -xy$ with $y(0) = 1$ upto third approximation. (b) 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]