Numerical Methods & Transform Calculus 2018 Question Paper — Governme…

Instructions

• Attempt five questions in all selecting at least two questions from each section.
• All carry equal marks.
• Use of calculator is allowed.

Questions

SECTION - A

• Q1. a) A cubic polynomial $y = f(x)$ satisfies the following data: \begin{tabular}{c|ccccc} $x$ & 0 & 1 & 3 & 4 \\ \hline $f(x)$ & 1 & 4 & 40 & 85 \end{tabular} Determine the function $f(x)$ and hence find $f''(2)$. b) Evaluate $\int\limits_{0}^{1} \sqrt{(1-x^2)} dx$ by Trapezoidal rule, taking $h = 0.1$. c) Define the operators $\Delta, \nabla, \mu, \delta, E$ and $E^{-1}$. Also show that $\mu\delta = \sinh(hD)$.
• Q2. a) Find the missing terms from the following data: \begin{tabular}{c|cccccccc} $x$ & 2 & 2.1 & 2.2 & 2.3 & 2.4 & 2.5 & 2.6 \\ \hline $y$ & 0.135 & \dots & 0.111 & 0.1 & \dots & 0.082 & 0.074 \end{tabular} b) Using the method of false position, find a real root of $xe^x = 3$ correct to three decimal places. c) Using Eular's method, evaluate $y(0.1)$ for $\frac{dy}{dx} = x + y + xy$ with $y(0) = 1$, taking $h = 0.025$.
• Q3. a) Describe Newton-Raphson algorithm to find the cube root of a positive number $N$. Apply this method to compute the cube root of 18. b) Using Taylor's method, find the approximate value of $y$ at $x = 0.1$ from $\frac{dy}{dx} = x^2 + y^2$ with $y(0) = 1$. c) Using iteration method, find a real root of $x^3 + x^2 - 100 = 0$ correct to three decimal places.
• Q4. a) Using Runge-Kutta of $4^{\text{th}}$ order, compute $y(0.4)$, given that $\frac{dy}{dx} = \frac{y^2 - x^2}{y^2 + x^2}$, $y(0) = 1$ and $h = 0.2$. b) Find the first and second derivatives of $y$ at $x = 1.5$ for the following data: \begin{tabular}{c|cccccc} $x$ & 1.5 & 2 & 2.5 & 3 & 3.5 & 4 \\ \hline $y$ & 3.375 & 7 & 13.625 & 24 & 38.875 & 59 \end{tabular} c) Find the $8^{\text{th}}$ and the general term of the series: $0, 12, 51, 132, 270, 480, \dots\dots\dots$

SECTION - B

• Q5. a) Using Laplace transform techniques, solve the differential equation: $y'' + y = t \cos 2t$, with $y(0) = y'(0) = 0$. b) Find the inverse L.T. of $\frac{S+1}{S^2 + S + 1}$. c) Find the L.T. of $2 \cosh^3(2t)$.
• Q6. a) State and prove second shifting theorem. Also find the inverse L.T. of $\frac{(S^2 - 2S + 3)e^{-3s}}{(S+1)(S-1)^2}$. b) Find the L.T. of $t^2 e^{-2t} \cos 2t$. c) Find the inverse Fourier cosine transform of $\overline{f}(S) = \begin{cases} S^2 - 1, & 0 \le S < 1 \\ 0, & \text{otherwise} \end{cases}$
• Q7. a) Derive the Fourier integral representation of $f(x) = \begin{cases} 10, & 0 \le x \le 3 \\ 5, & 6 \le x \le 9 \\ 0, & \text{otherwise} \end{cases}$ at which points, if any, does the Fourier integral fail to converge to $f(x)$? To what value, does the integral converge at those points? b) State and prove convolution property for Fourier transforms. c) Using Laplace transform techniques, evaluate $\int\limits_{0}^{\infty} \frac{\sin^3 t}{t} dt$.
• Q8. a) Find the inverse Fourier transform of $\frac{S}{S^2 + 1}$. b) Find Fourier sine and cosine transform of $f(t) = \begin{cases} t, & 0 \le t < 1 \\ 2-t, & 1 \le t < 2 \\ 0, & \text{otherwise} \end{cases}$

Related pages

• Numerical Methods & Transform Calculus syllabus
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