Numerical Methods & Transform Calculus 2022 Question Paper — Governme…
The 2022 Numerical Methods & Transform Calculus previous year question paper (Government College of Engineering and Technology, Jammu), with every question and its marks.
- University: Government College of Engineering and Technology, Jammu
- Subject: Numerical Methods & Transform Calculus
- Course: NUMERICAL METHODS & TRANSFORM CALCULUS
- Branch: Electrical/Electronics and Comm. Engg.
- Year: 2022
- Paper: end-term
- Duration: 180 minutes
- Total marks: 100
Instructions
- Attempt five questions in all, selecting two questions from each section.
- Use of calculator is allowed.
Questions
Section-A
- Q1. (a) Using Newton's divided difference formula, evaluate $f(9)$ from the following data : $f(5) = 150, f(7) = 392, f(11) = 1452, f(13) = 2366, f(17) = 5202$. (b) Using Newton-Raphson method, find a real root of $2x - \log_{10}x - 7 = 0$ correct to three decimal places. (c) Using Trapezoidal rule, evaluate $\int_{0}^{5} \frac{1}{4x + 5} dx$ by taking 10 subintervals. [20 marks]
- Q2. (a) Using Lagrange interpolation formula, evaluate $f(10)$ from the following data : $f(5) = 12, f(6) = 13, f(9) = 14, f(11) = 16$. (b) Using Iteration method, find a real root of $x^3 + x - 1 = 0$ correct to three decimal places. (c) Evaluate $\int_{0}^{0.6} e^{-x^2} dx$ using Simpson's $1/3$ rule by taking 7 ordinates. [20 marks]
- Q3. (a) Find by Taylor's series method, the values of $y$ at $x = 0.1$ to four decimal places from $\frac{dy}{dx} = 3x + y^2, y(0) = 1$. (b) Using Picard's method, find the value of $y$ at $x = 1$ for the initial value problem $\frac{dy}{dx} = y + x, y(0) = 1$. (c) Compute $y(1.1)$ by using Runge-Kutta method of 4th order for the initial value problem $\frac{dy}{dx} = x - y, y(1) = 1$. [20 marks]
- Q4. (a) Using Regula-Falsi method, find a real root of $e^x - x^3 = 0$ between $1.75$ and $2$ correct to four decimal places. (b) Compute $y(0.3)$ by using Euler's modified method for the initial value problem $\frac{dy}{dx} = 1 - y, y(0) = 1$. (c) Show that $\Delta = E\Delta = \Delta E = \delta E_{1/2}$. [20 marks]
Section-B
- Q5. (a) Find the Laplace transform of $\frac{1 - \cos at}{t}$. (b) Find the inverse Laplace transform of $\frac{s - 4}{(s - 4)^2 + 9}$. (c) Using Laplace transform, find the solution of the initial value problem $y'' + 25y = 10 \cos 5t, y(0) = 2, y'(0) = 0$. [20 marks]
- Q6. (a) Find the Fourier sine and cosine transform of $f(x) = x^{n-1}$. (b) Find the Fourier sine integral for $f(x) = e^{-\beta x}$. (c) Find the inverse Laplace transform of $\frac{s + 2}{(s^2 - 4s + 13)}$. [20 marks]
- Q7. (a) Let $f(t) = \begin{cases} t, & 0 < t < 2 \\ 0, & \text{otherwise} \end{cases}$ Then express $f(t)$ in terms of unit step function and also find its Laplace transform. (b) Find the Laplace transform of $t^2 e^{-2t} \cos t$. (c) Find the Fourier sine transform of $f(x) = \frac{e^{-ax}}{x}$. [20 marks]
- Q8. (a) Find the Laplace transform of full wave rectified $\sin$ wave given by $f(t) = \sin wt, 0 < t < \frac{\pi}{w}$. (b) Find the inverse Laplace transform of $\frac{3s + 1}{(s - 1)(s^2 + 1)}$. (c) Find the Fourier cosine transform of $f(x) = e^{-2x} + 4e^{-3x}$. [20 marks]