Numerical Methods & Transform Calculus 2022 Question Paper — Governme…

Instructions

• Attempt any five questions selecting atleast two questions from each section. All questions carry equal marks.
• Assume missing data if any.
• Use of scientific Calculator is allowed.

Questions

Section A

• QQ1.(a). Find $x(0.150)$ from the following data using Lagrange's interpolating formula \begin{tabular}{|c|c|c|c|c|} \hline $x$ & $0.10$ & $0.15$ & $0.20$ & $0.25$ \\ \hline $y$ & $0.200$ & $0.300$ & $0.500$ & $0.600$ \\ \hline \end{tabular} [10 marks]
• Q(b). Find the fifth root of 3 correct to five decimal places using Newton-Raphson method. [10 marks]
• QQ2.(a). Given that \begin{tabular}{|c|c|c|c|c|c|c|c|} \hline $x$ & $4.0$ & $4.2$ & $4.4$ & $4.6$ & $4.8$ & $5.0$ & $5.2$ \\ \hline $\log x$ & $1.3863$ & $1.4351$ & $1.4816$ & $1.5261$ & $1.5686$ & $1.6094$ & $1.6487$ \\ \hline \end{tabular} evaluate $\int_{4}^{5.2} \log x \, dx$ by Trapezoidal rule and Simpson's $1/3$ rule. [10 marks]
• Q(b). Using Iteration method, find a real root of the equation $xe^x = 1$ between 0 and 1 correct to four decimal places. [10 marks]
• QQ3.(a). Find the value of $y(1.1)$ and $y(1.2)$, by Taylor's series method for the initial value problem $$\frac{dy}{dx} = \log(xy), \; y(1) = 2$$ [10 marks]
• Q(b). Using Picard's method, find the value of $y$ at $x = 0.2$ for the initial value problem $$\frac{dy}{dx} = x - y, \; y(0) = 1.$$ [10 marks]
• QQ4.(a). Compute $y(0.2)$ and $y(0.4)$ by using Runge-Kutta method of 4th order for the initial value problem $$\frac{dy}{dx} = \frac{x^2 + y^2}{10}, \; y(0) = 1$$ taking $h = 0.1$. [10 marks]
• Q(b). Using Euler's modified formula, solve $$\frac{dy}{dx} = 1 - 2xy, \; y(0) = 0$$ from $x = 0$ to $0.6$ taking the interval $h = 0.2$. [10 marks]

Section B

• QQ5.(a). Find the inverse Laplace transform of $\frac{1}{(s + 1)(s^2 + 2s + 2)}$. [10 marks]
• Q(b). Using Laplace transform, find the solution of the initial value problem $$y'' + 2y' + y = te^{-t}, \; y(0) = 1, \; y'(0) = -2.$$ [10 marks]
• QQ6.(a). Find the Fourier sine and cosine transform of $f(x) = ae^{-ax} + be^{-bx}$. [10 marks]
• Q(b). Find the Laplace transform of the periodic function $$f(t) = e^t \text{ for } 0 < t < 2\pi.$$ [10 marks]
• QQ7.(a). Express $f(t)$ in terms of unit step function and find its Laplace transform $$f(t) = \begin{cases} \sin t, & 0 < t < \pi \\ 1, & t > \pi \end{cases}$$ [10 marks]
• Q(b). Find the Laplace transform of $te^{-t} \cosh 2t$ and $\frac{1}{t} \sinh t$. [10 marks]
• QQ8.(a). Find the Laplace transform of the Periodic function (saw tooth wave) $$f(t) = \frac{kt}{T}, \; 0 < t < T, \; f(t + T) = f(t).$ [10 marks]
• Q(b). State and prove Fourier's integral theorem. [10 marks]

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