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Total No. of Questions: 08
Roll No.: ..........................
Total No. of Pages: 02
M.TECH FIRST SEMESTER EXAMINATION, APRIL – 2021
[Offline mode open book system]
(Course: Nanotechnology)
Paper Code: MNT-101
Mathematical Methods and Programming
Maximum Marks: 70
Minimum Marks: 28
Time: 3 hrs
Note: Total questions to be answered by candidate shall be only five. Each question have equal marks.
Q.1 (a)
Find the Fourier transform of \(f(x) = \begin{cases} 1-x^2, & \text{if } |x| < 1 \\ 0, & \text{if } |x| > 1 \end{cases}\) and use it to evaluate \(\int_{0}^{\infty} \frac{x \cos x - \sin x}{x^3} \cos \frac{x}{2} \,dx\).
(b)
Find the Fourier sine transform of \(\frac{e^{-ax}}{x}\).
Q.2 (a) (i)
Find \(L\left\{ \frac{\sin t}{t} \right\}\).
(ii)
Evaluate \(L^{-1}\left\{ \frac{1}{s^2(s^2+1)} \right\}\) using convolution theorem.
(b)
Solve the following differential equation using Laplace transform method \(\frac{d^2y}{dt^2} + 3 \frac{dy}{dt} + 2y = 4t + e^{3t}\) when \(y(0)=1\) and \(y'(0)=-1\).
Q.3 (a)
Prove that \(\int_{0}^{\pi} x J_n^2(x) \,dx = \frac{2}{\pi x} \sin x - \frac{3}{x^2} \cos x\).
(b)
Prove that \(\int_{0}^{x} x J_0^2(x) \,dx = \frac{1}{2} x^2 J_1^2(x)\).
Q.4 (a)
Prove that \((1 - x^2) P_n'(x) = (n + 1) [x P_n(x) - P_{n+1}(x)]\).
(b)
Three urns contain 6 red, 4 black: 4 red, 6 black and 5 red, 5 black balls respectively, one of the urns is selected at random and a ball is drawn from it. If the ball drawn is red, find the probability that it is drawn from the first urn.
Q.5 (a)
Calculate the co-efficient of correlation for the following age of husbands and wives:
| Husband's age x: | 23 | 30 | 28 | 28 | 29 | 30 | 31 | 33 | 35 | 36 |
| Wife's age y: | 18 | 20 | 22 | 27 | 21 | 29 | 27 | 29 | 28 | 29 |
(b)
A random variable X has the triangular probability density function given below:
\[ f(x) = \begin{cases} x, & \text{If } 0 \le x \le 1 \\ 2-x, & \text{If } 1 \le x \le 2 \\ 0, & \text{otherwise} \end{cases} \]
Calculate the following:
(i)
\(P \left( -1 < X < \frac{1}{2} \right)\)
(ii)
\(P \left( X < \frac{3}{2} \right)\)
Q.6 (a)
What do you mean by user defined Function in C? Explain the function definition, function declaration and function call with syntax and suitable example?
(b)
List the various types of loop available in C and explain any one with the syntax and suitable example.
Q.7 (a)
What do you mean by an Array? How to declare and initialize one dimensional array in C?
(b)
Discuss about script file and function file in writing MATLAB program with examples.
Q.8 (a)
Discuss about elementary mathematical function of MATLAB with proper commands.
(b)
Explain various features of MATLAB in details. Give five applications where it is used.