21MAT31 SET-2 Solved Model Question Paper with Answer

21MAT31 SET-2 Solved Model Question Paper with Answers of all Modules with answers of Subject Transform Calculus, Fourier Series, and Numerical Techniques

21MAT31 SET-2 Solved Model Question Paper with Answer

MODULE 1

Q.01	a	Find the Laplace transform of i) e-3t sin5t cos3t ii) \frac{1-e^2}{t}	06
	b	Find the Laplace transform of the square–wave function of period a given by f(t)= 1, 0<t<a/2 -1, a/2<t<2	07
	c	Using the convolution theorem find the inverse Laplace transform of \frac{1}{(s^2+1)(s^2+9)}	07

or

Q.02	a	Using the unit step function, find the Laplace transform of f(t)= cos𝑡, 0 ≤ 𝑡 ≤ 𝜋 cos2𝑡 , 𝜋 ≤ 𝑡 ≤ 2𝜋 cos3𝑡 , 𝑡 ≥ 2𝜋	06
	b	Find the inverse Laplace transform of \frac{2s^2-6s+5}{s^3-6s^2+11s-6}	07
	c	Solve by using Laplace transform techniques \frac{d^2x }{dt^2}-2\frac{d x}{dt}+x=e^t with x(0)=2 and x'(0)=-1

MODULE 2

Q.03	a	Find a Fourier series to represent 𝑓(𝑥) = 𝑥2 𝑖𝑛 − 𝜋 ≤ 𝑥 ≤ 𝜋	06
	b	Obtain the half-range cosine series for 𝑓(𝑥) = 𝑥 𝑠𝑖𝑛𝑥 in (0, 𝜋) and hence show that \frac{\Pi-2}{4}=\frac{1}{1.3}-\frac{1}{3.5}+\frac{1}{5.7} − ⋯ ∞	07
	c	The following table gives the variation of periodic current over a period. Show that there is a direct current part of 0.75 amp in the variable current and obtain the amplitude of the first harmonic.

t sec	0	T/6	T/3	T/2	2T/3	5T/6	T
A amp	1.98	1.30	1.05	1.30	-0.88	-0.25	1098

or

Q.04	a	Find the Fourier series expansion of 𝑓(𝑥) = 2𝑥 − 𝑥2 , 𝑖𝑛 ( 0, 3)	06
	b	Obtain half-range sine series for f(x)= 𝑘𝑥 , 0 ≤ 𝑥 ≤ 𝑙/2 𝑘(𝑙 − 𝑥) , 𝑙/2≤ 𝑥 ≤ 𝑙	07
	c	Expand y as a Fourier series up to the first harmonic if the values of y are given by	07

x	0o	30o	60o	90o	120o	150o	180o	210o	240	270	300	330
y	1.80	1.10	0.30	0.16	1.50	1.30	2.16	1.25	1.30	1.52	1.76	2.00

MODULE 3

Q.05	a	Find the Fourier transform of 𝑓(𝑥) = 1 , |𝑥| ≤ 1 0 , |𝑥| > 1 Hence evaluate \int_{0}^{\infty }\frac{sinx}{x}dx	06
	b	Find the Fourier cosine and sine transforms of 𝑒−𝑎𝑥	07
	c	Find the Z-transforms of (𝑖) (𝑛 + 1)2 and (𝑖𝑖)sin(3𝑛 + 5)	07

Q.06	a	Find the Fourier transform of e^{-a^{2}x^{2}} , 𝑎 > 0. Hence deduce that it is self-reciprocal in respect of the Fourier series	06
	b	Find the inverse z –transform of \frac{2z^2+3z}{(z+2)(z-4)}	07
	c	Using z-transformation, solve the difference equation 𝑢𝑛+2 + 4𝑢𝑛+1 + 3𝑢𝑛 = 3𝑛 , 𝑢0 = 0 ,𝑢1 = 1	07

MODULE 4

Q.07	a	Classify the following partial differential equations (i) 𝑢𝑥𝑥 + 4𝑢𝑥𝑦 + 4𝑢𝑦𝑦 − 𝑢𝑥 + 2𝑢𝑦 = 0 (ii) 𝑥2𝑢𝑥𝑥 + (1 − 𝑦2)𝑢𝑦𝑦 = 0 ,−1 < 𝑦 < 1 (iii) (1 + 𝑥2)𝑢𝑥𝑥 + (5 + 2𝑥2)𝑢𝑥𝑡 + (4 + 𝑥2)𝑢𝑡𝑡 = 0 (iv) 𝑦2𝑢𝑥𝑥 − 2𝑦𝑢𝑥𝑦 + 𝑢𝑦𝑦 − 𝑢𝑦 = 8𝑦	10
	b	Find the values of 𝑢(𝑥,𝑡) satisfying the parabolic equation 𝑢𝑡 = 4𝑢𝑥𝑥 and the boundary conditions 𝑢(0,𝑡) = 0 = 𝑢(8,0) and 𝑢(𝑥,0) = 4𝑥 − 𝑥2/2 at the points 𝑥 = 𝑖 ∶ 𝑖 = 0,1,2,…,8 and 𝑡 = 𝑗/8 ∶ 𝑗 = 0,1,2,3,4.

or

Q.08	a	Solve the equation 𝜕𝑢/𝜕𝑡=𝜕2𝑢/𝜕𝑥2 subject to the conditions 𝑢(𝑥,0) = sin𝜋𝑥, 0 ≤ 𝑥 ≤ 1 𝑢(0,𝑡) = 𝑢(1,𝑡) = 0, Carry out computations for two levels, taking ℎ = 1/34 𝑎𝑛𝑑 𝑘 = 1/36	10
	b	The transverse displacement u of a point at a distance x from one end and at any time t of a vibrating string satisfies the equation 𝑢𝑡𝑡 = 25 𝑢𝑥𝑥, with the boundary conditions 𝑢(𝑥,𝑡) = 𝑢(5,𝑡) = 0 and the initial conditions 𝑢(𝑥,0) =20𝑥, 0 ≤ 𝑥 ≤ 1 5(5 − 𝑥), 1 ≤ 𝑥 ≤ 5 and 𝑢𝑡(𝑥,0) = 0. Solve this equation numerically up to 𝑡 = 5 taking ℎ = 1,𝑘 = 0.2.	10

MODULE 5

Q.09	a	Using Runge –Kutta method of order four, solve 𝑑2𝑦/𝑑𝑥2 = 𝑦 + 𝑥 𝑑𝑦/𝑑𝑥 for x = 0.2 , Given that , 𝑦(0) = 1 ,𝑦′(0) = 0	06
	b	Find the extremals of the functional x1∫x2 [𝑦2 + (𝑦′)2 + 2𝑦𝑒𝑥]𝑑𝑥	07
	c	Find the path on which a particle in the absence of friction, will slide from one point to another in the shortest time under the action of gravity	07

or

Q.10	a	Apply Milne’s method to solve 𝑑2𝑦/𝑑𝑥2 = 1 +𝑑𝑦 𝑑𝑥 at x = 0.4. given that 𝑦(0) = 1 , 𝑦(0.1) = 1.1103, 𝑦(0.2) = 1.2427 , 𝑦(0.3) = 1.399 𝑦′(0) = 1 ,𝑦′(0.1) = 1.2103,𝑦′(0.2) = 1.4427,𝑦′(0.3) = 1.699	06
	b	Find the extremals of the functional \int_{x_1}^{x_2 }\frac{(y')^2}{x^3}dx	07
	c	Find the curve on which the functional o∫𝜋/2 [(𝑦′)2 + 12𝑥𝑦] 𝜋/2 0 𝑑𝑥 with 𝑦(0) = 0 𝑎𝑛𝑑 𝑦(𝜋/2) = 0 can be extremised.	07