21MAT31 SET-1 Solved Model Question Paper with Answer

21MAT31 SET-1 Solved Model Question Paper with Answer of all Modules with answers of Subject Transform Calculus, Fourier Series and Numerical Techniques

21MAT31 SET-1 Solved Model Question Paper with Answer

MODULE 1

Q.01	a	Find the Laplace transform of te-t sin2t + \frac{cos2t-cos3t}{t} .	06
	b	Find the Laplace transform of the triangular wave of period 2a given by f(t)= t, 0<t<a 2a-t, a<t<2a	07
	c	Using the convolution theorem find the inverse Laplace transform of \frac{s}{(s^2+a^2)^2}	07

or

Q.02	a	Find the inverse Laplace transform of i) \frac{(s^2-1)^2}{s^2} ii) \frac{s}{s^2+6s+13}	06
	b	Express the following function in terms of unit step function and hence find its Laplace transform f(t)= 1, 0<t<1 2t, 1<t<2 3t, 2<t<3	07
	c	Solve by using Laplace transform techniques 𝑦′′ − 3 𝑦′ + 2𝑦 = 𝑒3𝑡 ,𝑦(0) = 1, 𝑦′(0) = −1	07

MODULE 2

Q.03	a	Obtain the Fourier series for f(x) = \frac{\Pi-x}{2} in 0\le x\le 2\Pi	06
	b	Find half-range Fourier cosine series for the function f(x)= (x-1)2 , in 0<x<1, and hence show that \frac{\Pi^2}{8}=\frac{1}{1^2}+\frac{1}{3^2}+\frac{1}{5^2}+….	07
	c	Find Fourier series expansion of y up to first harmonic if it is given by	07

x	0	1	2	3	4	5
f(x)	9	18	24	28	26	20

or

Q.04	a	Obtain the Fourier series for 𝑓(𝑥) = |𝑥| , −𝜋 ≤ 𝑥 ≤ 𝜋	06
	b	Obtain half-range sine series for 𝑓(𝑥) = x, 0\le x\le \frac{\Pi}{2} \Pi-x, \frac{\Pi}{2}\le x\le \Pi	07
	c	Expand y as a Fourier series up to first harmonic if the values of y given by	07

x	0	𝜋/6	𝜋/3	𝜋/2	2𝜋/3	5𝜋/6
y	1.98	1.30	1.05	1.30	-0.88	-0.25

MODULE 3

Q.05	a	Find the Fourier transform of f(x)= 1-x2, \left| x \right|\le 1 0, |x|>1 Hence evaluate \int_{0}^{\infty }\frac{sinx-xcosx}{x^3}cos(\frac{x}{2})dx	06
	b	Find the Z-transforms of cosh𝑛𝜃 and sinh𝑛𝜃	07
	c	Using z –transformation, solve the difference equation 𝑢𝑛+2 + 6𝑢𝑛+1 + 9𝑢𝑛 = 2𝑛 , 𝑢0 = 0 ,𝑢1 = 0	07

Q.06	a	Find the Fourier sine transform of f(x)= x, 0<x<1 2-x, 1<x<2 0, x>2	06
	b	Find the inverse cosine transform of Fc(𝛼) = 1-𝛼, 0\le \propto \le 1 0, 𝛼>1 And hence evaluate \int_{0}^{\infty }\left( \frac{sint}{t} \right)^2 dt	07
	c	Find the inverse Z-transform of \frac{z^2-20z}{(z-2)(z-3)(z-4)}	07

MODULE 4

Q.07

a

Solve 𝑢𝑥𝑥 + 𝑢𝑦𝑦 = 0 for the square mesh with boundary values as given below. Iterate till the mesh values are correct to two decimal places

10

Q.07

b

Evaluate the pivotal values of the equation 𝑢𝑡𝑡 = 16𝑢𝑥𝑥 , taking h = 1 up to t = 1.25. The boundary condition are 𝑢(0,𝑡) = 𝑢(5,𝑡) = 0,𝑢𝑡(𝑥,0) = 0 and 𝑢(𝑥,0) = 𝑥2(5 − 𝑥)

10

or

Q.08

a

Given the values of u(x, y) on the boundary of the square as in the following figure. Evaluate the function u(x, y) satisfying the Laplace equation 𝑢𝑥𝑥 + 𝑢𝑦𝑦 = 0 at the pivotal points of the figure

10

Q.08

b

Find the solution of the parabolic equation 𝑢𝑥𝑥 = 2𝑢𝑡 when u(4,𝑡) = 0,𝑎𝑛𝑑 𝑢(𝑥,0) = 𝑥(4 − 𝑥),taking ℎ = 1. Find the values up to t = 5.

10

MODULE 5

Q.09	a	Using Runge –Kutta method of order four, solve \frac{d^2y}{dx^2}=x\left( \frac{dy}{dx} \right)^{2}-y^2 for x=0.2. Given, y(0)=1, y'(0)=1	06
	b	Find the external of the functional \int_{x_{0}}^{x_{1}}(1+x^2y')y' dx	07
	c	Show that the geodesies on a plane are straight lines	07

or

Q.10	a	Given 𝑦′′ = 1 + 𝑦′,𝑦(0) = 1,𝑦′(0) = 1, compute y(0.4) for the following data using Milne’s predictor-corrector method. 𝑦(0.1) = 1.1103, 𝑦(0.2).2427, 𝑦(0.3) = 1.344 𝑦′(0.1) = 1.2103, 𝑦′(0.2) = 1.4427, 𝑦′(0.3) = 1.699	06
	b	Derive the Euler’s equation	07
	c	Find the curves on which the functional 0∫1 [(𝑦′)2 + 12𝑥𝑦] 𝑑𝑥 with 𝑦(0) = 0 𝑎𝑛𝑑 𝑦(1) = 1	07