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.01aFind the Laplace transform of
i) e-3t sin5t cos3t
ii) \frac{1-e^2}{t} 		06
bFind 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
cUsing the convolution theorem find the inverse Laplace transform of \frac{1}{(s^2+1)(s^2+9)} 07

or

Q.02aUsing the unit step function, find the Laplace transform of
f(t)= cos𝑑, 0 ≀ 𝑑 ≀ πœ‹
cos2𝑑 , πœ‹ ≀ 𝑑 ≀ 2πœ‹
cos3𝑑 , 𝑑 β‰₯ 2πœ‹		06
bFind the inverse Laplace transform of \frac{2s^2-6s+5}{s^3-6s^2+11s-6} 07
cSolve 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.03aFind a Fourier series to represent 𝑓(π‘₯) = π‘₯2 𝑖𝑛 βˆ’ πœ‹ ≀ π‘₯ ≀ πœ‹06
bObtain the half-range cosine series for 𝑓(π‘₯) = π‘₯ 𝑠𝑖𝑛π‘₯ in (0, πœ‹) and hence show that \frac{\Pi-2}{4}=\frac{1}{1.3}-\frac{1}{3.5}+\frac{1}{5.7} βˆ’ β‹― ∞07
cThe 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 sec0T/6T/3T/22T/35T/6T
A amp1.981.301.051.30-0.88-0.251098

or

Q.04aFind the Fourier series expansion of 𝑓(π‘₯) = 2π‘₯ βˆ’ π‘₯2 , 𝑖𝑛 ( 0, 3)06
bObtain half-range sine series for
f(x)= π‘˜π‘₯ , 0 ≀ π‘₯ ≀ 𝑙/2
π‘˜(𝑙 βˆ’ π‘₯) , 𝑙/2≀ π‘₯ ≀ 𝑙		07
cExpand y as a Fourier series up to the first harmonic if the values of y are given by07
x0o30o60o90o120o150o180o210o240270300330
y1.801.100.300.161.501.302.161.251.301.521.762.00

MODULE 3

Q.05aFind the Fourier transform of
𝑓(π‘₯) = 1 , |π‘₯| ≀ 1
0 , |π‘₯| > 1
Hence evaluate \int_{0}^{\infty }\frac{sinx}{x}dx 		06
bFind the Fourier cosine and sine transforms of π‘’βˆ’π‘Žπ‘₯07
cFind the Z-transforms of (𝑖) (𝑛 + 1)2 and (𝑖𝑖)sin(3𝑛 + 5)07
Q.06aFind the Fourier transform of e^{-a^{2}x^{2}} , π‘Ž > 0. Hence deduce that it is self-reciprocal in respect of the Fourier series06
bFind the inverse z –transform of \frac{2z^2+3z}{(z+2)(z-4)} 07
cUsing z-transformation, solve the difference equation 𝑒𝑛+2 + 4𝑒𝑛+1 + 3𝑒𝑛 = 3𝑛 , 𝑒0 = 0 ,𝑒1 = 107

MODULE 4

Q.07aClassify the following partial differential equations
(i) 𝑒π‘₯π‘₯ + 4𝑒π‘₯𝑦 + 4𝑒𝑦𝑦 βˆ’ 𝑒π‘₯ + 2𝑒𝑦 = 0
(ii) π‘₯2𝑒π‘₯π‘₯ + (1 βˆ’ 𝑦2)𝑒𝑦𝑦 = 0 ,βˆ’1 < 𝑦 < 1
(iii) (1 + π‘₯2)𝑒π‘₯π‘₯ + (5 + 2π‘₯2)𝑒π‘₯𝑑 + (4 + π‘₯2)𝑒𝑑𝑑 = 0
(iv) 𝑦2𝑒π‘₯π‘₯ βˆ’ 2𝑦𝑒π‘₯𝑦 + 𝑒𝑦𝑦 βˆ’ 𝑒𝑦 = 8𝑦		10
bFind the values of 𝑒(π‘₯,𝑑) satisfying the parabolic equation 𝑒𝑑 = 4𝑒π‘₯π‘₯ and the boundary conditions 𝑒(0,𝑑) = 0 = 𝑒(8,0) and 𝑒(π‘₯,0) = 4π‘₯ βˆ’ π‘₯2/2 at the points π‘₯ = 𝑖 ∢ 𝑖 = 0,1,2,…,8 and 𝑑 = 𝑗/8 ∢ 𝑗 = 0,1,2,3,4.

or

Q.08aSolve the equation πœ•π‘’/πœ•π‘‘=πœ•2𝑒/πœ•π‘₯2 subject to the conditions 𝑒(π‘₯,0) = sinπœ‹π‘₯, 0 ≀ π‘₯ ≀ 1 𝑒(0,𝑑) = 𝑒(1,𝑑) = 0, Carry out computations for two levels, taking β„Ž = 1/34 π‘Žπ‘›π‘‘ π‘˜ = 1/3610
bThe 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 𝑒𝑑𝑑 = 25 𝑒π‘₯π‘₯, with the boundary conditions 𝑒(π‘₯,𝑑) = 𝑒(5,𝑑) = 0 and the initial conditions
𝑒(π‘₯,0) =20π‘₯, 0 ≀ π‘₯ ≀ 1
5(5 βˆ’ π‘₯), 1 ≀ π‘₯ ≀ 5 and 𝑒𝑑(π‘₯,0) = 0. Solve this equation numerically up to 𝑑 = 5 taking β„Ž = 1,π‘˜ = 0.2.		10

MODULE 5

Q.09aUsing Runge –Kutta method of order four, solve 𝑑2𝑦/𝑑π‘₯2 = 𝑦 + π‘₯ 𝑑𝑦/𝑑π‘₯ for x = 0.2 , Given that , 𝑦(0) = 1 ,𝑦′(0) = 006
bFind the extremals of the functional x1∫x2 [𝑦2 + (𝑦′)2 + 2𝑦𝑒π‘₯]𝑑π‘₯07
cFind 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 gravity07

or

Q.10aApply Milne’s method to solve 𝑑2𝑦/𝑑π‘₯2 = 1 +𝑑𝑦 𝑑π‘₯ 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
bFind the extremals of the functional \int_{x_1}^{x_2 }\frac{(y')^2}{x^3}dx 07
cFind the curve on which the functional oβˆ«πœ‹/2 [(𝑦′)2 + 12π‘₯𝑦] πœ‹/2 0 𝑑π‘₯ with 𝑦(0) = 0 π‘Žπ‘›π‘‘ 𝑦(πœ‹/2) = 0 can be extremised.07

Related Posts

2 Comments

Leave a Reply

Your email address will not be published. Required fields are marked *