Sathyabama University Model Question Paper : Engineering Mathematics

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Type of Announcement : Engineering Mathematics Model Question Paper

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Engineering Mathematics-IV (SMTX1010)
PART A :
1. What is the value of bn when the function f(x)=x2 expanded as a Fourier series in (-p, p)?
2. State Dirichlet conditions for a function to be expanded as a Fourier series.
3. Write the complex form of Fourier series of f(x) in 0<x<l.
4. State parseval’s identity on Fourier series
5. Write the formulas for finding the Euler’s constants in the Fourier series expansion of f(x) in (-p, p).
6. Find the root mean square value of the function f(x) = x in the interval (0,1).
7. Define Harmonic analysis
8. Form the partial differential equation from z = (x-a)2 + (y-b)2 +1 by eliminating a and b.
9. Find the partial differential equation by eliminating arbitrary constants a & b . from z = (x+a)(y+b)
10. Write the complete integral for the partial differential equation z = px +qy +pq .
11. Find the complete integral of p + q = pq.
12. The steady state temperature distribution is considered in a square plate with sides x = 0, y = 0, x = a and y = a. The edge y = 0 is kept at a constant temperature T and the other three edges are insulated. The same state is continued subsequently. Express the problem mathematically.
13. State any two laws which are assumed to derive one dimensional heat equation
14. State two-dimensional Laplace equation
15. What is meant by Steady state condition in heat flow?

PART B :
1. Expand the function f (x) = x sin x as a Fourier series in the interval 0 = x=2p .
2. Obtain the Fourier series of f (x) = cos x in -p = x = p
3. Find the singular integral of the PDE z = px + qy + p2- q2.
4. Form the partial differential equation by eliminating the arbitrary functions ‘f ’ and ‘g’ in z = f(2x + y) + g(3x-y)
5. Solve : (x2-yz)p + (y2- zx) q = z2 – xy.
6. Solve : x(y2 +z2 )p + y( z2 +x2 )q = z(y2 – x2 )
7. Solve (D2 – 6DD’ + 5D’2)z = ex sinhy.
8. A taut string of length L is fastened at both ends. The midpoint of the string is taken to a height of b and then released from rest in this position. Find the displacement of the string at any time t.
9. A tightly stretched string with fixed end points x = 0 and x = l is initially in a position given by y (x, 0) = Vo sin3 (px/l). If it is released from rest from this position. Find the displacement y at any distance x from one end at any time t.
10. A rod of length 30 cms has its ends A and B kept at 20°C and 80°C respectively until steady state conditions prevail. The temperature at each end is then suddenly reduced to 0°C and kept so. Find the resulting temperature function u(x,t).
11. A rod of length 30 cms has its ends A and B kept at 20°C and 80°C respectively until steady state conditions prevail. The temperature at end A is then suddenly increased to 60°C and at end B decreased to 60°C . Find the subsequent temperature function u(x,t).
12. State and prove Parseval’s identity on Fourier transform
13. State the prove convolution theorem on Fourier transform

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