Anna University :: Chennai – 600 025
Model Question Paper
B.E. / B.Tech Degree Examinations
Third Semester
MA 231 Mathematics III
Time : 3 Hrs Max. Marks : 100
Answer all Questions
Part – A ( 10 × 2 = 20 Marks)
 Form a partial differential equation by eliminating the arbitrary function f from .
 Find the complete integral of q = 2px.
 Find the half range sine series for f(x) = 2 in 0 < x < 4.
 If the cosine series for f(x) = x sin x for 0 < x < p is given by show that
 Classify the partial differential equation
.
 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.  Find the Laplace transform of
 Verify the initial value theorem for f(t) = 5 + 4 cos 2t.
 If Fourier transform of f(x) is F(s), prove that the Fourier transform of f(x) cos ax is .
10. Find the Fourier cosine integral representation of .

Part – B ( 5 × 16 = 80 Marks)
Question No. 11 has no choice; Questions 12 to 15 have one choice
(either – or type) each.
11. (i) Expand in Fourier series of periodicity 2p of . (8)
(ii) Find the halfrange cosine series for the function and hence deduce the sum of the series (8)
12. (a) (i) Find the complete solution and singular solution of z = px + qy + p^{2}– q^{2}. (8)
(ii) Find the general solution of (8)
(OR)
(b) (i) Solve: (8)
(ii) Solve : . (8)
13. (a) 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. (16)
(OR)
(b) A rod 30 cm long, has its ends A and B at 20ºC and 80ºC respectively, until steady state conditions prevail. The temperature at the end B is then suddenly reduced to 60º C and at the end A is raised to 40º C and maintained so. Find the resulting temperature u (x,t). (16)
14. (a) (i) Find the Laplace transform of the function
and extending periodically with period 2p. (8)
(ii)Apply the Convolution theorem to find (8)
(OR)
(b) (i) Solve by using Laplace transform technique, , given that y(0) = 2 and . (8)
(ii) Find the inverse Laplace transform of (8)
15. (a) (i) Find the Fourier transform of . (6)
Hence evaluate the following integral:
(ii) (5)
(iii) (5)
(OR)
(b) (i) Find the Fourier sine and cosine transform of . (6)
Hence find the value of the following integrals:
(ii) . (5)
(iii) (5)
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