# Anna University Model Question Paper – Information Technology – Mathematics III – III Semester

**Model Question Paper**

**B.E. / B.Tech Degree Examinations**

**Third Semester**

** Mathematics III**

##### Time : 3 Hrs

##### 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 half-range 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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