MATHEMATICS-III
PAPER NO. 1          
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                                               CS-203 (Old)
                                       MATHEMATICS-III
                                  (B.Tech 3rd Semester,2121)
Time : 3 Hours                                                                                        Maximum Marks : 60
NOTE:- This paper consist of Three Sections. Section A is compulsory. Do any Four questions from
                 Section B and any two questions from Section C

                                     Section-A                                         Marks : 20

1(a) Evaluate
            
(b) State Taylor's theorm.
(c) State any one property satisfied by analytic functions.
(d) Show that:
            
(e) Discuss briefly the transformation, w = 1/z.
(f) Find the ared bounded by the parabola y2 = 4ax and its latus rectum.
(g) State cauchy's integral theorem.
(h) Determine the poles of the function,
            
(i) A rectangular plate with insulated surface is 10 cm wide and so long compared to its width that it may be
    considered infinite in length without introducing an appreciable error. If the temperature of the short edge
    y = 0 is given by, u = 20x for 0 < x < 5 and u = 20(10 - x) for 5 < x < 10 and the two long edges x = 0,
    x = 10 as well as the other short edge are kept at 0o C, then write down the initial and boundary conditions
    for this problem if it is required to find the temperature u at any point (x, y).
(j) State the fundamental theorm of integral calculus.

                                              Section-B                                          Marks:5 Each

2. Using Euler's method, find an appropriate value of y corresponding to x = 1, given that dy/dx = x + y and y = 1
    when x = 0.
3. If f(z) is a regular function of z, prove that:
            
4. Discuss the applicabililty of Rolle's theorem to f(a) = (x - a)m(x - b)n in the interval [a, b].
5. Find the centroid of the area enclosed by the curves y2 = ax, x2 = ay.
6. Using method of seperation of variables, solve,
            
    where u = (x, 0) = 6 e-3x.

                                               Section-C                                        Marks : 10 Each


7. If a string of length l is initially at rest in equillibrium position and each of its points is given the velocity.
            
    find the displacement y(x,t).
8. By the method of contour integration,show that
            
9. State and prove Cauchy's integral formula.
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