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INSTRUCTIONS
(1) Assume appropriate data and state your reasons
(2) Marks are given to the right of every question
(3) Draw neat diagrams wherever necessary

1 (a) Obtain s solution upto the third approximation of y for x=0.2 by Picard's method, given that $\dfrac {dy}{dx}+y=e^x; \ y(0)=1$
6 M
1 (b) Apply Runge kutta method of order 4, to find an approximate value of y for x=0.2 in step of 0.1, if dy/dx=x+y2 given that y=1 when x=0.
7 M
1 (c) Using Adams-Bashforth formulae, determine y(0,4) given the differential equation dy/dx=1/2 xy and the data, y(0)=1, y(0.1)=1.0025, y(0.2)=1.0101, y(0,3)=1.0228. Apply the corrector formula twice.
7 M

2 (a) Apply Picard's method to find second approximation to the values of 'y' and 'z' given that $\dfrac {dy}{dx}=z, \dfrac {dz}{dx}=x^3(y+z), \ given \ y=1, z=\dfrac {1}{2}\ when \ x=0$
6 M
2 (b) Using Runge-kutta method solve $\dfrac {d^2y}{dx^2}-x\left (\dfrac {dy}{dx} \right )^2+y^2=0 \ for \ x=0.2$ correct to four decimal places. Initial conditions are x=0, y=1, y'=0.
7 M
2 (c) Obtain the solution of the equation $\dfrac {2d^2 y}{dx^2}=4x+\dfrac {dy}{dx}$ at the point x=1.4 by applying Milne's method given that y(1)=2, y(1.1)=2.2156, y(1.2)=2.4649, y(1.3)=2.7514, y'(1)=2, y'(1.1)=2.3178, y'(1.2)=2.6725 and y'(1.3)=3.0657
7 M

3 (a) Define a anallytic function in a region R and show that f(z) is constant, if f(z) is an analytic function with constant modulus.
6 M
3 (b) Prove that $u=x^2-y^2 \ and \ v=\dfrac {y}{x^2+y^2}$ are harmonic functions of (x,y) but are not harmonic conjugate.
7 M
3 (c) Determine the analytic function $f(z)=u + iv, \ if \ u-v=\dfrac {\cos x +\sin x-e^{-y}}{2 (\cos x - \cosh y)}\ and \ f(\pi/2)=0$
7 M

4 (a) Find the image of the circle |z|=1 and |z|=2 under the conformal transformation $w=z+\dfrac {1}{z}$ and sketch the region.
6 M
4 (b) Find the bilinear transformation that transforms the point 0, i, ? onto the point 1, -i, -1 respectively.
7 M
4 (c) State and prove Cauchy's integral formula and hence generalized Cauchy's integral formula.
7 M

5 (a) Obtain the solution of the equation $x^2 \dfrac {d^2 y}{dx^2}+ x \dfrac {dy}{dx}+ \left ( x^2 - \dfrac {1}{4} \right )y=0$
6 M
5 (b) Obtain the series solution of Legendre's differential equation, $(1-x^2)\dfrac {d^2y}{dx^2}-2x \dfrac {dy}{dx}+ n (n+1)y=0$
7 M
5 (c) State Rodrigue's formula for Legendre polynomials and obtain the expression for P4(x) from it. Verify the property of Legendre polynomials in respect of P4(x) and also find $\int_{-1}^{1} P_4(x) dx$
7 M

6 (a) Two fair dice are rolled. If the sum of the numbers obtained is 4, find the probability that the numbers obtained on both the dice are even.
6 M
6 (b) Give that $P(\bar{A}\cap\bar{B})= \dfrac {7}{12}, \ P(A\cap \bar{B})= \dfrac {1}{6}=P(\bar{A} \cap B).$ Prove that A and B are neither independent nor mutually disjoint. Also compute P(A/B)+P(B/A) and P( A/B)+ P( B/ A).
7 M
6 (c) Three machine M1, M2 and M3 produces identical items, Of their respective output 5%, 4% and 3% of items are faulty. On a certain day, M1 has produced 25% of the total output, M2 has produced 30% and M3 the remainder. An item selected at random is found to be faulty. What are the chances that it was produced by the machine with the highest output?
7 M

7 (a) In quiz contest of answering Yes" or "No"
7 M
7 (b) Define exponential distribution and obtain the mean and standard deviation of the exponential distribution.
7 M
7 (c) If X is a normal variate with mean 30 and standard deviation 5, find the probabilities that (i) 26?X?40, (ii) X?45, (iii) [X-30]>5. [Give that ?(0.8)=0.2881, ?(2,0)=0.4772, ?(3,0)=0.4987, ?(1.0)=0.3413]
6 M

8 (a) Certain tubes manufactured by a company have mean life time of 800 hrs and standard deviation of 60 hrs. Find the probability that a random sample of 16 tubes taken from the group will have a mean life time (i) between 790 hrs and 810 hrs. (ii) less than 785 hrs, (iii) more than 820 hrs. [?(0.67)=0.2486, ?(1)=0.3413, ?(1.33)=0.4082].
6 M
8 (b) A set of five similar coins is tossed 320 times and the result is
 No. of heads: 0 1 2 3 4 5 Frequency: 6 27 72 112 71 32

Test the hypothesis that the data follow a binomial distribution [Give that ?2 0.05(5)=11.07]
7 M
8 (c) It required to test whether the proportion of smokers among students is less than that among the lectures. Among 60 randomly picked students, 2 were smokers. Among 17 randomly picked lectures, 5 were smokers. What would be your conclusion?
7 M

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