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+y

^{2}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.

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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 P

_{4}(x) from it. Verify the property of Legendre polynomials in respect of P_{4}(x) and also find \[ \int_{-1}^{1} P_4(x) dx \]
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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 M

_{1}, M_{2}and M_{3}produces identical items, Of their respective output 5%, 4% and 3% of items are faulty. On a certain day, M_{1}has produced 25% of the total output, M_{2}has produced 30% and M_{3}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?
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7 (a)
In quiz contest of answering Yes" or "No"

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7 (b)
Define exponential distribution and obtain the mean and standard deviation of the exponential distribution.

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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 φ(0.8)=0.2881, φ(2,0)=0.4772, φ(3,0)=0.4987, φ(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. [φ(0.67)=0.2486, φ(1)=0.3413, φ(1.33)=0.4082].

6 M

8 (b)
A set of five similar coins is tossed 320 times and the result is

Test the hypothesis that the data follow a binomial distribution [Give that Ψ

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 Ψ

^{2}_{0.05}(5)=11.07]
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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 lecturers, 5 were smokers. What would be your conclusion?

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