1 (a)
Obtain y(0.2) using Picard's methods up to second iteration for the initial value problem.

\[\dfrac{dy}{dx}=x^{2}-2y y(0)=1\]

\[\dfrac{dy}{dx}=x^{2}-2y y(0)=1\]

6 M

1 (b)
Solve by Euler's modified method to obtain y(1.2) given \[y=\dfrac{y+x}{y-x}\ y(1)=2\]

7 M

1 (c)
Using Adm Bash forth method obtain y at x=0.8 given

\[\dfrac{dy}{dx}=x-y^{2},y(0)=0, y(0.2)=0.02, y(0.4)=0 .0795 \ and\ y(0.6)=0.1762.\]

\[\dfrac{dy}{dx}=x-y^{2},y(0)=0, y(0.2)=0.02, y(0.4)=0 .0795 \ and\ y(0.6)=0.1762.\]

7 M

2 (a)
Solve by 4

\[\dfrac{dx}{dt}=y-t,\dfrac{dy}{dt}=x+t\ with\ x=1=y\ at\ t=0,\ obtain\ y(0.1)\ and\ x(0.1)\]

^{th}order Runge-Kutta method simultaneous equation given by.\[\dfrac{dx}{dt}=y-t,\dfrac{dy}{dt}=x+t\ with\ x=1=y\ at\ t=0,\ obtain\ y(0.1)\ and\ x(0.1)\]

6 M

2 (b)
Solve \[\dfrac{d^{2}y}{dx^{2}}-x\left ( \dfrac{dy}{dx} \right )^{2}+y^{2}=0,y(0)=1,y(0)=0\] Evaluate y(0.2) correct to four decimal place using Runge-Kutta method of fourth order.

7 M

2 (c)
Solve for x=0.4 using Milne's predictor corrector formula for the differential equation y'+xy''+y=0 with y(0)=1, y(0.1)=0.995, y(0.2)=0.9802 and y(0.3)=0.956. Also z=(0)=0,z(001)=0.0995,z(0.2)=0.196,z(0.3)=0.2863.

7 M

3 (a)
Verify whether f(z)=sin2z is analytic, hence obtain the derivative.

6 M

3 (b)
Determine the analytic function f(z) whose imaginary part is \[\dfrac{y}{x^{2}+y^{2}}\].

7 M

3 (c)
Determine harmonic function. Prove that real and imaginary parts of an analytic function are harmonic.

7 M

4 (a)
Under the mapping w=e

^{x}, find the image of i) 1≤x≤2 ii) π/3
6 M

4 (b)
Find the bilinear transformation which maps the points 1,i,-1 from z plane to 2, i-2 into w plane. Also find the fixed points.

7 M

4 (c)
State and prove Cauchy's integral formula.

7 M

5 (a)
Prove \[J_{n}(x)=\dfrac{x}{2n}\left [ J_{n-1}(x)+J_{n-1}(x) \right ]\].

6 M

5 (b)
Prove (n+1) P

_{n}(x)=(2n+1)xP_{n}(x)-nP_{n-1}(x).
7 M

5 (c)
Explain the following in terms of Legendre's polynomials.

7 M

6 (a)
A class has 10 boys and girls. There students are selected at random one after another. Find the probability that i) first and third are boys, second a girl ii) first and second are of the same sex and third is of opposite sex.

6 M

6 (b)
If P(A)=0.4, P(B/A)=0.9, \[P(\bar{B}/\bar{A})=0.6 Find\ P(A/B), P(A/\bar{B}).\]

7 M

6 (c)
In a bolt factors machines A,B and C manufacture 20%, 35% and 45% of the total of their outputs 5%,4% and 2% are defective. A bolt is drawn at random found to be defective, what is the probability that it is from machine B.

7 M

7 (a)
A random variable x has the following distribution

Find k mean and S.D of the distribution.

X: | -2 | -1 | 0 | 1 | 2 | 3 | 4 |

P(x) | 0.1 | 0.1 | k | 0.1 | 2k | k | k |

Find k mean and S.D of the distribution.

6 M

7 (b)
The Probability that a bomb dropped hits the largest is 0.2. find the probability that out of 6 bombs dropped i) Exactly 2 will hit the largest ii) Atleast 3 will hit the target.

7 M

7 (c)
Find the mean and variance of the exponential distribution.

7 M

8 (a)
A die is tossed 960 times and 5 appear 184 times, is the die biased?

6 M

8 (b)
Nine times have value 45,47,50,52,48,47,49,53,51. Does the mean of these differ significantly from assumed of mean of 47.5 (γ=8, t

_{0.05}=2.31).
7 M

8 (c)
A set of 5 similar coins tossed 320times gives following table.

Test the hypothesis that data follows binomial distribution (Given γ=5, X

No of heads | 0 | 1 | 2 | 3 | 4 | 5 |

Freq | 6 | 27 | 72 | 112 | 71 | 32 |

Test the hypothesis that data follows binomial distribution (Given γ=5, X

^{2}_{0.05}=11.07)
7 M

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