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VTU Computer Science (Semester 4)
Engineering Mathematics 4
June 2015
Total marks: --
Total time: --
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 y(0.2) using Picard's methods up to second iteration for the initial value problem.
$\dfrac{dy}{dx}=x^{2}-2y y(0)=1$
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
$\dfrac{dy}{dx}=x-y^{2},y(0)=0, y(0.2)=0.02, y(0.4)=0 .0795 \ and\ y(0.6)=0.1762.$
7 M

2 (a) Solve by 4th 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=ex, 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) Pn(x)=(2n+1)xPn(x)-nPn-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
 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, t0.05=2.31).
7 M
8 (c) A set of 5 similar coins tossed 320times gives following table.
 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, X20.05=11.07)
7 M

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