Engineering Mathematics 4 - Jun 2015

Electronics & Communication (Semester 4)

TOTAL MARKS: 100
TOTAL TIME: 3 HOURS (1) Question 1 is compulsory.
(2) Attempt any four from the remaining questions.
(3) Assume data wherever required.
(4) Figures to the right indicate full marks. 1 (a) Obtain y(0, 2) using Picard's method upto second iteration for the initial value problem $$ \dfrac {dy}{dx} = x^2 - 2y \ \ y(0)=1 $$(6 marks) 1 (b) Solve by Euler's modified method to obtain y(1, 2) given $$ y' = \dfrac {y+x} {y-x} \ \ y(1)=2. $$(7 marks) 1 (c) Using Adam Bash forth method obtain y at x=0.8 given $$ \dfrac {dy} {dx} = x -y^2, \ \ y(0)=0, \ \ y(0.2)=0.02, \ \ y(0.4)= 0.0795 \ and \ y(0.6) = 0.1762.\lt/span\gt\ltspan class='paper-ques-marks'\gt(7 marks)\lt/span\gt \lt/span\gt\ltspan class='paper-question'\gt\ltspan class='paper-ques-desc'\gt\ltb\gt2 (a)\lt/b\gt Solve by 4\ltsup\gtth\lt/sup\gt order Runge-Kutta method simultaneous equations given by $$ \dfrac {dx} {dt} = y-t, \ \ \dfrac {dy} {dt} = x+t \ with \ x=1 = y at \ t=0, $$ obtain y(0.1) and x(0.1).\lt/span\gt\ltspan class='paper-ques-marks'\gt(6 marks)\lt/span\gt \lt/span\gt\ltspan class='paper-question'\gt\ltspan class='paper-ques-desc'\gt\ltb\gt2 (b)\lt/b\gt Solve $$ \dfrac {d^2y}{dx^2} - x \left ( \dfrac {dy}{dx} \right )^2 + y^2 = 0, \ \ y(0)=1 , \ y'(0)=0 $$ Evaluate y(0.2) correct to four decimal places, using Runge method of fourth order.\lt/span\gt\ltspan class='paper-ques-marks'\gt(7 marks)\lt/span\gt \lt/span\gt\ltspan class='paper-question'\gt\ltspan class='paper-ques-desc'\gt\ltb\gt2 (c)\lt/b\gt Solve for x=0.4 using Milnes predictor corrector formula for the differential equation y''+xy'+y=0 with y(0)=1, y(0.1)=0.995, y(0.2)=0.9802 and y(0.3)=0.956. Also Z(0)=0, z(0.1)=-0.0995, z=(0.2)=-0.196, z(0.3)=0.2863.\lt/span\gt\ltspan class='paper-ques-marks'\gt(7 marks)\lt/span\gt \lt/span\gt\ltspan class='paper-question'\gt\ltspan class='paper-ques-desc'\gt\ltb\gt3 (a)\lt/b\gt Verify whether f(z)=sin2z is analytic, hence obtain the derivative.\lt/span\gt\ltspan class='paper-ques-marks'\gt(6 marks)\lt/span\gt \lt/span\gt\ltspan class='paper-question'\gt\ltspan class='paper-ques-desc'\gt\ltb\gt3 (b)\lt/b\gt Determine the analytic function f(z) whose imaginary part is $$ \dfrac {y}{x^2+y^2}. $$\lt/span\gt\ltspan class='paper-ques-marks'\gt(7 marks)\lt/span\gt \lt/span\gt\ltspan class='paper-question'\gt\ltspan class='paper-ques-desc'\gt\ltb\gt3 (c)\lt/b\gt Define a harmonic function. Prove that real and imaginary parts of an analytic function are harmonic.\lt/span\gt\ltspan class='paper-ques-marks'\gt(7 marks)\lt/span\gt \lt/span\gt\ltspan class='paper-question'\gt\ltspan class='paper-ques-desc'\gt\ltb\gt4 (a)\lt/b\gt Under the mapping w=e\ltsup\gtz\lt/sup\gt, find the image of $$ i) \ 1\le x \le 2 \ ii) \ \pi /3 < y < \dfrac {\pi } {2 } $$\lt/span\gt\ltspan class='paper-ques-marks'\gt(6 marks)\lt/span\gt \lt/span\gt\ltspan class='paper-question'\gt\ltspan class='paper-ques-desc'\gt\ltb\gt4 (b)\lt/b\gt Find the bilinear transformation which maps the points 1, i -1 from z plane to 2, i, -2 into w plane. Also find the fixed points.\lt/span\gt\ltspan class='paper-ques-marks'\gt(7 marks)\lt/span\gt \lt/span\gt\ltspan class='paper-question'\gt\ltspan class='paper-ques-desc'\gt\ltb\gt4 (c)\lt/b\gt State and prove Cauchy's integral formula.\lt/span\gt\ltspan class='paper-ques-marks'\gt(7 marks)\lt/span\gt \lt/span\gt\ltspan class='paper-question'\gt\ltspan class='paper-ques-desc'\gt\ltb\gt5 (a)\lt/b\gt Prove $$ J_n (x) = \dfrac {x} {2n} [ J_{n-1} (x) + J_{n+1} (x)] $$(6 marks) 5 (b) Prove (n+1) P_n(x) = (2n+1) × P_n(x)- n P_n-1(x).(7 marks) 5 (c) Explain the following in terms of Legendre's polynomials.
x⁴+3x³-x²+5x-2.(7 marks) 6 (a) A class has 10 boys and 6 girls. Three students are selected at random one after another. Find the probability that i) first and third are boys, second a girls ii) first and second are of same sex and third is of opposite sex.(6 marks) 6 (b) If P(A)=0.4, P(B/A)=0.9, P(B/A)=0.6. Find P(A/B), P(A/B).(7 marks) 6 (c) In a bolt factory machines A, B and C manufacture 20%, 35% and 45% of the total of their output 5%, 4% and 2% are defective. A bolt is drawn at random found to be defective. What is the probability that is is from machine B?(7 marks) 7 (a) A random variable x has the following distribution: