## 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.

x^{4}+3x^{3}-x^{2}+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:

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 marks)

**7 (b)**The probability that a bomb dropped hits the target is 0.2. Find the probability that our out of 6 bombs dropped

i) Exactly 2 will hit the target

ii) At least 3 will hit the target.(7 marks)

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

**8 (a)**A die is tossed 960 times and 5 appear 184 times. Is the die biased?(6 marks)

**8 (b)**Nine items 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 marks)

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

No. of head | 0 | 1 | 2 | 3 | 4 | 5 |

Frequency | 6 | 27 | 72 | 112 | 71 | 32 |

Test the hypothesis that data follows binomial distribution (Give γ=5, x

^{2}

_{0.05}= 11.07.(7 marks)