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## Engineering Mathematics 4 - Dec 2014

### 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)** Employ Taylor's series method to find an approximate solution to find y at x=0.1 given dy/dx=x-y^{2}, y(0)=1 by considering upto fourth degree term.(6 marks)
**1 (b)** Solve the following by Euler's modified method dy/dx=log (x+y), y(0)=2 to find y(0.4) by taking h=0.2.(7 marks)
**1 (c)** Given dy/dx=x^{2} (Hy) and y(1)=1, y(1,1)=1.233, y(1.2)=1.548, y(1.3)=1.979. Evaluate y(1.4) by Adams-Bash forth method. Apply corrector formula twice.(7 marks)
**2 (a)** $$ Solve \ \dfrac {dy}{dx}=1+xz \ and \ \dfrac {dz}{dx}=-xy \ for \ x=0.3 $$ by applying Runge Kutta method given y(0)=0 and z=(0)=1. Taken h=0.3.(6 marks)
**2 (b)** Use Picard's method to obtain the second approximation to the solutions of $$ \dfrac {d^2 y}{dx^2}- x^3 \dfrac {dy}{dx}- x^3 y=0 $$ given y(0)=1, y'(0)=0.5. Also find y(0,1).(7 marks)
**2 (c)** Apply Milne's method to compute y(0,4) given y+xy'+y=0(7 marks)
**3 (a)** Derive Cauchy-Riemann equations in Cartesian form.(6 marks)
**3 (b)** Find an analytic function f(z) whose real part is $$ \dfrac {\sin 2x}{\cosh 2y-\cos 2x} $$ and hence find its imaginary part.(7 marks)
**3 (c)** If f(z) is a holomorphic function of z, then show that $$ \left \{ \dfrac {\partial }{\partial x} |f(z) \right \}^2 + \left \{ \dfrac {\partial }{\partial y} |f(z) \right \}^2 = |f'(z)|^2 $$(7 marks)
**4 (a)** Discuss the transformation w=z+1/z.(6 marks)
**4 (b)** Find the BLT which maps the point z=1, i, -1 to w=i, 0, -i. Find image of |z|?1.(7 marks)
**4 (c)** Evaluate $$ \int_C\left \{ \dfrac {\sin \pi z^2 + \cos \pi z^2}{(z-1)^2 (z-2)} \right \}dz $$ where 'C' is circle |z|=3(7 marks)
**5 (a)** Express f(x)=x^{4}+3x^{3}-x^{2}+5x-2 in terms of Legendre polynomials.(6 marks)
**5 (b)** Obtain the solution of x^{2}y"+xy'+(x^{2}-n^{2})y=0 in terms of J_{n}(x) and J_{-n}(x).(7 marks)
**5 (c)** Derive Rodrique's formula $$ P_n(x)= \dfrac {1}{2^nn!} \dfrac {d}{dx^n} \left [ (x^2-1)^x \right ] $$(7 marks)
**6 (a)** State the axioms of probability. For any two events A and B, Prove that P(A?B)=P(A)+P(B)-P(A?B).(6 marks)
**6 (b)** A box 'A' contains 2 white and 4 black balls. Another box 'B' contains 5 white and 7 black balls. A ball is transferred from the box A to the box B. Then a ball is drawn from the box B. Find the probability that it is white.(7 marks)
**6 (c)** In a certain college 4% of boys and 1% of girls are taller than 1.8m. Further more 60% of the students are girls. If a student is selected at random and if found to be taller than 1.8 m, what is the probability that the student is a girl?(7 marks)
**7 (a)** The probability density of a continuous random variable is given by P(x)=y_{0} e^{-|x|}, -10<x<?. find="" y<sub="">0 also find the mean.</x<?.></span>(6 marks)
**7 (b)** Obtain the mean and variance of binomial distribution.(7 marks)
**7 (c)** In a test on 2000 electric bulbs, it was found that the life of a particular make was normally distributed with an average life of 2040 hours ans SD of 60 hours. Estimate the number of bulbs likely to burn for.

i) More than 2150 hours.

ii) Less than 1950 hours

iii) More than 1920 hours but less than 2160 hours.

Given A(1.5)=0.4332, A(1.83)=0.4664, A(2)=0.47772.(7 marks)
**8 (a)** In a city 'A' 20% of a random sample of 900 school boys had a certain slight physical defect. In another city B, 18.5% of a random sample of 1600 school boys had the same defect. Is the difference between the proportions is significant? Why?(6 marks)
**8 (b)** A manufacture claimed that atleast 95% of the equipment which he supplied to a factory conformed to specifications. An examination of a sample of 200 pieces of equipment revealed that 18 of them were faulty. Test his claim at a significance level of 1% and 5%.(7 marks)
**8 (c)** A set of five similar coins is tossed 320 times and the result is

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 [ x

_{0.05}

^{2}=11.07 for 5df].(7 marks)