## Engineering Mathematics 4 - Jun 2014

### Mechanical Engineering (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 s solution upto the third approximation of y for x=0.2 by Picard's method, given that $$ \dfrac {dy}{dx}+y=e^x; \ y(0)=1 $$(6 marks)
**1 (b)** Apply Runge kutta method of order 4, to find an approximate value of y for x=0.2 in step of 0.1, if dy/dx=x+y^{2} given that y=1 when x=0.(7 marks)
**1 (c)** Using Adams-Bashforth formulae, determine y(0,4) given the differential equation dy/dx=1/2 xy and the data, y(0)=1, y(0.1)=1.0025, y(0.2)=1.0101, y(0,3)=1.0228. Apply the corrector formula twice.(7 marks)
**2 (a)** Apply Picard's method to find second approximation to the values of 'y' and 'z' given that $$ \dfrac {dy}{dx}=z, \dfrac {dz}{dx}=x^3(y+z), \ given \ y=1, z=\dfrac {1}{2}\ when \ x=0 $$(6 marks)
**2 (b)** Using Runge-kutta method solve $$ \dfrac {d^2y}{dx^2}-x\left (\dfrac {dy}{dx} \right )^2+y^2=0 \ for \ x=0.2 $$ correct to four decimal places. Initial conditions are x=0, y=1, y'=0.(7 marks)
**2 (c)** Obtain the solution of the equation $$ \dfrac {2d^2 y}{dx^2}=4x+\dfrac {dy}{dx} $$ at the point x=1.4 by applying Milne's method given that y(1)=2, y(1.1)=2.2156, y(1.2)=2.4649, y(1.3)=2.7514, y'(1)=2, y'(1.1)=2.3178, y'(1.2)=2.6725 and y'(1.3)=3.0657(7 marks)
**3 (a)** Define a anallytic function in a region R and show that f(z) is constant, if f(z) is an analytic function with constant modulus.(6 marks)
**3 (b)** Prove that $$ u=x^2-y^2 \ and \ v=\dfrac {y}{x^2+y^2} $$ are harmonic functions of (x,y) but are not harmonic conjugate.(7 marks)
**3 (c)** Determine the analytic function $$ f(z)=u + iv, \ if \ u-v=\dfrac {\cos x +\sin x-e^{-y}}{2 (\cos x - \cosh y)}\ and \ f(\pi/2)=0 $$(7 marks)
**4 (a)** Find the image of the circle |z|=1 and |z|=2 under the conformal transformation $$ w=z+\dfrac {1}{z} $$ and sketch the region.(6 marks)
**4 (b)** Find the bilinear transformation that transforms the point 0, i, ? onto the point 1, -i, -1 respectively.(7 marks)
**4 (c)** State and prove Cauchy's integral formula and hence generalized Cauchy's integral formula.(7 marks)
**5 (a)** Obtain the solution of the equation $$ x^2 \dfrac {d^2 y}{dx^2}+ x \dfrac {dy}{dx}+ \left ( x^2 - \dfrac {1}{4} \right )y=0 $$(6 marks)
**5 (b)** Obtain the series solution of Legendre's differential equation, $$ (1-x^2)\dfrac {d^2y}{dx^2}-2x \dfrac {dy}{dx}+ n (n+1)y=0 $$(7 marks)
**5 (c)** State Rodrigue's formula for Legendre polynomials and obtain the expression for P_{4}(x) from it. Verify the property of Legendre polynomials in respect of P_{4}(x) and also find $$ \int_{-1}^{1} P_4(x) dx $$(7 marks)
**6 (a)** Two fair dice are rolled. If the sum of the numbers obtained is 4, find the probability that the numbers obtained on both the dice are even.(6 marks)
**6 (b)** Give that $$ P(\bar{A}\cap\bar{B})= \dfrac {7}{12}, \ P(A\cap \bar{B})= \dfrac {1}{6}=P(\bar{A} \cap B). $$ Prove that A and B are neither independent nor mutually disjoint. Also compute P(A/B)+P(B/A) and P( A/B)+ P( B/ A).(7 marks)
**6 (c)** Three machine M_{1}, M_{2} and M_{3} produces identical items, Of their respective output 5%, 4% and 3% of items are faulty. On a certain day, M_{1} has produced 25% of the total output, M_{2} has produced 30% and M_{3} the remainder. An item selected at random is found to be faulty. What are the chances that it was produced by the machine with the highest output?(7 marks)
**7 (a)** In quiz contest of answering Yes" or "No"(7 marks)
**7 (b)** Define exponential distribution and obtain the mean and standard deviation of the exponential distribution.(7 marks)
**7 (c)** If X is a normal variate with mean 30 and standard deviation 5, find the probabilities that (i) 26?X?40, (ii) X?45, (iii) [X-30]>5. [Give that ?(0.8)=0.2881, ?(2,0)=0.4772, ?(3,0)=0.4987, ?(1.0)=0.3413](6 marks)
**8 (a)** Certain tubes manufactured by a company have mean life time of 800 hrs and standard deviation of 60 hrs. Find the probability that a random sample of 16 tubes taken from the group will have a mean life time (i) between 790 hrs and 810 hrs. (ii) less than 785 hrs, (iii) more than 820 hrs. [?(0.67)=0.2486, ?(1)=0.3413, ?(1.33)=0.4082].(6 marks)
**8 (b)** 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 [Give that ?

^{2}

_{0.05}(5)=11.07](7 marks)

**8 (c)**It required to test whether the proportion of smokers among students is less than that among the lectures. Among 60 randomly picked students, 2 were smokers. Among 17 randomly picked lectures, 5 were smokers. What would be your conclusion?(7 marks)