### 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)** Use modified Euler's method to solve dy/dx=x+y, y(0)=1 at x=0.1 for three iterations taking h=0.1.(6 marks)
**1 (b)** Solve dy/dx=x+y, x=0, y=1 at x=0.2 using Runge-Kutta method. Take h=0.2(7 marks)
**1 (c)** Using Milne's predictor-corrector method find y(0.3) correct to three decimals given.

x | -0.1 | 0 | 0.1 | 0.2 |

y | 0.908783 | 1.0000 | 1.11145 | 1.25253 |

(7 marks)
**2 (a)** Approximate y and z at x=0.2 using Picard's method for the solution of $$ \dfrac {dy}{dx}=z \ \dfrac {dz}{dx}=x^3 (y+z) $$ with y(0)=1, z(0)=1/2. Perform two steps (y_{1}. y_{2}, z_{1}.z_{2}).(10 marks)
**2 (b)** Using Runge-Kutta method solve y=x(y')^{2}-y^{2} at x=0.2 with x_{0}=0(10 marks)
**3 (a)** If f(z)=u+iv is analytic prove that Cauchy-Reimann equations u_{x}=v_{y}, u_{y}=-v_{x} are true.(6 marks)
**3 (b)** If w=z^{3} find dw/dz(7 marks)
**3 (c)** If the potential function is $$ \phi =\log \sqrt{x^2+y^2} $$ Find the stream function.(7 marks)
**4 (a)** Find the bilinear transformation which maps the points z=1, i, -1 onto the points w=j, o, -i.(6 marks)
**4 (b)** Discuss the conformal transformation w=e^{z}. Any horizontal strip of height 2? in z-plane will map what portion of w-plane.(7 marks)
**4 (c)** State and prove Cauchy's integral formula.(7 marks)
**5 (a)** Prove that $$ \int^{x}_{1/2}=\sqrt{\dfrac {2}{\pi x}}\sin x. $$(6 marks)
**5 (b)** State and prove Rodrigues formula for Legendre's polynomials.(7 marks)
**5 (c)** Express f(x)=x^{4}+3x^{3}-x^{2}+5x-2 in terms of Legendre polynomials.(7 marks)
**6 (a)** The probabilities of four persons A, B, C, D hitting target are respectively 1/2, 1/3, 1/4, 1/5. What is the probability that target is hit by atleast one person if all hit simultaneously?(6 marks)
**6 (b)** i) State addition law of probability for any two events A and B.

ii) Two different digits from 1 to 9 are selected. What is the probability that the sum of the two selected digits is odd if '2' one of the digits selected.(7 marks)
**6 (c)** Three machine A, B, C produce 50%, 30%, 20% of the items. The percentage of defective items are 3, 4, 5 respectively. If the item selected is defective what is the probability that it is from machine A? Also find the total probability thatn an item is defective.(7 marks)
**7 (a)** The p.d.f of x is

x | 0 | 1 | 2 | 3 | 4 | 5 | 6 |

p(x) | k | 3k | 5k | 7k | 9k | 11k | 13k |

Find k. Also p(x?5), p(3<x?6).< a="">

</x?6).<></span>(6 marks)
**7 (b)** A die is thrown 8 times. Find the probability that '3' falls,

i) Exactly 2 times

ii) At least once

iii) At te most 7 times.(7 marks)
**7 (c)** In a certain town the duration of shower has mean 5 minutes. What is the probability that shower will last for i) 10 minutes or more; ii) less than 10 minutes; iii) between 10 and 12 minutes.(7 marks)
**8 (a)** What is null hypothesis, alternative hypothesis significance level?(6 marks)
**8 (b)** The nine items of a sample have the following values: 45, 47, 50, 52, 48, 47, 49, 53, 51. Does the mean of these differ significantly from the assumed mean 47.5. Apply student's t-distribution at 5% level of significance. (t_{0.05} for 8df=2.31).(7 marks)
**8 (c)** In experiments on a pea breading. The following frequencies of seeds were obtained:

is the experiment is in the agreement of theory which predicts proportion of frequencies 9:3:3:1 (x^{2} _{0.05}, 3df=7.815).(7 marks)