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 (y1. y2, z1.z2).(10 marks)
2 (b) Using Runge-Kutta method solve y=x(y')2-y2 at x=0.2 with x0=0(10 marks)
3 (a) If f(z)=u+iv is analytic prove that Cauchy-Reimann equations ux=vy, uy=-vx are true.(6 marks)
3 (b) If w=z3 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=ez. 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)=x4+3x3-x2+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. (t0.05 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 (x2 0.05, 3df=7.815).(7 marks)
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