Mechanical Engineering (Semester 4)
TOTAL MARKS: 80
TOTAL TIME: 3 HOURS
(1) Question 1 is compulsory.
(2) Attempt any three from the remaining questions.
(3) Assume data if required.
(4) Figures to the right indicate full marks.
1 (a) Find the fourier expansion of f(x) = 4 - x2 in the interval (0,2)(5 marks)
1 (b) Find the probability that at most 5 defective fuses will be found in a box of 200 fuses if experience shows that 2% of such fuses are defective.(5 marks)
1 (c) Given 6y = 5x + 90, 15x = 8y + 130, (σx)2 = 16. Find :
(i) ¯x and ¯y
(ii) r
(iii) (σy)2(5 marks)
1 (d)
Solve the two dimensional heat equation$\dfrac{d^{2}u}{dx^{2}}+\dfrac{d^{2}u}{dy^2}=0$ which satisfies the conditions u(0, y)=u(l, y)=u(x, 0) and u(x, a) = sin $\dfrac{n \pi x}{l}$
(5 marks)
2 (a) Obtain fourier series for f(x) = x - x2, -π < x < π. Hence deduce that:
(7 marks)
2 (b) Seven dice are thrown 729 times. How many times do you expect atleast 4 dices to show three or five?(7 marks)
2 (c) A continuous random variable X had pdf f(x) = kx2 e-x, x ≥ 0.Find k mean and variance.(6 marks)
3 (a) Using normal distribution find the probability that in a group of 100 persons there will be 55 males assuming that the probability of a person being male is 1/2(7 marks)
3 (b) Derive wave equation for vibration of string.(7 marks)
3 (c) Obtain fourier expansion of f(x)= sin ax in the interval (-l,l) where a is not an integer.(6 marks)
4 (a) Calculate the correlation coefficient from the following data.
|
X : |
23 |
27 |
28 |
29 |
30 |
31 |
33 |
35 |
36 |
39 |
|
Y : |
18 |
22 |
23 |
24 |
25 |
26 |
28 |
29 |
30 |
32 |
(7 marks)
4 (b) A die was thrown 132 times and the following frequencies were observed
|
No. obtained : |
1 |
2 |
3 |
4 |
5 |
6 |
|
Frequency : |
15 |
20 |
25 |
15 |
29 |
28 |
(7 marks)
4 (c) Obtain complex form of fourier series for f(x) = cosh 3x + sinh 3x in (-3,3)(6 marks)
5 (a) A homogenous rod of conducting material of length l has ends kept at zero temperature and the temperature at centre is T and falls uniformly to zero at the two ends. Find the temperature u(x,t) at any time.(7 marks)
5 (b) Obtain half range sine series for f(x) when
f(x) = x, for 0 < x < π/2
f(x) = π-x,for π/2 < x < π
Hence deduce
(7 marks)
5 (c) Two independent samples of sizes 8 and 7 gave the following results
|
Sample 1 : |
19 |
17 |
15 |
21 |
16 |
18 |
16 |
14 |
|
Sample 2 : |
15 |
14 |
15 |
19 |
15 |
18 |
16 |
|
(6 marks)
6 (a) Find the expansion of f(x) = x(π-x), 0 < x < π as a half range cosine series. Hence show that
(i)
(ii)
(7 marks)
6 (b) The diameter of a semicircular plate of radius a is kept at 0°C and the temperature at the semicircular boundary T°C. Find the steady state temperature u(r,θ)(7 marks)
6 (c) The average of marks scored by 32 boys is 72 with standard deviation 8 while that of 36 girls is 70 with standard deviation 6. Test at 1% level of significance. Whether the boys perform better than the girls? (6 marks)
7 (a) Show that the functions f1(x)=1; f2(x)=x are orthogonal on (-1,1). Determine the constants a and b such that the function f3(x)=-1+ax+bx2is orthogonal to both f1 and f2 on that interval.(7 marks)
7 (b) Find fourier integral represention of
f(x) = x for 0 < x < a
f(x) = 0 for x > a.
and f(-x) = f(x)(7 marks)
7 (c) If u=x-y, v=x+y and if x,y are uncorrelated, prove that:
(6 marks)
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