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)
Using Green's theorem evaluate ∫c(xy+y2)dx+x2dy
where c is the closed curve of the region bounded by y=x and y=x2
(5 marks)
1 (b) Use Cayley-Hamilton theorem to find A5-4A4-7A 3 +11A 2-A-10 I in terms of A where A=[1423]
(5 marks)
1 (c) A continuous random variable has probability density function f(x)=6(x-x2) 0≤x≤1. Find mean and variance.(5 marks)
1 (d) A random sample of 900 items is found to have a mean of 65.3cm. Can it be regarded as a sample from a large population whose mean is 66.2cm and standard deviation is 5cm at 5% level of significance.(5 marks)
2 (a) Calculate the value of rank correlation coefficient from the following data regarding marks of 6 students in statistics and accountancy in a test.
Marks in Statistics: | 40 | 42 | 45 | 35 | 36 | 39 |
Marks in Accountancy: |
46 | 43 | 44 | 39 | 40 | 43 |
(6 marks)
2 (b) If 10% of bolts produced by a machine are detective. Find the probability that out of 5 bolts selected at random at most one will be defective.(6 marks)
2 (c) Show that the matrix A=[8−62−67−42−43]
is diagonalisable. Find the transforming matrix and the diagonal matrix.(8 marks)
3 (a) In a laboratory experiment two samples gave the following results.
Sample | size | mean |
sum of square of deviations from the mean |
1 2 | 10 13 | 15 14 | 90 108 |
Test the equality of sample at 5% level of significance.(6 marks)
3 (b) Find the relative maximum or minimum of the function z=x21+x22+x23−6x1−10x2−14x3+130
(6 marks)
3 (c) Prove that ˉF=(y2cosx+z3)i+(2ysinx−4)j+(3xz2+2)k
is a conservative field. Find the scalar potential for F and the work done in moving an object and this field from (0, 1, -1) to (π/2, -1, 2).(8 marks)
4 (a) The weights of 4000 students are found to be normally distributed with mean 50kgs. And standard deviation 5kg. Find the probability that a student selected at random will have weight (i) less than 45kgs. (ii) between 45 and 60 kgs.(6 marks)
4 (b) Use Gauss's Divergence theorem to evaluate ∬sˉN⋅ˉFds where ˉF=4xˆi+3yˆj−2zˆk
and s is the surface bounded by x=0, y=0, z=0 and 2x+2y+z=4(6 marks)
4 (c) Based on the following data, can you say that there is no relation between smoking and literacy.
| Smokers | Nonsmokers |
Literates Illiterates | 83 45 | 57 68 |
(8 marks)
5 (a) A random variable X follows a Poisson distribution with variance 3 calculate p(x=2) and p(x≥4).(6 marks)
5 (b) Use Stroke's theorem to evaluate ∫cˉF.dˉr where ˉF=x2i+xyj
and c is the boundary of the rectangle x=0, y=0, x=a, y=b(6 marks)
5 (c) Find the equations of the two lines of regression and hence find correlation coefficient from the following data.
x | 65 | 66 | 67 | 67 | 68 | 69 | 70 | 72 |
y | 67 | 68 | 65 | 68 | 72 | 72 | 69 | 71 |
.(8 marks)
6 (a) 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 | |
Is the difference between sample means significant.(6 marks)
6 (b) If A=[23−3−7] find A50
(6 marks)
6 (c) Use the Kuhn-Trucker Conditions to solve the following N.L.P.P Maximise z=2x21−7x22+12x1x2Subject to 2x1+5x2≤98x1x2≥0
(8 marks)