Question Paper: Applied Mathematics - 4 : Question Paper Dec 2014 - Civil Engineering (Semester 4) | Mumbai University (MU)
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## Applied Mathematics - 4 - Dec 2014

### Civil 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 $$\int_c (xy+y^2)dx+x^2dy$$ 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 $$A= \begin{bmatrix} 1 & 4\\2 &3 \end{bmatrix}$$(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 $$A=\begin{bmatrix} 8 & -6 & 2\\ -6& 7 &-4 \\ 2&-4 & 3 \end{bmatrix}$$ 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 1013 15 14 90108

Test the equality of sample at 5% level of significance.
(6 marks)
3 (b) Find the relative maximum or minimum of the function $$z=x^2_1+x^2_2+x^2_3-6x_1 -10x_2-14x_3+130$$(6 marks) 3 (c) Prove that $$\bar{F}= (y^2 \cos x +z^3 )i + (2y\sin x -4)j+(3xz^2+2)k$$ 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 $$\iint_s \bar{N}\cdot \bar{F}ds \ where \ \bar{F}= 4x\widehat{i} + 3y\widehat{j}-2z\widehat{k}$$ 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 LiteratesIlliterates 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 $$\int_c \bar{F}.d\bar{r} \ where \ \bar{F}=x^2i+xyj$$ 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) $$If \ A=\begin{bmatrix}2 &3 \\-3 &-7 \end{bmatrix} \ find \ A^{50}$$(6 marks) 6 (c)

Use the Kuhn-Trucker Conditions to solve the following N.L.P.P \begin {align*} Maximise \ z =&2x_1^2 -7x_2^2+12x_1x_2 \\ Subject \ to \ & 2x_1 +5x_2 \le 98 \\ & x_1x_2\ge 0 \end{align*}

(8 marks)