Question Paper: Applied Mathematics 4 : Question Paper Dec 2016 - Mechanical Engineering (Semester 4) | Mumbai University (MU)
0

## Applied Mathematics 4 - Dec 2016

### 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. $\int _C\bar{F}.d\bar{r}\ \ \test{Where}\bar{F}=x^2\hat{i}-xy\hat{j}\lt/span\gt/ and c is the triangle having vertices A(0,2), B(2,0), C(4,2).\lt/span\gt\ltspan class='paper-ques-marks'\gt(5 marks)\lt/span\gt \lt/span\gt\ltspan class='paper-question'\gt\ltspan class='paper-ques-desc'\gt\ltb\gt1(b)\lt/b\gt Use Cayley - Hamilton theorem to find \ltspan class="math-tex"\gt$ 2A^4-5A^3-7A+61 \ \ \text {where} \ \ A=\begin{bmatrix} 1 &2 \\ 2 & 2 \end{bmatrix} $\lt/span\gt/\lt/span\gt\ltspan class='paper-ques-marks'\gt(5 marks)\lt/span\gt \lt/span\gt\ltspan class='paper-question'\gt\ltspan class='paper-ques-desc'\gt\ltb\gt1(c)\lt/b\gt If the mean of the following distribution is 16 finf m,n and variance \ltbr\gt \lttable border="1" cellpadding="10" cellspacing="1" style="width: 500px;"\gt \lttbody\gt \lttr\gt \lttd\gt X\lt/td\gt \lttd\gt 8\lt/td\gt \lttd\gt 12\lt/td\gt \lttd\gt 16\lt/td\gt \lttd\gt 20\lt/td\gt \lttd\gt 24\lt/td\gt \lt/tr\gt \lttr\gt \lttd\gt P(X=X)\lt/td\gt \lttd\gt 1/8\lt/td\gt \lttd\gt m\lt/td\gt \lttd\gt n\lt/td\gt \lttd\gt 1/4\lt/td\gt \lttd\gt 1/12\lt/td\gt \lt/tr\gt \lt/tbody\gt \lt/table\gt\lt/span\gt\ltspan class='paper-ques-marks'\gt(5 marks)\lt/span\gt \lt/span\gt\ltspan class='paper-question'\gt\ltspan class='paper-ques-desc'\gt\ltb\gt1(d)\lt/b\gt 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 girls.\lt/span\gt\ltspan class='paper-ques-marks'\gt(5 marks)\lt/span\gt \lt/span\gt\ltspan class='paper-question'\gt\ltspan class='paper-ques-desc'\gt\ltb\gt2(a)\lt/b\gt Calculate speaman's coefficient of rank correlation from the data on height and weight of 8 students \ltbr\gt \lttable border="1" cellpadding="10" cellspacing="1" style="width: 500px;"\gt \lttbody\gt \lttr\gt \lttd\gt Height (in inches)\lt/td\gt \lttd\gt 60\lt/td\gt \lttd\gt 62\lt/td\gt \lttd\gt 64\lt/td\gt \lttd\gt 66\lt/td\gt \lttd\gt 68\lt/td\gt \lttd\gt 70\lt/td\gt \lttd\gt 72\lt/td\gt \lttd\gt 74\lt/td\gt \lt/tr\gt \lttr\gt \lttd\gt Weight (in lbs)\lt/td\gt \lttd\gt 92\lt/td\gt \lttd\gt 83\lt/td\gt \lttd\gt 101\lt/td\gt \lttd\gt 110\lt/td\gt \lttd\gt 128\lt/td\gt \lttd\gt 119\lt/td\gt \lttd\gt 137\lt/td\gt \lttd\gt 146\lt/td\gt \lt/tr\gt \lt/tbody\gt \lt/table\gt\lt/span\gt\ltspan class='paper-ques-marks'\gt(6 marks)\lt/span\gt \lt/span\gt\ltspan class='paper-question'\gt\ltspan class='paper-ques-desc'\gt\ltb\gt2(b)\lt/b\gt It is known that the probability of an item produced by a certain machine will be defective is 0.05. if the produced items are sent to the market in packets of 20, find the number of packets containing i) at least 2 ii) exactly 2 iii) at most 2 defective items in a consignment of 1000 packets using poisson distribution.\lt/span\gt\ltspan class='paper-ques-marks'\gt(6 marks)\lt/span\gt \lt/span\gt\ltspan class='paper-question'\gt\ltspan class='paper-ques-desc'\gt\ltb\gt2(c)\lt/b\gt Find the eigen values and eigen vectors of the matrix. \ltspan class="math-tex"\gt$ A=\begin{bmatrix} 8 &-8 & -2\\ 4 & -3& -2\\ 3 & -4& 1 \end{bmatrix} $\lt/span\gt/\lt/span\gt\ltspan class='paper-ques-marks'\gt(8 marks)\lt/span\gt \lt/span\gt\ltspan class='paper-question'\gt\ltspan class='paper-ques-desc'\gt\ltb\gt3(a)\lt/b\gt Two different processes A and B are used to manufacture light bulbs. Samples were drawn from these two population and following results were obtained. \ltbr\gt\lttable border="1" cellpadding="10" cellspacing="1" style="width: 500px;"\gt \lttbody\gt \lttr\gt \lttd\gt Population\lt/td\gt \lttd\gt A\lt/td\gt \lttd\gt B\lt/td\gt \lt/tr\gt \lttr\gt \lttd\gt Sample Size\lt/td\gt \lttd\gt 20\lt/td\gt \lttd\gt 17\lt/td\gt \lt/tr\gt \lttr\gt \lttd\gt Sample Standard deviation\lt/td\gt \lttd\gt 60\lt/td\gt \lttd\gt 50\lt/td\gt \lt/tr\gt \lt/tbody\gt \lt/table\gt Test the hypothesis that variance of A is greater than variance of B.\lt/span\gt\ltspan class='paper-ques-marks'\gt(6 marks)\lt/span\gt \lt/span\gt\ltspan class='paper-question'\gt\ltspan class='paper-ques-desc'\gt\ltb\gt3(b)\lt/b\gt Using the method of lagrange's multipliers solve the following N. L. P.P. Optimize \ltspan class="math-tex"\gt$Z=6x_1^{2}+5X_2^{2} $\lt/span\gt/ suject to \ltspan class="math-tex"\gt$ x_1+5x_2=7\ \ \text{and }x_1,x_2\geq 0 $\lt/span\gt/\lt/span\gt\ltspan class='paper-ques-marks'\gt(6 marks)\lt/span\gt \lt/span\gt\ltspan class='paper-question'\gt\ltspan class='paper-ques-desc'\gt\ltb\gt3(c)\lt/b\gt Prove that \ltspan class="math-tex"\gt$ \bar{F}=\left ( 2xy+z \right )\hat{i}+\left ( x^2+2yz^3 \right )\hat{j}+\left ( 3y^2z^2+x \right )\hat{k}$\lt/span\gt/ is irrotational. Find the sclar potential for $$\bar{F}$$ and the work done in moving an object in this field from {1,2,0} to {2,2,1}(8 marks) 4(a) In an intelligence test administered to 1000 students the average score was 42 and standard deviation was 24. Find the number of students(i) exceeding the score 50 ii) between 30 and 54.(6 marks) 4(b) Use Gauss's divergence theorem to evaluate$ \iint_s\bar{N}. \bar{F}\ \ \text{where} \bar{F}=2x\hat{i}+xy\hat{j}+zk $/ over region bounded by the cylinder $$x^2+y^2=4, z=0, z=6$$(6 marks) 4(c) A sample of 400 students of undergraduates and 400 students of post graduate classes was taken to know their opinion about autonomus colleges. 290 of the undergraduate and 310 of the post graduate students favored the autonomus status. Present these facts in the form of a table and test at 5% level that the opinion regarding Autonomus status of colleges is independent of the level of classes of students.(8 marks) 5(a) Seven dice are thrown 729 times. How many times do you expect at least for dice to show three or five?(6 marks) 5(b) L'se stoke's theorem of evaluate$ \int _c\bar{F}.d\bar{r}\ \text{where}\bar{F}=4xz\hat{i}-y^2\hat{j}+yz\hat{k} \$/ and C is the boundary of x=0, y=0 and X2+y2=1 in the plane z=0(6 marks) 5(c) A chemical engineer is investigating the effect of process operating temperature X on product yield Y. The results in the following data

 X 100 110 120 130 140 150 160 170 180 190 Y 45 51 54 61 66 70 74 78 85 89
Find the equation of regression line which will be enable to predict yield on the basis of Temperature. Find also the correlation coefficient between X and Y.(8 marks) 6(a) Ten individual are chosen at random from a population and teir heights are found to 62, 63, 64, 65, 66, 69, 69,70, 71,70 inches. Discuss the suggestion that the mean height of the population is 65 inches.(6 marks) 6(b) Show that matrix A is derogatory and find its minimal polynomial $$A=\begin{bmatrix} 2 & -3 &3 \\ 0 & 3 &-1 \\ 0 & -1 & 3 \end{bmatrix}$$(6 marks) 6(c) Using the Kuhn-Tucker conditions solve the following problem Maximize $$Z=10x_1+10x_2-x_1^2-x_2^2$$ Subject to $$x_1+x_2\leq 8 \ltbr\gt-x_1+x_2\leq 5 \ltbr\gt x_1,x_2\geq 0$$(8 marks)