Question Paper: Applied Mathematics 4 : Question Paper Dec 2015 - Computer Engineering (Semester 4) | Mumbai University (MU)
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## Applied Mathematics 4 - Dec 2015

### Computer 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) Evaluate the line integral $\int^{1+i}_{0} (x^2 iy)dz$ along the path y=x.(5 marks) 1 (b) State Cayley-Hamilton theorem & verify the same for $A= \begin{bmatrix} 1 &3 \\\\2 &2 \end{bmatrix}$(5 marks) 1 (c) The probability density function of a random variable x is

 x -2 -1 0 1 2 3 p(x) 0.1 k 0.2 2k 0.3 K

Find i) k ii) mean iii) variance(5 marks) 1 (d) Find all the basic solutions to the following problem.
Maximum
z=x1+x2+3x3
Subject to
x1+2x2+3x3=4
2x1+3x2+5x3=7
and x1, x2, x3 ≥0.
(5 marks)
2 (a) Find the Eigen values and the Eigen vectors of the matrix $\begin{bmatrix} 4 &6 &6 \\\\1 &3 &2 \\\\-1 &-5 &-2 \end{bmatrix}$(6 marks) 2 (b) Evaluate $\oint_c \dfrac {dz}{z^3 (z+4)}$ where c is the circle |z|=2.(6 marks) 2 (c) If the heights of 500 students is normally distributed with mean 68 inches and standard deviation of 4 inches, estimate the number of students having heights
i) less than 62 inches, ii) between 65 and 71 inches.
(8 marks)
3 (a) Calculate the coefficient of correlation from the following data:
 x 30 33 25 10 33 75 40 85 90 95 65 55 y 68 65 80 85 70 30 55 18 15 10 35 45
(6 marks)
3 (b) In sampling a large number of parts manufactured by a machine, the mean number of defectives in a sample of 20 is 2. Out of 100 such samples, how many would you expect to contain 3 defectives i) using the Binomial distribution, ii) Poisson distribution.(6 marks) 3 (c) Show that the matrix $\begin{bmatrix} -9 &4 &4 \\\\-8 &3 &4 \\\\-16 &8 &7 \end{bmatrix}$ is diagonalizable. Find the transforming matrix and the diagonal matrix.(8 marks) 4 (a) Fit a Poisson distribution to the following data:
 x 0 1 2 3 4 5 6 7 8 f 56 156 132 92 37 22 4 0 1
(6 marks)
4 (b) Solve the following LPP using Simplex method
Maximize z= 6x1-2x2 + 3x3
Subject to
2x1-x2+2x3 ≤2
x1+4x3 ≤4
x1, x2, x3 ≥ 0.
(6 marks)
4 (c) Expand $f(z) = \dfrac {2}{(z-2)(z-1)}$ in the regions
i) |z| <1, ii) 1<|z|<2, iii) |z|>2.
(8 marks)
5 (a) Evaluate using Cauchy's Residue theorem $\oint_c \dfrac {1-2z}{z(z-1)(z-2)}dz$ where is |z|=1.5.(8 marks) 5 (b) The average of marks scored by 32 boys is 72 with standard deviation 8 while that of 36 girls is 70 with standard 6. Test at 1% level of significance whether the boys perform better than the girls.(6 marks) 5 (c) Solve the following LPP using the Dual Simplex method.
Minimize
z=2x1+2x2+4x3
Subject to
2x1 + 3x2 + 5x3 ≥ 2
3x1+ x2+7x3≤3.
x1+4x2+6x3≤5
x1, x2, x≥0
(8 marks)
6 (a) Solve the following NLPP using Kuhn-Tucker conditions
Maximum $z=10x_1+ 4x_2 - 2x^2_1 - x^2_1$
subjected to 2x1+x2≤5; and x1, x2≥0.
(6 marks)
6 (b) In an experiment on immunization of cattle from Tuberculosis the following results were obtained.
Use X2 Test to determine the efficacy and vaccine in preventing tuberculosis.
 Affected Not Affected Total Inoculated 267 27 294 Not Inoculated 757 155 912 Total 1024 182 1206
(6 marks)
6 (c) (i) The regression lines of a sample are x+6y=6 and 3x+2y=10 find (a) sample means $\overline x$ and $\overline y$ (b) coefficient of correlation between x and y.(4 marks) 6 (c) (ii) If two independent random samples of sizes 15 & have respectively the mean and population standard deviations as $\overline x_1 = 980, \ \overline x_2=1012: \ \sigma_1 = 75, \ \sigma_2 = 80$
Test the hypothesis that μ12 at 5% level of significance.
(4 marks)

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