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Applied Mathematics 4 Question Paper - Dec 17 - Computer Engineering (Semester 4) - Mumbai University (MU)
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Applied Mathematics 4 - Dec 17

Computer Engineering (Semester 4)

Total marks: 80
Total time: 3 Hours
INSTRUCTIONS
(1) Question 1 is compulsory.
(2) Attempt any three from Q.2 to Q.6.
(3) Use of statistical table permitted.

(4) Figures to the right indicate full marks.

Q1

a) Evaluate $\int_{c}^{} log z dz$ , where C is the unit circle in the z-plane.
(5 marks) 00

b) Find the eigen values of the adjoint of A = $ \begin{bmatrix} 2 & 0 & -1 \\ 0 & 2 & 0 \\ -1 & 0 & 2 \end{bmatrix}. $ .
(05 marks) 00

c) If the arithmetic mean of regression coffecient is p and their difference is 2p. find the correlation coeffecient.
(05 marks) 00

d) Write dual of the following L.P.P .
(05 marks) 00

Maximize Z = 2$X_1$ - $X_2$ + 4$X_3$

Subject to $X_1$ + 2$X_2$ - $X_3$ < 5.

2$X_1$ - $X_2$ + $X_3$ < 6

$X_1$ + $X_2$ + 3$X_3$ < 10

4$X_1$ + $X_3$ < 12

$X_1$, $X_2$, $X_3$ > 0.

Q2

a) Evaluate $\int_{c}^{} $/frac{cotz}{z}$ dz$, where C s the ellipse 9$x^2$ + 4$y^2$ = 1.
(06 marks) 00

b) Show that A = $ \begin{bmatrix} 1 & 2 & 3 \\ 2 & 3 & 4 \\ 3 & 4 & 5 \end{bmatrix} $ is non-derogatory.
(06 marks) 00

c) If X is a normal variable with mean 10 and standard deviation 4. find
(06 marks) 00

i) P(|X-14|), ii P(5<X<18), iii) P(X<15)</p>

Q3

a) Find the expectation of number of failures preceeding the first success in an infinite series of independent trials with constant probablities p & q of success and failure respectively.
(06 marks) 00

b) Using simplex Method solve the following L.P.P
(06 marks) 00

Maximize Z - 10$x_1$ + $x_2$ + $x_3$

subejct to $x_1$ + $x_2$ - 3$x_3$ < 10

4$x_1$ + $x_2$ + $x_3$ < 20

$X_1$, $X_2$, $X_3$ > 0.

c) Expand f(z) = $\frac{1}{z(z+1)(z-2)}$
(08 marks) 00

i) Within the unit circle about the origin.

ii) with in the anulus region between the concebtric circles about the origin having radii 1 and 2 respectively.

iii) In the exterior of the circle with centre at the origin and radius 2.

Q4

a) If X is a Binomial distributed with mean = 2 and variance 4.3 . find the probablity distribution of X.
(06 marks) 00

b) Calculate the value of rank correlation coefficient from the following data regrading score of 6 students in physics & chemistry test.
(06 marks) 00

Marks in Physics : 40, 42, 45, 35, 36, 39

Marks in chemistry : 46, 43, 44, 39, 40, 43.

c) Is the matrix A = $ \begin{bmatrix} 2 & 1 & 1 \\ 1 & 2 & 1 \\ 0 & 0 & 1 \end{bmatrix} $ diagonalisable? If so find the diagonal form and the transforming matrix.
(08 marks) 00

Q5

a) A random sample of 50 items gives the mean 6.2 and standard deviation 10.24. can it be regraded as drawn from a normal population with mean 5.4 at 5% level of significance.
(06 marks) 00

b) Evaluate $\int_{0}^{oo} \frac{dx}{(x^2 + a^2)^3} $, a>0. Using Cauchy's residue theorem.
(06 marks) 00

c) Using Kuhn-Tucker condition to solve the following NLPP
(08 marks) 00

Maximize Z = 8$x_1$ + 10$x_2$ - $(x_1)^2$ - $(x_2)^2$

Subject to 3$x_1$ + 2$x_2$ < 6

$x_1$>,$x_2$ > 0

Q6

a) The following table gives the number of accidents in a ity during a week. Find wheter the accidents are uniformly distributed over a week.
(06 marks) 00

Day Sun Mon Tue Wed Thu Fri Sat Total
No of acc 13 15 9 11 12 10 14 84

b) If two independent random samples of sizes 15 & 8 have respectively the following means and population standard deviations,
(06 marks) 00

$X_1$ = 980 $X_2$ = 1012

$a_1$ = 75 $a_2$ = 80

c) Using Penally (Big M) method solve the following L.P.P
(08 marks) 00

Maximize Z = 2$x_1$ + $x_2$

subject to 3$x_1$ + $x_2$

4$x_1$ + 3$x_2$ > 6

$x_1$ + 2$x_2$ < 3

$x_1$,$x_2$ > 0

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