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Applied Mathematics 4 Question Paper - May 17 - Computer Engineering (Semester 4) - Mumbai University (MU)
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Applied Mathematics 4 - May 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 questions.
(3) Use Statistical table permitted.
Q1
a)
Evaluate $\int_{C}^{} (z - z^2) dx$, where C is the upper half of the circle |z| = 1
(5 marks)
00
b)
If A = $ \begin{bmatrix}
2 & 4 \\
0 & 3
\end{bmatrix} $, then find the eigen values of 6$A_{-1}$ + $A_3$ + 2I
(6 marks)
00
c)
State whether the following statements are true or false with reasoning: "The Regression coefficients between 2x and 2y are the same as those bet x and y"
(6 marks)
00
d)
Construct the dual of the following L.P.P
(6 marks)
00
Maximize Z = 3$x_1$ + 17$x_2$ + 9$x_3$
Subject to $x_1$ - $x_2$ + $x_3$ > 3
-3$x_1$ + 2$x_2$ < 1
2$x_1$ + $x_2$ - 5$x_3$ =1
$x_1$,$x_2$,$x_3$ >0
Q2
a)
Evaluate $\int_{C}^{} \frac{e^{2z}}{(z+1)^4} dz$, where C is the circle |z-1| = 3
(6 marks)
00
b)
Show that the matrix A = $ \begin{bmatrix}
5 & -6 & -6 \\
-1 & 4 & 2 \\
3 & -6 & -4
\end{bmatrix} $ is derogatory
(6 marks)
00
c)
A manufacturer knows from his expereince that the resistors he produces is normal with U=100 ohms and standard deviation a=2 ohms. What percentage of resistors will have resistance bet 98ohms and 102 ohms?
(8 marks)
00
Q3
a)
A discrete random variable has the probablity distribution given below:
(6 marks)
00
| x | -2 | -1 | 0 | 1 | 2 | 3 |
|---|---|---|---|---|---|---|
| p(x) | 0.2 | k | 0.1 | 2k | 0.1 | 2k |
Find k, the mean and variance.
b)
Solve the following L.P.P by simplex method
(6 marks)
00
Maximise Z=3$x_1$ +2$x_2$
Subject to $x_1$ + $x_2$ <4
$x_1$ - $x_2$ <2
$x_1$,$x_2$ >0
c)
Expand f(z) = $\frac{z^2 - 1}{z^2 + 5z + 6}$ around z=0, indicating region of convergence
(8 marks)
00
Q4
a)
Find the first two moments about the origin of Possion distribution and hence find mean and variance
(6 marks)
00
b)
Calculate R and r from the following data:
(6 marks)
00
| x | 12 | 17 | 22 | 27 | 32 |
|---|---|---|---|---|---|
| y | 113 | 119 | 117 | 115 | 121 |
c)
Show that the matrix A = $ \begin{bmatrix}
8 & -8 & -2 \\
4 & -3 & -2 \\
3 & -4 & 1
\end{bmatrix} $ is diagonaliasable
(8 marks)
00
Find the transforming matrix and the diagonal matrix.
Q5
a)
A tyre company claims that the lives of tyres has mean 42,000 kms with S.D of 4000kms. A change in the production process is believed to result in better product. A test sample of 81 new tyers has a mean life of 42,500kms. test at 5% level of significance that the new product is significantly better than the old one
(6 marks)
00
b)
Evaluate $\int_{0}^{2\pi} \frac{d0}{5+3sin0}$ using Cauchy's residue theorem
(6 marks)
00
c)
Using kuhn - Tucker conditions solve the following N.L.P.P
(8 marks)
00
Minimize Z= 7$(x_1)^2$ + 5$(x_2)^2$ - 6$x_1$
Subject to $x_1$ + 2$x_2$ <10
$x_1$ + 3$x_2$ <9
$x_1$,$x_2$>0
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