Question Paper: Applied Mathematics 4 : Question Paper Dec 2016 - Information Technology (Semester 4) | Mumbai University (MU)
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## Applied Mathematics 4 - Dec 2016

### Information Technology (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) Find the Eigenvalues and eigenvectors of the matrix.
A=$\begin{bmatrix} 2 & 2& 0\\\\ 0 & 2& 1\\\\ 0& 0& 2 \end{bmatrix}$/
(5 marks)
1(b) Evaluate the line integral $$\int_{0}^{l+i}\left ( x^2+iy \right )$$ dz along the path y=x(5 marks) 1(c) Find k and then E (x) for the p.d.f.
$f(x)=\left\{\begin{matrix} k(x-x^2),0\leq x\leq 1,k> 0& \\\\ 0, & otherwise \end{matrix}\right.$/
(5 marks)
1(d) Calculate Karl person's coefficient of correlation from the following data.

 x 100 200 300 400 500 y 30 40 50 60 70
(5 marks) 2(a) Show that the matrix $A=\begin{bmatrix} 2 & -2& 3\\\\ 1& 1& 1\\\\ 1& 3& -1 \end{bmatrix}$/ is non-derogatory.(6 marks) 2(b) Evaluate $$\int \frac{e^2^z}{\left ( z+1 \right )^4}$$ dz where C is the circle |z-1|=3(6 marks) 2(c) If x is a normal variate with mean 10 and standard deviation 4 find
i) P(|x-14|<1)
ii) P(5≤x≤18)
iii) P(x≤12)
(8 marks)
3(a) Find the relative maximum of minimum (if any) of the $$Z=X_{1}^{2}+X_{2}^{2}+X_{3}^{2}-4X_1-8X_2-12X_3+100$$(6 marks) 3(b) If x is Binomial distributed with E(x)=2 and V(x)=4/3,find the probability distribution of x.(6 marks) 3(c) If $A=\begin{bmatrix} 2& 1\\\\ 1 & 2 \end{bmatrix}$/,
find A50.
(8 marks)
4(a) Solve the following L.P.P by simplex method Minimize
z=3x1+2x2 Subject to 3x1+2x2≤18
0≤x1≤4
0≤x2≤6
x1,x2≥0.
(6 marks)
4(b) The average of marks scored by 32 boys is 72 with statndard deviation 8 while that of 36 girls is 70 with standard deviation 6. Test 1% level significance whether the boys perform better than the girls.(6 marks) 4(c) Find Laurent's series which represents the function
$$f(z)=\frac{2}{\left ( Z-1 \right )\left ( z-2 \right )}$$ When
i) |z| <1,
ii) 1<|z|<2
iii) |z|>2
(8 marks)
5(a) Evaluate $$\int \frac{Z^2}c_{\left ( z-1 \right )^2\left (z+1 \right )}$$ dz where C is|z| =2 using residue theorem(6 marks) 5(b) The regression lines of a sample are x+6y=6 and 3x+2y=10 Find
i) Sample means
$$\bar{x} \ \text{and}\ \bar{y}$$
ii) Correlation coefficient between x ad y. Also estimate y When x=12
(6 marks)
5(c) A die was thrown 132 times and the following frequencies were observed
 No.obtained 1 2 3 4 5 6 Total Frequency 15 20 25 15 29 28 132
Using χ2-test examine the hypothesis that the die is unbiased.
(8 marks)
6(a) Evaluate $$\int ^\infty _\\-\infty\frac{x^2+x+2}{x^4+10x^2+9}$$ dx using contour integration.(6 marks) 6(b) If a random variable x follows Poisson distribution such that P(x-1)=2(x=2) Find the mean the variance of the distribution Also find P(x=3).(6 marks) 6(c) Use Penalty method to solve the following L.P.P. Minimize
z=2x,sub>1+3x2
x1+x2≥5
x1+2x2≥6 x1, x2≥0.
(8 marks)