Question Paper: Applied Mathematics 4 : Question Paper Dec 2016 - Information Technology (Semester 4) | Mumbai University (MU)
0
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 |
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 |
z=2x,sub>1+3x2
x1+x2≥5
x1+2x2≥6 x1, x2≥0.(8 marks)
