## 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 |

**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 A

^{50}.(8 marks)

**4(a)**Solve the following L.P.P by simplex method Minimize

z=3x

_{1}+2x

_{2}Subject to 3x

_{1}+2x

_{2}≤18

0≤x

_{1}≤4

0≤x

_{2}≤6

x

_{1},x

_{2}≥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 |

^{2}-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+3x

_{2}

x

_{1}+x

_{2}≥5

x

_{1}+2x

_{2}≥6 x

_{1}, x

_{2}≥0.(8 marks)