## Applied Mathematics 4 - May 2014

### 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)**

If A =$\begin{bmatrix} x & 4x\\\\ 2 & y \end{bmatrix}\$ $has eigen values 5 and -1 then find values of x and y.

(5 marks)**1(b)**

Evaluate $\int_{c}(\bar{Z}+2z)dz$ along the circle c: $ x^{2}+y^{2}=1$

(5 marks)**1(c)**State true or false with justification: If the two lines of regression are x+3y-5y =0 and 4x+3y-8 =0 then the correlation coefficient is +0.5(5 marks)

**1(d)**

Find dual of following LP model

max z= 2x_{1} +3x_{2}+5x_{3}

Subjected to

x_{1}+x_{2}-x_{3}$\ge$ -5

x_{1}+x_{2}+4x_{3}=10

-6x_{1}+7x_{2}-9x_{3}$\le$4

x_{1},x_{2}$\ge$0 and x_{3} is unrestricted.

**2(a)**

Using Cauchy's integral formula,evaluate $\int_{c} \dfrac{(12z-7)dz}{(z-1)^{2}(2z+3)}$where C: |z+i|=$\sqrt{3}\\\\$\lt/span\gt\lt/p\gt \lt/span\gt\ltspan class='paper-ques-marks'\gt(6 marks)\lt/span\gt \lt/span\gt\ltspan class='paper-question'\gt\ltspan class='paper-ques-desc'\gt\ltb\gt2(b)\lt/b\gt \ltp\gtDetermine whether matrix a is derogatory A =\ltspan class="math-tex"\gt$\begin{bmatrix} 2 &1 &0 \\ 0&2 &1 \\ 0&0 &2 \end{bmatrix}\\$

(6 marks)**2(c)**In a competitive examination,the top 15% of the students appeared will get grade 'A' while the bottom 20% will be declared fail.If the grades are normally distributed with mean % of marks 75 and S.D 10,determine the lowest % of marks to receive grade A and the lowest % of marks that passes.(8 marks)

**3(a)**

The daily consumption of electric power (in millions of kwh)is r.v with PDF f(x) =k xe^{-x/30} ,x>0.find k and the probability that on a given day the electricity consumption is more than expected electricity consumption.

**3(b)**

Using Simplex method,solve the following LPP

max z = 15x_{1}+6x_{2}+ 9x_{3}+2x_{4}

s.t2x_{1}+x_{2}+5x_{3}+6x_{4} $\le$ 20

3x_{1} +x_{2}+3x_{3}+25x_{4}$\le$24

7x_{1}+x_{4}$\le$70

& x_{1},x_{2},x_{3},x_{4}$\ge$ 0

**3(c)**

Obtain ALL Taylor's and Laurent series expansion of function $\dfrac{(z-1)(z+2)}{(z+1)(z+4)}about z=0$

(8 marks)**4(a)**Find the moment generating function of Poission distributed and hence find mean and variance.(6 marks)

**4(b)**Obtain the equation of the line of regression of cost on age from the following table giving the age of a car of certain make and the annual maintenance cost.Also find maintenance cost if age of the car is 9years.

Age of car(in years):x | 2 | 4 | 6 | 8 |

maintenance cost :y (in thousands) | 5 | 7 | 8.5 | 11 |

**4(c)**

Show that the matrix A is diagonalizable,Find its diagonal form and transforming matrix,if A =$\begin{bmatrix} -9 &4 &4 \\\\ -8& 3 &4 \\\\ -16&8 &7 \end{bmatrix}\\\\$\lt/span\gt\lt/p\gt \lt/span\gt\ltspan class='paper-ques-marks'\gt(8 marks)\lt/span\gt \lt/span\gt\ltspan class='paper-question'\gt\ltspan class='paper-ques-desc'\gt\ltb\gt5(a)\lt/b\gt a sample of 8student of 16years each shown up a mean systolic blood pressure of 118.4 mm of Hg with S.D of 12.17 mm.While a sample of 10student of 17years each showed the mean systolic BP of 121.0mm with S.D of 12.88 mm during in investigation.The investigator feels that the systolic BP is related to age.Do you think that the data provides enough reasons to support investigator's feeling at 5% Los? Assume the distribution of systolic BP to be normal.\lt/span\gt\ltspan class='paper-ques-marks'\gt(6 marks)\lt/span\gt \lt/span\gt\ltspan class='paper-question'\gt\ltspan class='paper-ques-desc'\gt\ltb\gt5(b)\lt/b\gt \ltp\gtUsing Cauchy's residue theorem,show that \ltspan class="math-tex"\gt$\int_{0}^{2\pi}\dfrac{cos 2\theta}{5+4cos\theta}d\theta=\dfrac{\pi}{6}\\$

(6 marks)**5(c)**

Using Dual simplex method,solve

max z = -2x_{1} -x_{3}

s.t x_{1} +x_{2} -x_{3} $\ge
$5

x_{1} -2x_{2} +4x_{3} $\ge$8

& x_{1} ,x_{2} ,x_{3} $\ge$ 0

**6(a)**A total of 3759 individuals were interviewed in a public opinion survey on a political proposal.of them,1872 were men and the rest women.A total of 2257 individuals were in favour of the proposal and 917 were opposed to it. A total of 243 men were undecided and 442 women were opposed to the proposal.Do you justify on the hypothesis that there is no association between sex and attitude,at 5% LoS.(6 marks)

**6(b)**Using Kuhn-Tuker's method solve

Maximize Z=2x

_{1 }

^{2}

_{ }

^{ }+12x

_{1}x

_{2}-7x

_{1}

^{2}

subject to the constraints 2x

_{1}+5x

_{2}? 98 and x

_{1}, x

_{2}7 ? 0(6 marks)

**6(c) (i)**Average mark scored by 32boys is 72 with standard deviation of 8while that for 36girl is 70 with standard deviation of 6.Test at 1% LoS whether the boys perform better than the girls.(4 marks)

**6(c) (ii)**If the first four moments of a distribution about the value 4 of the random variable are-1.5,17,-30 and 108 then find first four raw moments.(4 marks)