## Applied Mathematics 4 - May 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 Eigen values of A^{2} + 2I, where $ A=\begin{bmatrix}
1 & 0 & 0\\\\
2 & -2 & 0\\\\
3 & 5 & 3
\end{bmatrix} $ and I is the Identity matrix of order 3.(5 marks)
**1(b)** Evaluate the line integral $ \int ^{1+l}_0(x^2+iy)dz $ along the path y = x.(5 marks)
**1(c)** If x is a continuous random variable with the probability density function given by $$\left\{\begin{matrix}
k(x-x^3) & 0\leq x\leq 1\\
0 & \text{otherwise}
\end{matrix}\right.$$

Find i) k ii) the mean of the distribution.(5 marks)
**1(d)** Compute Spearman's rank correlation coefficient from the following data

X |
18 | 20 | 34 | 52 | 12 |

Y |
39 | 23 | 35 | 18 | 46 |

**2(a)**Is the following matrix Derogatory? Justify. $$\begin{bmatrix} 5 & -6 & -6\\ -1 & 4 & 2\\ 3 & -6 & -4 \end{bmatrix}$$(6 marks)

**2(b)**Evaluate $ \oint _c \dfrac{e^{2z}}{(z-1)^4}dz $ where c is the circle |z| = 2(6 marks)

**2(c)**The marks of 1000 students in an Examination are found to be normally distributed with mean 70 and standard deviation 5, estimate the number of students whose marks will be i) between 60 and 75 ii) more than 75.(8 marks)

**3(a)**Solve the following non-linear programming problem using kuhn-tucker conditions

Maximum z = 10x

_{1}+4x

_{2}-2x

^{2}

_{1}-x

^{2}

_{2}

Subject to 2x

_{1}+x

_{2}≤5 ; and x

_{1}, x

_{2}≥0(6 marks)

**3(b)**Fit a binomial distribution to the following data

X |
0 | 1 | 2 | 3 | 4 | 5 | 6 |

F |
5 | 18 | 28 | 12 | 7 | 6 | 4 |

**3(c)**Is the following matrix diagonalizable? If yes, find the transforming matrix abd the diagonal matrix. $$\begin{bmatrix} 8 & -8 & -2\\ 4 & -3 & -2\\ 3 & -4 & 1 \end{bmatrix}$$(8 marks)

**4(a)**Solve the following LPP using Simplex method.

Maximize z = 4x

_{1}+x

_{2}+3x

_{3}+5x

_{4}

Subject to

-4x

_{1}+6x

_{2}+5x

_{3}+4x

_{4}≤20

-3x

_{1}-2x

_{2}+4x

_{3}+x

_{4}≤10

-8x

_{1}-3x

_{2}+3x

_{3}+2x

_{4}≤ 20

x

_{1}, x

_{2}, x

_{3}, x

_{4}≥0(6 marks)

**4(b)**If a random variable X follows the Poisson distribution such that P(x = 1) = 2P(X=2), find the mean, the variance of the distribution and p(X=3)(6 marks)

**4(c)**Expand $ f(e)=\dfrac{1}{z(z-2)(z+1)} $ in the regions

i) |z| < 1, ii) 1 < |z| < 2, iii) |z| > 2(8 marks)

**5(a)**Evaluate using Cauchy's Residue theorem $ \oint _c \dfrac{2z-1}{z(2z+1)(z+2)}dz $ where c is |z| = 1.(6 marks)

**5(b)**A certain administered to each of 12 patients resulted in the following change in blood pressure:

5, 2, 8, -1, 3, 0, -2, 1, 5, 0, 4, 6

Can it be concluded that the stimulus will increas the blood pressure (at 5% level of significance)?(6 marks)

**5(c)**Solve the following LPP using the Dual Simplex Method

Maximise z = -3x

_{1}- 2x

_{2}

Subject to

x

_{1}+ x

_{2}≥ 1

x

_{1}+ x

_{2}≤ 7

x

_{1}+ 2x

_{2}≥ 10

x

_{2}≤ 3

x

_{1}, x

_{2}≥ 0(6 marks)

**6(a)**Find the equations of lines of regression for the following data

x |
5 | 6 | 7 | 8 | 9 | 10 | 11 |

y |
11 | 14 | 14 | 15 | 12 | 17 | 16 |

**6(b)**Evaluate $ \int ^{\infty}_{-\infty}\dfrac{x^2}{(x^2+1)(x^2+4)}dx $ using contour integration.(6 marks)

**6(c)**In an experiments on pea breeding, the following frequencies of seeds were obtained

Round and Yellow |
Wrinkled and yellow |
Round and green |
Wrinkled and green |
Total |

315 | 101 | 108 | 32 | 556 |

theory predicts that the frequencies should be in proportions 9:3:3:1.

Examine the correspondence between theory and experiment using Chi-square Test(8 marks)