## Applied Mathematics 4 - Dec 2012

### 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)** A sample of 100 students is taken from a large population the mean heights of students in the sample is 160 cm. Can it be reasonably regarded that in the population the mean height is 165 cm. And standard deviation is 10 cm.

Given: Z_{tab} = 1.96 (5% 2 tailed)(5 marks)
**1 (b)** Find all basic solution of following problem

Maximise Z = x_{1}+3x_{2}+3x_{3}

Subject to x_{1}+2x_{2}+3x_{3}=4

2x_{1}+3x_{2}+5x_{3}=7

x_{1},x_{1},x_{1}≥0.

Also find the basic feasible, non-degenerate, infeasible basic, optimal basic feasible solution.(5 marks)
**1 (c)** The first four moments of a distribution about the value 4 are -1.5, 17, -30 and 108. Calculate the moments about the mean.(5 marks)
**1 (d)** If f(1)=2, f(2)=4, f(3)=8, f(4)=16, f(7)=129. Find f(5) using Lagranges interpolation formula.(5 marks)
**2 (a)** A pair of fair dice is rolled once. Let X be the random variable whose value for any outcome is the sum of two numbers on dice.

(i) Find the probability function for X and construct the probability table.

(ii) Find the probability that X is an odd number.

(iii) Find the probability that X lies between 3 and 9.(6 marks)
**2 (b)** Using bisection method, find a positive root of xe^{x}=1 lying between 0 and 1. Solve upto two decimals.(6 marks)
**2 (c)** Fit the second degree parabolic curve to the following data:

x | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 |

y | 2 | 6 | 7 | 8 | 10 | 11 | 11 | 10 |

**3 (a)**Samples of two types of electric bulbs were tested for length of life and the following data were obtained -

No. of Sample | Mean Sample | Standard Deviation | |

Sample 1 | 8 | 1134 hrs | 36 hrs |

Sample 2 | 7 | 1024 hrs | 40 hrs |

Test at 5% level of significance whether the difference in the sample means is significant.

(Table value of t for 13 d.f. is 2.16, for 14 d.f. is 2.15 and for 15 d.f. is 2.13)(6 marks)

**3 (b)**Find mean and variance of Binomial Distribution.(6 marks)

**3 (c)**(i) Using forward difference formula find y when x = 0.5 from the following data:-

x | 0 | 1 | 2 | 3 |

y | -1 | 1 | 1 | -2 |

(ii) Show that

(8 marks)

**4 (a)**Apply Guass-Seidel iteration method to solve the equations

20x + y - 2z = 17;

3x + 20y - z = -18

2x - 3y + 20z = 25(6 marks)

**4 (b)**A skilled typist on routine work kept a record of mistakes made per day during 300 working days. If she made 1 mistake on 143 days, 2 mistakes on 110 days. Find the number of days on which she made 3 mistakes using Poisson distribution. (6 marks)

**4 (c)**The following table shows the height of a sample of 12 fathers and their sons. Find rank correlation coefficients.

x | 65 | 63 | 67 | 64 | 68 | 62 | 70 | 66 | 68 | 67 | 69 | 71 |

y | 68 | 66 | 68 | 65 | 69 | 66 | 68 | 65 | 71 | 67 | 68 | 80 |

**5 (a)**The PDF of random variable X is given by

f(x) = kx

^{2}(2 - x) for 0≤x≤2

f(x)=0 otherwise

Find: (i) k (ii) Mean (iii) Variance(6 marks)

**5 (b)**Fitting of binomial distribution for the following data and test the goodness of fit.

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

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

**5 (c)**Evaluate

by using

(i) Trapezoidal Rule.

(ii) Simpons 1/3

^{rd}rule.

(iii) Simpons 3/8

^{th}rule.

Take h = 0.25. Compare the results with exact value.(8 marks)

**6 (a)**Find the real root of x

^{3 }(6 marks)

**6 (b)**A factory turns out at an article by mass production method from past experience it appears that 20 articles on an average are rejected out of every batch of 100. Find variance of the number of rejected articles. What is the probability that the number of rejects in a batch exceeded 30?

Given: Area (z = 0 to z = 2.5) = 0.4938)(6 marks)

**6 (c)**The following marks have been obtained by a class of students in stats (out of 100).

Paper I | 45 | 55 | 56 | 58 | 60 | 65 | 68 | 70 | 75 | 80 | 85 |

Paper II | 56 | 50 | 48 | 60 | 62 | 64 | 65 | 70 | 74 | 82 | 90 |

Compute the coefficient of correlation for the above data. Find also the equations of lines of regression.(8 marks)

**7 (a)**For a Poisson distribution, P(x=2) = 9P(x=4) + 90P(x=6), then find mean and variance of distribution.(6 marks)

**7 (b)**The following data is collected on two characters. Based on this, can you say that there is no relation between smoking and literacy:-

Smokers | Non-Smokers | |

Literates | 83 | 57 |

Illiterates | 45 | 68 |

_{fab}

^{2}=5.991 (ν=2, 5% L.O.S)(6 marks)

**7 (c)**Using Simplex Method, solve the following L.P.P.

Maximize z = 3000x

_{1}+ 2500x

_{2}

Subject to, 2x

_{1}+ x

_{2}≤ 40

x

_{1}+ 3x

_{2}≤ 45

x

_{1}≤ 12

x

_{1}, x

_{2}≥ 0.(8 marks)