## Applied Mathematics 4 - May 2013

### 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 fair coin is tossed till head appears. What is the expectation of the number of tosses required?(5 marks)
**1 (b)** Solve using Bisection method

x - cosx = 0. Find the positive root.(5 marks)
**1 (c)** Solve graphically the following L.P.P.

Maximize, z = x - 2y

Subject to, -x + y ≤ 1

6x + 4y ≥ 24

0 ≤ x ≤ 5, 2 ≤ y ≤ 4(5 marks)
**1 (d)** The mean value of random sample of 60 items was found to be 145 with standard deviation of 40. Find the 95% confidence limits for the population mean.(5 marks)
**2 (a)** If p.d.f. of a random variable is given by,

f(x) = x for 0 ≤ x ≤ 1

f(x)=2 - x for 1 ≤ x ≤ 2

f(x)=0 Otherwise

Find the m.g.f. and hence find mean and variance.(6 marks)
**2 (b)** If x_{1} and x_{2} are independent normal variates with mean 30 and 25 and variance 16 and 12 respectively and y = 3x_{1} -x_{2}. Find P(60 ≤ y ≤ 80)(6 marks)
**2 (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)

**3 (a)**Test significance of difference between the means of samples drawn from two normal populations with the following data:

Size | Mean | s.d. | |

Sample I | 100 | 61 | 4 |

Sample II | 200 | 63 | 6 |

**3 (b)**If x = au + b, y = cv + d; a, b, c, d are constants then prove r

_{xy}=r

_{uv}

where r

_{xy}is coefficient of correlation between x and y.(6 marks)

**3 (c)**Fit a second degree curve for the following data:

x | 1 | 2 | 3 | 4 | 5 |

y | 1250 | 1400 | 1650 | 1950 | 2300 |

**4 (a)**Using Gauss-Seidel method, solve the equations

10x + y + z = 12;

2x + 10y + z = 13

2x + 2y + 10z = 14(6 marks)

**4 (b)**According to theory of proportion of commodity in the four classes A, B, C, D should be 9 : 2 : 4 : 1. In a survey of 1600 items of this commodity the numbers in four classes were 882, 432, 168 and 118. Does this survey support the theory?(6 marks)

**4 (c)**Find the coefficient of correlation for the following data:

x | 2 | 4 | 5 | 6 | 8 | 11 |

y | 18 | 12 | 10 | 8 | 7 | 5 |

**5 (a)**Let X be a random variable with the following p.d.f.

x | -3 | 6 | 9 |

P(x) | 1/6 | 1/2 | 1/3 |

Find the mean, variance and E(2X+1)

^{2}.(6 marks)

**5 (b)**Explain:-

(i) Null Hypothesis.

(ii) Alternate Hypothesis.

(iii) Critical Region.

(iv) Level of Significance.

(v) Types of errors.

(vi) One Tailed Two Tailed Tests.(6 marks)

**5 (c)**Find f(8) from the data -

x | 5 | 7 | 11 | 13 | 17 |

f (x) | 150 | 392 | 1452 | 2366 | 5202 |

**6 (a)**Solve using Gauss-Jordan method

2x + y + 4z = 16;

3x + 2y + z = 10

1x + 3y + 3z = 16.(6 marks)

**6 (b)**How many tosses of a coin are needed so that the probability of getting at least one head is 87.5%?(6 marks)

**6 (c)**Solve:

Maximize: z = 4x

_{1}+x

_{2}+ 3x

_{3}+ 5x

_{4};

Subject to, 4x

_{1}- 6x

_{2}- 5x

_{3}- x

_{4}≤ 2

-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.(8 marks)

**7 (a)**Find mean and variance of Binomial Distribution.(6 marks)

**7 (b)**Two batches of 12 animals are taken for test of inoculation. One batch was inoculated and other was not from the data can it be regarded as effective against the disease?

Dead | Survived | Total | |

Inoculated | 2 | 10 | 12 |

Non-Inoculated | 8 | 4 | 12 |

Total | 10 | 14 | 24 |

**7 (c)**Show that R=r for the following data:

x | 60 | 62 | 64 | 66 | 68 | 70 | 72 | 74 |

y | 92 | 83 | 101 | 110 | 128 | 119 | 137 | 146 |