1 (a)
Solve the partial differential equation

by the method of separation of variables.

by the method of separation of variables.

5 M

1 (b)
Obtain the complex form of Fourier series f(x) = cosh2x + sinh 2x in the interval (-2,2)

5 M

1 (c)
The distribution function of a random variable X is given by F(x)=1-(1+x)e

^{-x}, x ≥ 0. Find the density function, mean and variance of X.
5 M

1 (d)
A normal population has mean 0.1 and S.D. 2.1. Find the probability that the mean of the sample of size 900 will be negative.

5 M

2 (a)
For a binomial distribution mean is 6 and S.D. √2. Find the first two terms of the distribution.

6 M

2 (b)
Find the half-range fourier sine series for f(x)=x

^{2}in the interval 0 ≤ x ≤ 3.
6 M

2 (c)
A rod of length 30 cm has its ends A and B kept at 20°C and 80°C respectively until steady state conditions prevail. The temperature at each end is then suddenly reduced to 0°C and kept so. Find the resulting temperature function u(x,t) taking x=0 at A.

8 M

3 (a)
If X and Y are independent random variables following N(8,2) and N(12,4√3) respectively such that P(2X-Y ≤ 2λ) = P(X+2Y ≥ λ)

6 M

3 (b)
Obtain fourier series for f(x)=e

^{-x}in (-π, π). Hence derive the series for π/(sinh π)
6 M

3 (c)
Find the coefficient of correlation and obtain the lines of regression for the data.

x: | 62 | 64 | 65 | 69 | 70 | 71 | 72 | 74 |

y: | 126 | 125 | 139 | 145 | 165 | 152 | 180 | 208 |

8 M

4 (a)
The marks obtained by a number of students in a certain subject are approximately normally distributed with mean 65 and SD 5. If 3 students are selected at random from this group, what is the probability that at least one of them would have scored more than 75%?

6 M

4 (b)
Find Fourier series for f(x) in (0, 2π).

f(x)=x, 0 < x < π

f(x)=2π-x, π < x < 2π

f(x)=x, 0 < x < π

f(x)=2π-x, π < x < 2π

6 M

4 (c)
The bounding diameter of semi circular plate of radius 10 cm is kept at 0°C and the temperature along the boundary is given by

u(10, θ) = 50θ , 0 < θ < π/2

u(10, θ) = 50(π-θ) , π/2 < θ < π

u(10, θ) = 50θ , 0 < θ < π/2

u(10, θ) = 50(π-θ) , π/2 < θ < π

8 M

5 (a)
The mean height and S.D. height of 8 randomly chosen soldiers are 166.9 and 8.29 cms respectively. The corresponding values of 6 randomly chosen sailors are 170.3 and 8.5 cms respectively. Based on this data, can we conclude that soldiers are shorter than sailors?

6 M

5 (b)
Find the rank correlation for the indices of supply and price of an article.

Supply Index | 124 | 100 | 105 | 112 | 102 | 93 | 99 | 115 | 123 | 104 | 99 | 113 | 121 | 103 | 101 |

Price Index | 80 | 100 | 102 | 91 | 100 | 111 | 109 | 100 | 89 | 104 | 111 | 102 | 98 | 111 | 123 |

6 M

5 (c)
If the independent random variables X and Y have the variances 36 and 16 respectively, find the correlation between (X+Y) and (X-Y).

8 M

6 (a)
Fit a poisson distribution for the following data:

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

f : | 142 | 156 | 69 | 27 | 5 | 1 |

6 M

6 (b)
Fit a first degree curve to the following data and estimate the value of y when x = 73.

x : | 10 | 20 | 30 | 40 | 50 | 60 | 70 | 80 |

y : | 1 | 3 | 5 | 10 | 6 | 4 | 2 | 1 |

6 M

6 (c)
Express e

^{-x}cos x as fourier cosine integral and show that:
8 M

7 (a)
Show that the polynomials P

_{0}(x) = 1, P_{1}(x) = x, P_{2}(x) = 0.5(3x^{2}-1) from an orthogonal set over the interval [-1,1]. Hence find the corresponding orthonormal set.
6 M

7 (b)
A total number of 3759 individuals were interviewed in a public opinion survey on a political survey. Of them 1872 were men and rest women. A total 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 or contradict the hypothesis that there is no association between sex and attitude.

6 M

7 (c)
A tightly stretched string with fixed end points x=0 and x=l in the shape defined by y = kx(l-x) where k is a constant is released from this position of rest. Find y(x,t), the vertical displacement if:

8 M

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