1 (a)
The length of time (in minutes) a lady speaks on the telephone is found to be a random variable with probability density function:

Find 'A' and the probability:

(i) that she speaks more than 1 minutes

(ii) that she speaks less than 5 minutes

Find 'A' and the probability:

(i) that she speaks more than 1 minutes

(ii) that she speaks less than 5 minutes

5 M

1 (b)
If f(x)=2x

i) Test whether the inverse function exists for both f and g.

ii) Find fog."

^{2}+ 3; g(x)=4x + 3 where f: R → R and g: R → Ri) Test whether the inverse function exists for both f and g.

ii) Find fog."

5 M

1 (c)
Prove that the eigen values of orthogonal matrix is ±1.

5 M

1 (d)
Find Taylor's series expansion of f(z) = 1/(z)(z-1) about z=2. Find raidus of convergence.

5 M

2 (a)
Fit a Binomial Distribution to the following data and test the goodness of it:

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

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

7 M

2 (b)
Find the rank and signature of real quadratic form:

by expressing in conical form.

by expressing in conical form.

7 M

2 (c)
Determine the nature of poles of the following function.Also find the residue at each pole.

6 M

3 (a)
Verify Cayley Hamilton Theorem and find A

Hence find A

^{-1}forHence find A

^{5}-4A^{4}-7A^{3}+11A^{2}-A-101 in term of 'A'
7 M

3 (b)
Let L = {1, 2, 3, 5, 30} and the relation 'is divisible by'.Test whether:

(i) L is a distributive lattice

(ii) L is a complemented lattice

(i) L is a distributive lattice

(ii) L is a complemented lattice

7 M

3 (c)
Two independent random samples of size '8' and '10' have means 950 and 1000. The standard deviation of two populations are 80 and 100. Test the hypothesis that the two population have same mean.

6 M

4 (a)
The height of 1000 soldiers in a regiment are distributed normally with mean 172 cms and standard deviation = 5 cm. How many soldiers have height > 180 cms.

7 M

4 (b)
Prove that (R, t,

^{.}) where R = {0, 1, 2, 3, 4}, is a ring under 'addition modulo 5' and 'mutiplication modulo 5'. Is it integral domain. Give reasons.
7 M

4 (c)
Determine A

^{10}and A^{4}if
6 M

5 (a)
Intelligence tests of two groups of boys and girls obtained from two normal population having the same standard deviation gave the following data:

Is the difference between means significant at 5% LOS.

Mean | Std. Deviation | No. | |

Girls | 84 | 10 | 121 |

Boys | 81 | 12 | 81 |

7 M

5 (b)
Evaluate ∮

(i) is the circle |z|=2

(ii) is the circle |z|=1

_{c}tan z dz where C is:(i) is the circle |z|=2

(ii) is the circle |z|=1

7 M

5 (c)
Let R be the relation defined on Z by xRy iff |x - y| is divisible by 7.

i) Show R in an equivalence relation.

ii) Find the equivalence classes.

i) Show R in an equivalence relation.

ii) Find the equivalence classes.

6 M

6 (a)
Which probablity distribution in appropriate to describe the situation where 100 misprints are randomly distributed over 100 pages of a book. For this distribution, find the prob that a page selected at random will contain at least 3 misprints.

7 M

6 (b)
Is the following matrix diagonalizable?

Justify your answer.

Justify your answer.

7 M

6 (c)
A box contains 2 red and 3 black balls. Three balls are drawn at random. Let 'X' denote total number of red balls drawn from this box. Find:

i) The M. G. F. of X

ii) Hence find E(X) and Var(X)

i) The M. G. F. of X

ii) Hence find E(X) and Var(X)

6 M

7 (a)
State Central Limit Theorem. Use the theorem to estimate P(180<S

_{n}<250) where S_{n}= ∑_{ }^{n}_{i=1 }X_{i }and n = 70
7 M

7 (b)
Expand f(z)=1/z(z+1)(z+2) in Laurent's series where

(i) |z| < 1 (ii) 1 < |z| < 2 (iii) |z| > 2

(i) |z| < 1 (ii) 1 < |z| < 2 (iii) |z| > 2

7 M

7 (c)
For a normal distribution 5% of students get below 60 marks and 40% students obtain between 60 and 65 marks. Find the mean and variance of Normal distribution.

6 M

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