1 (a)
"A random variable X has the probability function:

Find: (i) k (ii) P(x ≤ 1) (iii) P(-2 < x < 1) (iv) Obtain the distribution function of X
"

X: | -2 | -1 | 0 | 1 | 2 | 3 |

P (X = x): | 0.1 | k | 0.2 | 2k | 0.3 | 3k |

5 M

1 (b)
In the set of natural numbers, prove that the relation xRy if and only if x

^{2}- 4xy + 3y^{2}=0, is reflexive, but neither symmetric nor transitive.
5 M

1 (c)
Find the characteristic roots of A

^{30}-9A^{28}where
5 M

1 (d)
Find Laurent's series about z = -2 for:

5 M

2 (a)
If X, Y are independent Poisson variates such that P(X=1) = P(X=2) and P(Y=2) = P(Y=3) find the variance of 2X - 3Y.

7 M

2 (b)
Find the Residues of

at its poles.

7 M

2 (c)
If

find cosA.

find cosA.

6 M

3 (a)
Check whether A = {2, 4, 12, 16} and B = {3, 4, 12, 24} are lattices under divisibility? Draw their Hasse diagrams.

7 M

3 (b)
Nine items of a sample had the following values.

45, 47, 50, 52, 48, 47, 49, 53, 51.

Does the mean of 9 items differ significantly from the assumed population mean 47.5 ?

45, 47, 50, 52, 48, 47, 49, 53, 51.

Does the mean of 9 items differ significantly from the assumed population mean 47.5 ?

7 M

3 (c)
Find characteristic equation of the matrix A and hence find matrix represented by A

^{8}-5A^{7}+7A^{6}-3A^{5}+A^{4}-5A^{3}+8A^{2}-2A^{1}+I where:
6 M

4 (a)
The average of marks scored by 32 boys is 72 with standard deviation 8 while that of 36 girls is 70 with standard deviaiton 6. Test at 1% level of significance whether the boys perform better than the girls.

7 M

4 (b)
Let

and + and · be the matrix addition and matrix multiplication. Is ( S, +, · ) an integral domain? Is it a field?

and + and · be the matrix addition and matrix multiplication. Is ( S, +, · ) an integral domain? Is it a field?

7 M

4 (c)
Show that ∫

_{c}dz/(z+1) = 2πI, where C is the circle |z| = 2. Hence deduce that:
6 M

5 (a)
The number of defects in printed circuit board is hypothesised to follow Poisson distribution. A random sample of 60 printed boards showed the following data.

Number of Defects: | 0 | 1 | 2 | 3 |

Observed Frequency: | 32 | 15 | 9 | 4 |

7 M

5 (b)
"If f and g are defined as

f: R → R, f(x) = 2x - 3 g: R → R, g(x) = 4 - 3x

i) Verify that (fog)

ii) Solve fog(x) = g of(1)"

f: R → R, f(x) = 2x - 3 g: R → R, g(x) = 4 - 3x

i) Verify that (fog)

^{-1}= g^{-1}of^{-1}ii) Solve fog(x) = g of(1)"

7 M

5 (c)
For a distribution the mean is 10, variance is 16, γ

_{1}is 1 and β_{2}is 4. Find the first four moments about the origin. Comment on the nature of this distribution.
6 M

6 (a)
Prove that the set A={0, 1, 2, 3, 4, 5} is a finite abelian group under addition modulo 6.

7 M

6 (b)
If

where C is the circle x

(i) f(3) (ii) f'(1-i) (iii) f"(1-i)

where C is the circle x

^{2}+ y^{2}= 4. Find the values of(i) f(3) (ii) f'(1-i) (iii) f"(1-i)

7 M

6 (c)
A manufacturer known from his experience that the resistance of resistors he produces is normal with µ = 100Ω and standard deviation σ=2Ω. What percentage of resistors will have resistance between 98Ω and 102Ω?

6 M

7 (a)
By using residue theorem evaluate

where C is |z|=1

where C is |z|=1

7 M

7 (b)
The ratio of the probability of 3 successes in 5 independent trials to the pobability of 2 successes in 5 independent trials is 1/4. What is the probability of 4 successes in 6 independent trials?

7 M

7 (c)
Prove that both A and B are not diagonalisable but AB is diagonalisable.

6 M

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