STUPIDSID
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 x2 - 4xy + 3y2=0, is reflexive, but neither symmetric nor transitive.
5 M
1 (c)
Find the characteristic roots of A30-9A28 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 A8-5A7+7A6-3A5+A4-5A3+8A2-2A1+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)-1 = g-1 of-1
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, γ1 is 1 and β2 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 x2 + y2 = 4. Find the values of
(i) f(3) (ii) f'(1-i) (iii) f"(1-i)
where C is the circle x2 + y2 = 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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