Question Paper: Applied Mathematics 4 : Question Paper May 2013 - Electronics Engineering (Semester 4) | Mumbai University (MU)
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## Applied Mathematics 4 - May 2013

### Electronics Engineering (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 random variable X has the probability function:

 X: -2 -1 0 1 2 3 P (X = x): 0.1 k 0.2 2k 0.3 3k
Find: (i) k  (ii) P(x ≤ 1)  (iii) P(-2 < x < 1)  (iv) Obtain the distribution function of X " (5 marks) 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 marks) 1 (c) Find the characteristic roots of A30-9A28 where (5 marks)
1 (d) Find Laurent's series about z = -2 for: (5 marks)
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 marks) 2 (b) Find the Residues of <bt> at its poles.</bt>
(7 marks)
2 (c) If find cosA.
(6 marks)
3 (a) Check whether A = {2, 4, 12, 16} and B = {3, 4, 12, 24} are lattices under divisibility? Draw their Hasse diagrams.(7 marks) 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 ?
(7 marks)
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 marks)
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 marks) 4 (b) Let and + and · be the matrix addition and matrix multiplication. Is ( S, +, · ) an integral domain? Is it a field?
(7 marks)
4 (c) Show that ∫c dz/(z+1) = 2πI, where C is the circle |z| = 2. Hence deduce that: (6 marks)
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 marks)
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)"
(7 marks)
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 marks) 6 (a) Prove that the set A={0, 1, 2, 3, 4, 5} is a finite abelian group under addition modulo 6.(7 marks) 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)
(7 marks)
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 marks) 7 (a) By using residue theorem evaluate where C is |z|=1
(7 marks)
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 marks) 7 (c) Prove that both A and B are not diagonalisable but AB is diagonalisable. (6 marks)

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