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 |

**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 marks)

**1 (c)**Find the characteristic roots of A

^{30}-9A

^{28}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 A

^{8}-5A

^{7}+7A

^{6}-3A

^{5}+A

^{4}-5A

^{3}+8A

^{2}-2A

^{1}+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 |

**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, 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 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 x

^{2}+ y

^{2}= 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)