Third Semester (C.B.S.)

**Time- 3 Hours**

**Max. Marks : 80**

**Notes : 1. All questions carry marks as indicated.**

**2. Solve Question 1 OR Questions No. 2.**

**3. Solve Question 3 OR Questions No. 4.**

**4. Solve Question 5 OR Questions No. 6.**

**5. Solve Question 7 OR Questions No. 8.**

**6. Solve Question 9 OR Questions No. 10.**

**7. Solve Question 11 OR Questions No. 12.**

**8. Assume suitable data whenever necessary.**

**9. Illustrate your answers whenever necessary with the help of neat sketches.**

**10. Use of non programmable calculator is permitted.**

**11. Use of normal distribution table is permitted.**

1.a) If L{f(t)=F(s) then show that Lleft { frac{f(t)}{t} right }=int_{S}^{infty }F(s)ds hence find Lleft { frac{sin t}{t} right } **[06 M]**

b. Find L^{-1}left { frac{s}{(s^{2}+a^{2})^{2}} right } by using convolution theorem. **[06 M]**

**OR**

2. a. Express begin{array}{l}f(x)=left{begin{array}{l}(t-1);;,;1<t<2\(3-t);,;;;2<t<3end{array}right.\\\\end{array} in terms of unit step function and hence find its Laplace transform. **[06 M]**

b. Solve frac{d^{2}t}{dt^{2}}+2frac{dy}{dt}+5y=e^{-t}sin t given y(0)=0, y'(0)=1 by using Laplace trnsform method. **[06 M]**

3. a. Find the Fourier series to represent f(x)=x^{2}-2, -2≤x≤2. **[06 M]**

b. Find Fourier sine transform of frac{e^{-ax}}x, a>0. **[06 M]**

**OR**

4. a. Using the Fourier Cosine integral show that

int_{0}^{infty }frac{coslambda x}{1+lambda ^{2}}dlambda =frac{pi }{2}e^{-x} **[06 M]**

b) Find the half range cosine series for sin x when 0<x<π, hence deduce that

1-frac{1}{3}+frac{1}{5}-frac{1}{7}+.......=frac{pi }{4} **[06 M]**

5. a) If z{f(n)}=F(z) then show that zleft { frac{f(n)}{n+k} right }=z^{k}int_{z}^{infty }frac{F(z)}{z^{k+1}}dz hence find zleft { frac{1}{n+1} right } **[06 M]**

b) Prove that frac{1}{n!}ast frac{1}{n!}=frac{2^{n}}{n!} where ∗ is a convolution operation. **[06 M]**

**OR**

6. a) Find Z-Transform of frac{(n+1)(n+2)}{2!}a^{n} . **[06 M]**

b) Solve y_{n+2}-2cosα y_{n+1}+y_{n}=0 given y_{o}=0, y_{1}=1 by using Z-Tranform. **[06 M]**

7. a) If f(z) is analytic function with constant modulus. Show that f(z) is constant. **[07 M]**

b) Evaluate int _{C}frac{z-1}{(z+1)^{2})(z-2)}dz where C is a circle |z-i|=2 by Cauchy Integral formula. **[07 M]**

OR

8. a) Evaluate int_{0}^{2pi }frac{cos 2theta }{5+4 costheta }dtheta bu using Contour Integration. **[07 M]**

b) Expand in Taylor’s series f(z)=frac{z}{(z+1)(z+2)} about Z = 2. Also find the region of convergence. **[07 M]**

9. a) Investigate the linear dependence of vectors

X_{1}= (2, -1, 3, 2), X_{2}= (1, 3, 4, 2), X_{3} =(3,-5, 2, 2) and if so find the relation. **[06 M]**

b) ) Find the modal matrix B corresponding to matrix A=begin{bmatrix}1&2\3&2end{bmatrix} nd verify that B^{-1}AB is a diagonal form. **[06 M]**

c) By using Cayley Hamilton’s theorem find A^{8} if A=begin{bmatrix}1&2\2&-1end{bmatrix} **[06 M]**

**OR**

10. a) If A=begin{bmatrix}-1&3\1&1end{bmatrix} verify 2 sinA=(sin 2)A by Sylvester’s theorem. **[06 M]**

b) Find the largest eigen value and corresponding eigen vector for the matrix A=begin{bmatrix}1&6&1\1&2&0\0&0&3end{bmatrix} by iteration method. **[06 M]**

c) Solve frac{d^{2}x}{dt^{2}}+4x=0 , x(0)=1, x'(0)=0 by matrix method.**[06 M]**

11. a. Each of the three identical Jewellery boxes has two drawers. In each drawer of the first box there is a gold watch. In each drawer of the second box there is a silver watch. In one of the drawer of the third box there is a gold watch while in the other there is silver watch. If we

select a box at random, open one of the drawer and find it to contain a silver watch. What is the probability that the other drawer has gold watch. **[06 M]**

b) The distribution function of a random variable X is given by F(x)=left{begin{matrix} cx^{3} & 0leq x<3\1 &xgeq 3 \0 & x<0end{matrix}right.

Find: i) Probability density function

ii) C

iii) p(x>1) **[06 M]**

**OR**

12. a) A random variable X can assume the value 1 and –1 with probability 1/2 each. Find (i) moment generating function (ii) first two moments about origin and about mean. **[06 M]**

b) A car hire firm has two cars which it hires out day by day. The number of demands for a car on each day is distributed as a Poisson distribution with mean 1.5. Calculate the proportion of days on which neither car is used and the proportion of days on which some

demand is refused. **[06 M]**

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