# Win-17 Copy

## Applied Mathematics – III

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)=x2-2, -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 yn+2-2cosα yn+1+yn=0 given yo=0, y1=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
X1= (2, -1, 3, 2),  X2= (1, 3, 4, 2), X3 =(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-1AB is a  diagonal form. [06 M]

c) By using Cayley Hamilton’s theorem find A8 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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