# Sum-18 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. Find Laplace Transform of frac{sin^{2}t}{t} and hence evaluate int_{0}^{infty }e^{-t}; frac{sin^{2}t}{t}.dt     [06 M]

b) Use convolution theorem to find L^{-1}left { frac{s^{2}}{(s^{2}+s^{2})(s^{2}+b^{2})} right }  [06 M]

OR

2.a.  Express the function f(t) in terms of unit step function & find its Laplace Transform :
f(t)=;left{begin{array}{l}t^2;,;;;0<t<1\4t,;;;t>1end{array}right. [06 M]

b. Solve frac{d^{2}y}{dt^{2}}+9y=sin t ,; y(0)=1,; y(frac{pi }{2})=-1 , using Laplace Transform. [06 M]

3. (a) Obtain Fourier series for    f(x)=left{begin{matrix}pi x& 0leq xleq 1\pi (2-x)& 1leq xleq 2end{matrix}right.    Hence show that  frac{pi ^{2}}{8}=frac{1}{1^{2}}+frac{1}{3^{2}}+frac{1}{5^{2}}+-----        [06 M]

b) Show that the Fourier sine integral of  f(x)=;left{begin{array}{l}mathrmpi/2;,;;;;;;;;;0leq xleqmathrmpi\0,;;;;;;;;;;;;;;;;;x>mathrmpiend{array}right.  is

int_{0}^{infty }frac{(1-cospilambda )sin lambda x}{lambda }.dlambda [06 M]

OR

4. (a) Obtain half range sine series for f(x) =πx-x2 in the interval (0,π)    [06 M]

b) Find the Fourier cosine transform of f(x)=frac{e^{-ax}}{x} , a>0.  [06 M]

5. If z{f(n)}=F(z), then prove that

z{f(n+k)}=z^{k}left [ F(z)-sum_{i=0}^{k-1}f(i).z^{-i} right ]      [06 M]

b) Using convolution theorem , find inverse Z-Transform of frac{z^{2}}{(z-a)(z-b)}
[06 M]

OR

6.a.  Find inverse Z-transform of frac{z^{3}-z^{2}+z}{(z+2)(z^{2}-1)}  [06 M]

b. Solve the difference equation yn+2 +5yn+1+6yn=6n, y0=0, y1=1.  using Z transform.  [06 M]

7. a) Given harmonic function u=e-x(x sin y-y cos y). Find v such that f(z)=u+iv is analytic and express f(z) in terms of z.  [07 M]

b) Expand f(z)=(z2+4z+3)-1 by Laurent’s series valid for:  [07 M]

i) 1< |z| <3    ii) |z| < 1   iii) |z| >3

OR

8. a) Find the value of oint frac{12z-7}{(z-1)^{2}(2z+3)}.dz , where C is a circle

|z+i| = sqrt{3}     [07 M]

b) Evaluate int_{0}^{2pi }frac{1}{5-4sin theta }.dtheta  by contour integration.  [07 M]

9. a) Determine modal matrix for A=begin{bmatrix}1&0&-1\1&2&1\2&2&3end{bmatrix} [06 M]

b) Are the following vectors linearly dependent? If so, the relation between them

X1=[1,1,1,3] , X2=[1,2,3,4], X3=[2,3,4,7].  [06 M]

c) By using Cayley Hamilton theorem find A8, if A=begin{bmatrix}1&2\2&-1end{bmatrix} [06 M]

OR

10. a) If m=begin{bmatrix}2&1\3&4end{bmatrix} find the value of m3-3m+I and verify the result by Sylvester’s theorem. [06 M]

b) Find the largest eigen value and corresponding eigen vectors for the matrix

A=begin{bmatrix}-4&-5\1&2end{bmatrix} [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. Three machines A, B and C produce respectively 60%, 30% and 10% of the total numberof times of a factory. The percentages of defective output of these machines are respectively2%, 3% and 4%. An item is selected at random and is found defective. Find the probability
that the item was produced by machine C.  [06 M]

b) A random variable X has density function f(x) = left{begin{matrix}kx^{2} & 1leq xleq 2\kx& 2< x< 3\0 & otherwise end{matrix}right.. Find k and the distribution functions. [06 M]

OR

12. a) Find mean, variance and moment generating function for exponential distribution

begin{array}{l}f(x)=;left{begin{array}{l}alpha e^{-alpha x},;x>0\0;;;;;;;;,;xleq0end{array}right.\\\\end{array}  [06 M]

b) In a normal distribution, 31% of the items are under 45 and 8% are over 64. Find the mean
and standard deviation of the distribution. [06 M]

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