Sum-17 Copy

B.E. (Computer Science & Engineering (New) / Computer Technology)
Third Semester (C.B.S.)

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)}=overline{f}(s) , then prove that L{f'(t)}=soverline{f}(s)-f(0) and hence find   Lleft { frac{d}{dt} left ( frac{sint}{t} right )right } .  [06 M]

b) Use Convolution theorem to find

L^{-1}left { frac{S}{(S+2)(S^{2}+9)} right } [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

f(t)= t^{2}+int_{0}^{t}f(u); sin(t-u)du  [06 M]

3. a) Find Fourier Series for begin{array}{l}f(x)=left{begin{array}{l}pi+x;,;pi<xleq0\pi-x;,;;;0leq x<piend{array}right.\\\\end{array} and Hence show that  frac{pi ^{2}}{8}=frac{1}{1^{2}}+frac{1}{3^{2}}+frac{1}{5^{2}}+-----     [06 M]

b) Solve the integral equation

int_{0}^{infty }f(x): cos lambda x ; dx=e^{-lambda }, lambda >0 [06 M]

OR

4. a) Obtain half range cosine series for f(x)=(2x-1); 0<x<1.  [06 M]

b) Using Fourier integral, show that int_{0}^{infty } frac{w sin(xw)}{1+w^{2}}; dw = frac{pi }{2} e^{-x} , x>0    [06 M]

5. a) If Z{f(n)}=F(z), prove that z{f(n+k)}=z^{k}left [ F(z)-sum_{i=0}^{k-1}f(i).z^{-i} right ]   k>0  and hence find Zleft { frac{1}{(n+1)!} right }    [06 M]

b) By using convolution theorem find frac{z^{2}}{(z-1)(z-3)}    [06 M]

OR

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

b) Solve x_n+2 -3x_n+1+2x_n=4n, x₀=0, x₁=1         [06 M]

7. a) If u=y³-3x²y,  show that u is harmonic function. Find V and the corresponding analytic
function f(z)=u+iv           [07 M]

b) Evaluate using Cauchy’s integral formula oint _{c}frac{(4-3z)}{z(z-1)(z-2)}.dz where c is a circle |z|=3/2.     [07 M]

OR

8. a) Find Laurent’s series expansion of f(z)=frac{(z^{2}-4)}{(z+1)(z+4)} valid for  [07 M]

(i) |z|<1          ii) 1< |z|<4        iii) |z| >4

b) Use residue theorem to evaluate oint _{C}frac{e^{zt}}{(z(z^{2}+1)}. dz, t>2 , where C is an ellipse  |z-sqrt{5}|+|z+sqrt{5}|=6 .   [07 M]

9. . a) Find whether the following set of vectors are linearly dependent. If so, find relationship.
X₁= (1, 2, 1, 3), X₂ =(2, -1, 3, 2)  and X₃= ( -1, 8, -9, 5)  .   [07 M]

b) Reduce the matrix begin{array}{l}A=begin{bmatrix}1&-2\-5&4end{bmatrix}\\\\end{array} to the diagonal form.

 [06 M]

c) Find the largest eigen value and corresponding eigen vector for the matrix

begin{array}{l}A=begin{bmatrix}1&6&1\0&2&0\0&0&3end{bmatrix}\\\\end{array} [06 M]

OR

10. a) Verify Cayley – Hamilton theorem for the matrix A and find A^-1 , where begin{array}{l}A=begin{bmatrix}2&-1&1\-1&2&-1\1&-1&2end{bmatrix}\\\\end{array}  [06 M]

b) If begin{array}{l}A=begin{bmatrix}cos;alpha&sin;alpha\-sin;alpha&cos;alphaend{bmatrix}\\\\end{array} find Aⁿ, using Sylvester’s theorem.  [06 M]

c) Solve frac{d^{2}y}{dt^{2}}-3frac{dy}{dt}-10y=0 , given y(0)=3, y'(0)=15 by matrix method. [06 M]

11. a) An insurance company insured 2000 scooter drivers, 4000 car drivers and 6000 truck
drivers. The respective probabilities of an accident are 0.01, 0.03 and 0.15. Out of the
insured persons meets an accident. What is the probability that he is a scooter driver?  [06 M]

b) Let X be a random variable having density function

Find (i) the constant C, (ii) Pleft ( frac{1}{2}<x< frac{3}{2} right )  and  iii) The distribution function    [06 M]

OR

12. a) Find moment generating function and first four moments about the origin for random variable X given by

X;=;left{begin{array}{l}1/2;;;prob.;1/2\-1/2;;prob.;1/2end{array}right.  [06 M]

b) A machine produces bolts which are 10% defective. Find the probability that in a random  sample of 400 bolts produced by this machine (i) between 30 and 50 and (ii) at the most 30 bolts will be defective. (use normal approximation).  [06 M]