 Applied Mathematics -III | Nagpur University | Summer-2018 - Grad Plus

# Applied Mathematics -III | Nagpur University | Summer-2018

## Applied Mathematics – III

NJR/KS/18/4352/4357/4362/4367
Time : Three 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. Use of non programmable calculator is permitted.

1. a) If L{f(t)} = F(S)  then show that L\left \{f'(t) \right \}= SL\left \{ f(t) - f(0)\right \}
hence find L\left \{ \frac{d}{dt} \left ( \frac{sin t}{t} \right )\right \}   [06 M]

b) Find L-1 { tan-1 (2/S2)}.       [06 M]

OR

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

b) Solve \frac{dy}{dt}+2y+\int_{0}^{t}y \; dt= sin t  given y(0) =1 by using Laplace Transform method. [06 M]

3. a) Find the Fourier Series for the function f(x)=\left\{\begin{matrix} \pi +x & -\pi <x<0 \\ \pi -x & 0<x<\pi \end{matrix}\right. [06 M]

b) Find the Fourier transform of
f(x)=\left\{\begin{matrix} 1& |x|<1\\ 0 & |x|>1 \end{matrix}\right.
and hence evaluate \int_{0}^{\infty }\frac{sin x}{x} dx         [06 M]

OR

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

b) Express f(x)=3x-1,  0<x<1
as a half range cosine series and hence show that
\frac{\pi ^{2}}{8}=\frac{1}{1^{2}}+\frac{1}{3^{2}}+\frac{1}{5^{2}}+----- .   [06 M]

5. Find the curve passing through the points (x1,y1) and (x2, y2)  which when rotated about the x-axis gives a minimum surface. [06 M]
OR

6. Find the extremals of the functional
I=\int_{0}^{\pi /2}\left \{ y'^{2} -y^{2}+2xy\right \}dx  given y(0)=0, y(π/2)=0.     [06 M]

7. a) If u+v=ex[cos y + sin y]  find analytic function f(z)=u+iv  in terms of z. [06 M]

b) Evaluate  \oint _{C}\frac{e^{2z}}{(z+1)^{4}}dz  where C is a circle |z|=3 by using Cauchy Integral formula.  [06 M]

c) Expand in Taylor’s Series f(z)=\frac{z}{(z+1)(z+2)} about z=2. Also find the region of convergence. [06 M]

OR

8.a)  Prove that f(z) =sin z  is an analytic function and hence find its derivative. [06 M]

b) Evaluate \oint _{C}\frac{1}{sin hz}dz  where C is a circle | z | =4  by using Cauchy Residue Theorem.  [06 M]

c) Evaluate \int_{0}^{\pi }\frac{sin^{2}\theta }{5-4cos\theta }d\theta by using contour integration.   [06 M]

OR

9. a) Solve p + 3q =  5z + tan(y – 3x) .  [06 M]

b) Solve by using method of separation of variables 4\frac{\partial u}{\partial x}+\frac{\partial u}{\partial y}=3u given that u=3e-y – e-5y when x=0.     [08 M]

OR

10. a) Solve \frac{\partial u}{\partial x^{2}}-2\frac{\partial ^{2}z}{\partial x\partial y}+\frac{\partial ^{2}z}{\partial y^{2}}=tan (x+y) .    [07 M]

b) Solve by using Laplace transform method \frac{\partial u}{\partial t}+x\frac{\partial u}{\partial x}=x ,  x>0, t>0, U(x,0)=0, U(0,t)=0.       [07 M]

11. a) Investigate the linear dependence of vectors X1= (1,2,4), X2= (2, 1,3), X3= (0,1,2), X4= ( -3,7,2)  and if so find the relation.  [06 M]

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

c) Verify Cayley Hamilton’s theorem for the matrix A= \begin{bmatrix} 3 & 1 & 1\\-1 & 5 & -1\\1 & -1 & 3 \end{bmatrix}  and hence find A-1.       [06 M]

OR

12. a) Determine the value of α, β , γ when the matrix \begin{bmatrix}0 & 2\beta & \gamma \\\alpha & \beta & -\gamma \\\alpha & -\beta & \gamma\end{bmatrix}  is orthogonal
[06 M]

b) If A=\begin{bmatrix}3 & 2\\2& 3\end{bmatrix}  verify that logeeA=A  by  Sylvester’s theorem.   [06 M]

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

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