AU First Year Engineering (Semester 2)
Mathematics 2
December 2012
Total marks: --
Total time: --
INSTRUCTIONS
(1) Assume appropriate data and state your reasons
(2) Marks are given to the right of every question
(3) Draw neat diagrams wherever necessary

1 Find the Wronskiam of y1, y2 of y''-2y'+y=ex log x
2 M

2 Find the particular integral of (D2 -4D+4)y=2x
2 M

3 Prove that $\overrightarrow{F}= yz\overrightarrow{i} + zx\overrightarrow{j}+xy\overrightarrow{k}$ is irrotational.
2 M

4 State Gauss divergence theorem.
2 M

5 Show that the function f(z)=z is nowhere differentiable.
2 M

6 Find the map of the circle |z|=3 under the transformation w=2z.
2 M

7 Evaluate $\int_c \dfrac{z \ dz}{(z-1)(z-2)}$ where C is the circle |z|=1/2
2 M

8 $If \ f(z)= \dfrac {-1}{z-1}-2 [1+(z-1)+(z-1)^2+ \cdots ]$ find the residue of f(z) at z=1.
2 M

9 Is the linearity property applicable to $L\left \{ \dfrac {1-\cos t}{t} \right \}?$ Reason out.
2 M

10 Find the inverse Laplace transform of $\dfrac {1}{(s+1)(s+2)}$
2 M

11 (a) (i) Solve the equation (D2+5D+4)y=e-x sin 2x.
8 M
11 (a) (ii) Solve the equation $\dfrac {d^2y}{dx^2}+y=cosec \ x$ by the method of variation of parameters.
8 M
11 (b) (i) $Solve \ \dfrac {dx}{dt}+y=e^t, \ x-\dfrac {dy}{dt}=t$
8 M
11 (b) (ii) Solve the equation $\dfrac {d^2y}{dx^2}+ \dfrac {1}{x} \dfrac {dy}{dx} = \dfrac {12 \log x}{x^2}$
8 M

Answer any one question from Q12 (a) & Q12 (b)
12 (a) (i) Show that $\bar{F}-(2xy-z^2)\bar{i}+ (x^2+2yz)\bar{j}+ (y^2 -2zx)\bar{k}$ is irrotational and find its scalar potential.
8 M
12 (a) (ii) Verify Green's theorem V=(x^2+y^2)i-2xyj taken around the rectangle bounded by the lines x=?a, y=0 and y=b.
8 M
12 (b) Verify Gauss's divergence theorem for $\overrightarrow {F}=4xz\overrightarrow{i}-y^2 \overrightarrow{j}+ yz\overrightarrow{k}$ over the cube bounded by x=0, x=1, y=0, y=1, z=0 and z=1.
16 M

Answer any one question from Q13 (a) & Q13 (b)
13 (a) (i) Find the bilinear transformation that maps the points z=&infty;, i, 0 onto w=0,i,&infty; respectively.
8 M
13 (a) (ii) Determine the analytic function whose real part is $\dfrac {\sin 2x}{cosh \ 2y - \cos 2x}$
8 M
13 (b) (i) Find the image of the hyperbola x2-y2=1 under the transformation w=1/z
8 M
13 (b) (ii) Prove that the transformation w=z/1-z maps the upper half of z plane on to the upper half of w-palne. What is the image of |z|=1 under this transformation?
8 M

Answer any one question from Q14 (a) & Q14 (b)
14 (a) (i) Evaluate $\int_c \dfrac {z+4}{z^2+2z+5}dz$ where C is the circle |z+1+i|=2, using Cauchy's integral formula.
8 M
14 (a) (ii) Find the residue of $f(x)= \dfrac {z^2}{(z-1)^2(z+2)^3}$at its isolated singularities using Laurent's series expansions. Also state the valid region.
8 M
14 (b) Evaluate $\int^{2 \pi}_{0} \dfrac {\sin^2 \theta}{\alpha + b \cos \theta}d\theta, a>b>0.$
16 M

Answer any one question from Q15 (a) & Q15 (b)
15 (a) (i) Find $L^{-1}\left [ \dfrac {s^2}{(s^2+4)^2} \right ]$ using convolution theorem.
8 M
15 (a) (ii) Find the Laplace transform of the Half wave rectifier $f(t)= \begin{cases}\sin \omega t & 0\lt t \lt \pi/ \omega \\0 & \pi / \omega \lt t \lt 2\pi/ \omega \end{cases}$ and $f(t+2\pi/ \omega)=f(t)$ for all t.
8 M
15 (b) (i) Find $L \left[\dfrac{\cos at-\cos bt}{t} \right]$
8 M
15 (b) (ii) Solve $\dfrac {d^2x}{dt^2}-3 \dfrac {dx}{dt}+2x=2$, given $x=0\ and \ \dfrac {dx}{dt}=5 \ for \ t=0$ using Laplace transform method.
8 M

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