Mathematics 2 - Dec 2012

First Year Engineering (Semester 2)

TOTAL MARKS:
TOTAL TIME: HOURS

1 Find the Wronskiam of y₁, y₂ of y''-2y'+y=e^x log x(2 marks) 10 Find the inverse Laplace transform of $$ \dfrac {1}{(s+1)(s+2)} $$(2 marks) 11 (a) (i) Solve the equation (D²+5D+4)y=e^-x sin 2x.(8 marks) 11 (a) (ii) Solve the equation $$ \dfrac {d^2y}{dx^2}+y=cosec \ x $$ by the method of variation of parameters.(8 marks) 11 (b) (i) $$ Solve \ \dfrac {dx}{dt}+y=e^t, \ x-\dfrac {dy}{dt}=t $$(8 marks) 11 (b) (ii) Solve the equation $$ \dfrac {d^2y}{dx^2}+ \dfrac {1}{x} \dfrac {dy}{dx} = \dfrac {12 \log x}{x^2} $$(8 marks)

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 marks) 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 marks) 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 marks)

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 marks) 13 (a) (ii) Determine the analytic function whose real part is $$ \dfrac {\sin 2x}{cosh \ 2y - \cos 2x} $$(8 marks) 13 (b) (i) Find the image of the hyperbola x²-y²=1 under the transformation w=1/z(8 marks) 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 marks)

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 marks) 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 marks) 14 (b) Evaluate $$ \int^{2 \pi}_{0} \dfrac {\sin^2 \theta}{\alpha + b \cos \theta}d\theta, a>b>0. $$(16 marks)

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 marks) 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 marks) 15 (b) (i) Find $$ L \left[\dfrac{\cos at-\cos bt}{t} \right] $$(8 marks) 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 marks) 2 Find the particular integral of (D² -4D+4)y=2^x(2 marks) 3 Prove that $$ \overrightarrow{F}= yz\overrightarrow{i} + zx\overrightarrow{j}+xy\overrightarrow{k} $$ is irrotational.(2 marks) 4 State Gauss divergence theorem.(2 marks) 5 Show that the function f(z)=z is nowhere differentiable.(2 marks) 6 Find the map of the circle |z|=3 under the transformation w=2z.(2 marks) 7 Evaluate $$ \int_c \dfrac{z \ dz}{(z-1)(z-2)} $$ where C is the circle |z|=1/2(2 marks) 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 marks) 9 Is the linearity property applicable to $$ L\left \{ \dfrac {1-\cos t}{t} \right \}? $$ Reason out.(2 marks)