Mathematics 2 - May 2014

First Year Engineering (Semester 2)

TOTAL MARKS:
TOTAL TIME: HOURS

1 Find the unit normal vector to the surface x²+y²=z at (1, -2, 5)(2 marks) 10 State Cauchy's residue theorem.(2 marks)

Answer any one question from Q11 (a) & Q11 (b)

11 (a) Verify Gauss divergence theorem for $$ \overrightarrow{F}=x^2\overrightarrow{i}+y^2\overrightarrow{j}+z^2\overrightarrow{k} $$ taken over the cube bounded by the planes x=0, y=0, z=0, x=1, y=1 and z=1.(16 marks) 11 (b) (i) Find the value of n such that the vector $$ r^n\overrightarrow{r} $$ is both solenoidal and irrotational.(8 marks) 11 (b) (ii) Verify Stokes' theorem for $$ \bar{F}= \left ( x^2-y^2 \right )\bar{i}+ 2xy\bar{j} $$ in the rectangular region of z=0 plane bounded by the lines x=0, y=0, x=a and y=b.(8 marks)

Answer any one question from Q12 (a) & Q12 (b)

12 (a) (i) Solve (D²-4D+3)y=cos 2x+2x²(8 marks) 12 (a) (ii) $$ Solve \ \dfrac {d^2y}{dx^2}+a^2y=\tan ax $$ using method of variation of parameters.(8 marks) 12 (b) (i) Solve $$ (x^2D^2-xD+1)y=\left ( \dfrac {\log x}{x} \right )^2 $$(8 marks) 12 (b) (ii) Solve the simultaneous equations $$ \dfrac {dx}{dt}+2y=-\sin t \ and \ \dfrac {dy}{dt}-2x=\cos t $$(8 marks)

Answer any one question from Q13 (a) & Q13 (b)

13 (a) (i) Find the Laplace transform of f(t), where $$ f(t)=\left\{\begin{matrix}\sin wt &for &0<t<\frac {\pi}{w} \\0 & for & \frac{\pi}{w} <t<\frac {2\pi}{w}\end{matrix}\right. $$ and f(t+2π)=f(t)(8 marks) 13 (a) (ii) Using convolution theorem find the inverse Laplace transform of $$ \dfrac {s^2}{(s^2+a^2)(s^2+b^2)} $$(8 marks) 13 (b) (i) Find the Laplace transform of f(t)=ie^-3tcos 2t(8 marks) 13 (b) (ii) Using Laplace transform, solve $$ \dfrac {d^2y}{dt^2}+4y=\sin 2t, $$ given y(0)=3 and y(0)=4(8 marks)

Answer any one question from Q14 (a) & Q14 (b)

14 (a) (i) Prove that the real and imaginary parts of an analytic functions are harmonic functions.(8 marks) 14 (a) (ii) Find the bilinear transformation that maps 1, i and -1 of the z-plane onto 0,1 and ∞ of the w-plane.(8 marks) 14 (b) (i) Show that v=e^-x(x cos y + y sin y) is harmonic functions. Hence find the analytic function f(z)=u+iv(8 marks) 14 (b) (ii) Find the image of |z+1|=1 under the map w=1/z(8 marks)

Answer any one question from Q15 (a) & Q15 (b)

15 (a) (i) Obtain the Laurent's series expansion of $$ f(z)= \dfrac {z^2-1}{(z+2)(z+3)} $$ in 2<|z|<3(8 marks) 15 (a) (ii) Evaluate $$ \int^{2\pi}_0 \dfrac {d\theta}{13+5 \sin \theta} $$(8 marks) 15 (b) (i) Using Cauchy's residue theorem evaluate $$ \int_c \dfrac {(z-1)}{(z-1)^2 (z-2)}dz $$ where C is |z-i|=2(8 marks) 15 (b) (ii) Evaluate by using contour integrating $$ \int^\infty_0 \dfrac {dx}{(1+x^2)^2} $$(8 marks) 2 Prove that curl(grad ϕ)=0(2 marks) 3 Solve the equation $$ \dfrac {d^2y}{dx^2}+2 \dfrac {dy}{dx}+ y=0 $$(2 marks) 4 Find the particular integral of the equation (D²-9)y=e^-3x(2 marks) 5 $$ Find \ L \left [\dfrac {\sin t}{t} \right ] $$(2 marks) 6 Evaluate $$ L^{-1} \left [ \dfrac {1}{s^2+6s+13} \right ] $$(2 marks) 7 Is the function f(z) = z analytic?(2 marks) 8 Find the invariant points of f(z)=z²(2 marks) 9 Evaluate $$ \int_c \dfrac {z}{z-2} dz $$ where C is (a) |z|=1, (b)|z|=3(2 marks)