Mathematics 2 - May 2013

First Year Engineering (Semester 2)

TOTAL MARKS:
TOTAL TIME: HOURS

1 Find the particular integral of (D²-2D+1)y=cosh x(2 marks) 10 Verify initial value theorem for the function f(t)=ae^-bt(2 marks)

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

11 (a) (i) Solve the differential equation $$ \dfrac {d^2y}{dx^2}+2\dfrac {dy}{dx}+ y=\dfrac {e^{-x}}{x^2} $$ by the method of variation of parameters.(8 marks) 11 (a) (ii) $$ Solve : \ (3x+2)^2 \dfrac {d^2y}{dx^2}+3 (3x+2)\dfrac {dy}{dx} -36y = 3x^2 +4x+1 $$(8 marks) 11 (b) (i) Solve the simultaneous differential equations: $$ \dfrac {dx}{dt}+5x-2y=t; \ \dfrac {dy}{dt}+2x+y=0 $$(8 marks) 11 (b) (ii) $$ Solve \ x^2 \dfrac {d^2 y}{dx^2}+ 4x \dfrac {dy}{dx}+2y=x^2 + \dfrac {1}{x^2} $$(8 marks)

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

12 (a) Verify Stokes' Theorem for the vector field $$ \overrightarrow{F}= (2x-y)\overrightarrow {i}-yz^2 \overrightarrow{j}-y^2z\overrightarrow{k} $$ over the upper half surface x²+y²+z²-1, bounded by its projection on the xy-plane.(16 marks) 12 (b) Verify divergence theorem for $$ \overrightarrow{F}=x^2 \overrightarrow{i}+z\overrightarrow{j} + yz\overrightarrow{k} $$ over the cube formed by the plane x=±1, y=±1, z=±1.(16 marks)

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

13 (a) (i) Prove that the function u-e^x(x cos y -y sin y) satisfies Laplace's equation and find the corresponding analytic function f(z)=u+iv.(8 marks) 13 (a) (ii) Find the Bilinear transformation which maps z=0, z=1, z=∞ into the points w=i, w=1, w=-i.(8 marks) 13 (b) (i) Find the image of |z-2i|=2 under the transformation w=1/z.(8 marks) 13 (b) (ii) If f(z) is an analytic function of z, prove that $$ \left ( \dfrac {\partial^2}{\partial x^2}+ \dfrac {\partial^2}{\partial y^2} \right )|f(z)|^2=4 |f(z)|^2 $$(8 marks)

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

14 (a) (i) Expand the function $$ f(z)= \dfrac {z^2-1}{z^2+5z+6} $$ in Laurent's series for |z|>3(8 marks) 14 (a) (ii) Evaluate $$ \int_c \dfrac {\sin \pi z^2+ \cos \pi z^2 }{(z+1)(z+2)}dz $$ where C is |z|=3(8 marks) 14 (b) (i) Evaluate $$ \int^{\infty}_0 \dfrac {x^2 dx}{(x^2 +a^2)(x^2+b^2)}, \ a>0,b>0 $$(8 marks) 14 (b) (ii) Evaluate $$ \int^{2\pi}_0 \dfrac {\cos 3\theta}{5-4\cos \theta}d\theta $$ using contour integration.(8 marks)

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

15 (a) (i) $$ Find \ L\left [ t^2e^{-3t} \sin 2t \right] $$(8 marks) 15 (a) (ii) Find the Laplace transform of the square-wave function (or Meoander function) of period α defined as $$ f(t)= \left\{\begin{matrix}1 & when & 0<t<\frac{a}{2} \\-1, &when & \frac {a}{2} <t<a\end{matrix}\right. $$(8 marks) 15 (b) (i) Using convolution theorem find the inverse Laplace transform of $$ \dfrac {4}{(s^2+2s+5)^2} $$(8 marks) 15 (b) (ii) Solve y''+5y'+6y=2 given y'(0)=0 and y(0)=0 using Laplace transform.(8 marks) 2 $$ Solve \ x^2\dfrac {d^2y}{dx^2}+4x \dfrac {dy}{dx}+2y=0 $$(2 marks) 3 Find the directional derivative of ?=xyz at (1,1,) in the direction of$$ \overrightarrow {i}+ \overrightarrow{j}+ \overrightarrow{k} $$(2 marks) 4 If $$ \overrightarrow{A} \ and \ \overrightarrow{B}$$ are irrotational, prove that $$ \overrightarrow{A}\times \overrightarrow{B} $$ is solenoidal.(2 marks) 5 Find the image of the line x=k under the transformation w=1/z(2 marks) 6 Find the fixed points of mapping $$ w=\dfrac {6z-9}{z} $$(2 marks) 7 Evaluate $$ \int_c \dfrac {3z^2 + 7z+1}{z+1}dz, \ where \ C \ is |z|=\dfrac {1}{2} $$(2 marks) 8 Find the residue of $$ \dfrac {1-e^{2z}}{z^4} \ at \ z=0 $$(2 marks) 9 Find the Laplace transform of $$ \dfrac {t}{e^t} $$(2 marks)