Mathematics 2 - May 2012

First Year Engineering (Semester 2)

TOTAL MARKS:
TOTAL TIME: HOURS

1 Transform the equation $$ (2x+3)^2 \dfrac {d^2y}{dx^2}-2 (2x+3)\dfrac {dy}{dx}-12 y =6x $$ into a different equation with constant coefficients.(2 marks) 10 verify initial value theorem for f(t)=1+e^t (sin t+ cos t).(2 marks)

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

11 (a) (i) Solve (D²+a²)y=sec ax using the method of variation of parameters.(8 marks) 11 (a) (ii) Solve: (D²-4D+3)y=e^x cos 2x.(8 marks) 11 (b) (i) Solve the differential equation $$ \left ( x^2 D^2 -xD +4 \right )y=x^2 \sin (\log x) $$(8 marks) 11 (b) (ii) Solve the simultaneous differential equations $$ \dfrac {dx}{dt}+2y = \sin 2t, \ \dfrac {dy}{dt}- 2x = \cos 2t. $$(8 marks)

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

12 (a) (i) Show that $$ \overrightarrow{F}= \left ( y^2 + 2xz^2 \right )\bar {i} \left (2xy -z \right )\overrightarrow{j}+ \left ( 2x^2z-y+2z \right )\overrightarrow{k} $$ is irrotational and hence find its scalar potential.(8 marks) 12 (a) (ii) Verify Green's theorem in a plane for $$ \displaystyle \int_c \left [ \left (3x^2 -8y^2 \right )dx+(4y-6xy)dy \right ], $$ where C is the boundary of the region defined by x=0, y=0 and x+y=1.(8 marks) 12 (b) (i) Using Stroke's theorem evaluate $$ \displaystyle \int_c \overrightarrow{F}\cdot d\overrightarrow{r} \ where \ \overrightarrow{F}= y^2 \overrightarrow{i}+ x^2 \overrightarrow{j}- (x+z) \overrightarrow{k} $$ and 'C' is the boundary of the triangle with vertices at (0,0,0), (1,0,0),(1,1,0).(8 marks) 12 (b) (ii) Find the work done in moving a particle in the force field given by $$ \overrightarrow{F}=3x^2\overrightarrow{i}+ (2xz-y)\overrightarrow{j}+ z\overrightarrow{k} $$ along the straight line from (0,0,0) to (2,1,3).(8 marks)

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

13 (a) (i) Prove that every analytic function w=u+iv can be expressed as a function of z alone, not as a function of z(8 marks) 13 (a) (ii) Find the bilinear transformation which maps the points z=0,1,? into w=i,1, -1 respectively.(8 marks) 13 (b) (i) If f(z) is an analytic function of z, prove that $$ \left (\dfrac{\partial^2}{\partial x^2}+ \dfrac {\partial^2}{\partial y^2} \right )\log |f(z)|=0 $$(8 marks) 13 (b) (ii) Show that the image of the hyperbola x²-y²=1 under the transformation ω=1/z is the lemniscate r² = cos 2θ(8 marks)

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

14 (a) (i) Evaluate $$ \displaystyle \int_c \dfrac {zdz}{(z-1)(z-2)^2} \ where \ C \ is \ |z-2| = \dfrac {1}{2} $$ by using Cauchy's integral formula.(8 marks) 14 (a) (ii) Evaluate $$ f(z)= \dfrac {1}{(z+1)(z+3)} $$ in Laurent series valid for the regions |z|>3 and 1<|z|<3.(8 marks) 14 (b) (i) Evaluate $$ \int_c \dfrac {z-1}{(z+1)^2(z-2)}dz $$ where C is the circle |z=i|=2 using Cauchy's residue theorem.(8 marks) 14 (b) (ii) Evaluate $$ \int^{\infty}_{0}\dfrac {\cos mx}{x^2+a^2}dx $$ using contour integration.(8 marks)

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

15 (a) (i) Apply convolution theorem to evaluate $$ L^{-1} \left [ \dfrac {s}{\left ( s^2+a^2 \right )^2} \right ] $$(8 marks) 15 (a) (ii) Find the Laplace transform of the following triangular wave function given by $$ f(t)\left\{\begin{matrix}t, & 0\le t \le \pi \\ 2\pi-t, & \pi \le t \le 2\pi\end{matrix}\right. \ and \ f(t+2\pi)=f(t) $$(8 marks) 15 (b) (i) Find the Laplace transform of $$ \dfrac {e^{at}-e^{-bt}}{t} $$(4 marks) 15 (b) (ii) Evaluate $$ \int^{\infty}_0 te^{-2t} \cos t \ dt $$ using Laplace transform.(4 marks) 15 (b) (iii Solve the differential equation $$ \dfrac {d^2 y}{dt^2}-3 \dfrac {dy}{dt}+2y=e^{-t} $$ with y(0)=1 and y'(0)=0 using Laplace transform.(8 marks) 2 Find the particular integral of (D-1)² y=e^x sin x.(2 marks) 3 Find the ? such that $$ \bar{F}= (3x+2y+z)\bar{z}+ (4x+\lambda y-z)\bar{j}+ (x-y+2z)\bar{k} $$ is solenoidal.(2 marks) 4 State Gauss divergence theorem.(2 marks) 5 State the basic difference between the limit of a function of a real variable and that of a complex variable.(2 marks) 6 Prove that a bilinear transformation has atmost two fixed points.(2 marks) 7 Define singular point.(2 marks) 8 Find the residue of the function $$ f(z)= \dfrac {4}{z^3 (Z-2)} $$ at a simple pole.(2 marks) 9 State the first shifting theorem on Laplace transforms.(2 marks)