Mathematics 2 - Dec 2013

First Year Engineering (Semester 2)

TOTAL MARKS:
TOTAL TIME: HOURS

1 Solve (D²-4)y=1(2 marks) 10 Find the Laplace transform of the function $$ f(t)\left\{\begin{matrix} 1,&t=0 \\0, &t \ne 0 \end{matrix}\right. $$(2 marks) 11 (a) (ii Solve by the method of variation of parameters
$$ 2 \dfrac {d^2y}{dx^2}+ 8y=\tan 2x $$(8 marks)

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

11 (a) (i) $$ Solve \ \dfrac {d^2y}{dx^2}- 2 \dfrac {dy}{dx}+ y=8xe^x \sin x $$(8 marks) 11 (b) (i) $$ Solve \ x^2 \dfrac {d^2y}{dx^2}+ 4x\dfrac {dy}{dx}+ y=e^{\log x} $$(8 marks) 11 (b) (ii) $$ Solve \ \dfrac {dx}{dt}+4x+3y=t; \ \dfrac {dy}{dt}+2x+5y=e^{2t} $$(8 marks)

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

12 (a) (i) Show that the vector field $$ \bar {F}= (x^2+xy^2)\bar{i}+ (y^2 + x^2y)\bar{j} $$ is irrotational. Find its scalar potential.(6 marks) 12 (a) (ii) Verify Stokes' Theorem for $$ \bar{F}= (x^2+y^2)\bar{i}-2xy\bar{j} $$ taken around the rectangle formed by the line x=-a; x=+a, y=0 and y=b.(10 marks) 12 (b) (i) Find a and b so that the surface ax³-by²z-(a+3)x²=0 and 4x²y-z³-11=0 cut orthogonally at the point (2, -1, -3)(6 marks) 12 (b) (ii) Verify Gauss Divergence theorem for $$ \bar{F}=4xz\bar{i}-y^2\bar{j}+yz\bar{k} $$ where S is the surface of the cube formed by the planes x=0, x=1, y=0, y=1, z=0 and z=1.(10 marks)

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

13 (a) (i) Prove that u=e^-2xy sin (x²-y²) is harmonic. Find the corresponding analytic function and the imaginary part.(8 marks) 13 (a) (ii) Find the bilinear map which maps the points z=0, -1, i onto the points w=i, 0, ∞. Also find the image of the unit circle of the z plane.(8 marks) 13 (b) (i) Prove that $$ w=\dfrac {z}{1-z} $$ maps the upper half of the z-plane to the upper half of the w-plane and also find the image of the unit circle of the z plane.(8 marks) 13 (b) (ii) Find the analytic function f(z)=u+iv where v=3r² sin 2θ-2r sin θ. Verify that u is a harmonic function.(8 marks)

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

14 (a) (i) Find the residue of $$ f(z)= \dfrac {z^2}{(z+2)(z-1)^2} $$ at its isolated singularities using Laurentz's series expansion.(8 marks) 14 (a) (ii) Evaluate $$ \int^{2\pi}_0 \dfrac{\cos 2 \theta} {5+4 \cos \theta}d \theta $$ using contour integration(8 marks) 14 (b) (i) show that $$ \int^{\infty}_{-\infty}\dfrac {x^2-x+2}{x^4+10x^2+9}dx = \dfrac {5\pi}{2} $$(8 marks) 14 (b) (ii) Evaluate $$ \int_c \dfrac {z+1}{(z^2+2z+4)^2}dz $$ where C is the circle |z+1+i|=2, by Cauchy's integral formula.(8 marks)

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

15 (a) (i) Evaluate $$ L^{-1} \left ( \dfrac {3s^2+16s+26}{s(s^2+4s+13)} \right ) $$(8 marks) 15 (a) (ii) Find the inverse Laplace transform of the following: $$ \log \left ( \dfrac {s+1}{s-1} \right ) $$(8 marks) 15 (b) (i) $$ Find \ L^{-1} \left [ \dfrac {s}{(s^2+a^2)^2} \right ] \ and \ find \ L^{-1} \left [ \dfrac {1}{(s^2+a^2)^2} \right ] \\ and \ find \ L^{-1} \left (\dfrac {1}{(s^2+9s+13)^2} \right ) $$(8 marks) 15 (b) (ii) Using Laplace transform, solve y''+y'=t²+2t, y(0)=4 and y(0)=-2(8 marks) 2 Convert (3x²D²+5xD+7)y=2/x log x into an equation with constant coefficients.(2 marks) 3 Define solenoidal vector function. $$ If \ \overrightarrow {V}= (x+3y)\overrightarrow{i}+ (y-2z)\overrightarrow{j}+ (x+2\lambda z)\overrightarrow{k} $$ is solenoidal, find the value of ?(2 marks) 4 State Green's theorem.(2 marks) 5 Find the constants a,b if f(z)=x+2ay+i(3x+by) is analytic.(2 marks) 6 Find the critical points of the transformation $$ w=1+\dfrac {2}{z} $$(2 marks) 7 Evaluate $$ \int_c \dfrac {z+4}{z^2+2z} $$ where C is the circle $$ \left |z-\dfrac {1}{2} \right |=\dfrac {1}{3} $$(2 marks) 8 Find the residue of $$ f(z)= \dfrac {1-e^{-z}}{z^3} \ at \ z=0 $$(2 marks) 9 Find the Laplace transform of $$ f(t)=\dfrac {1-e^{-t}}{t} $$(2 marks)