AU First Year Engineering (Semester 2)
Mathematics 2
December 2013
Total marks: --
Total time: --
INSTRUCTIONS
(1) Assume appropriate data and state your reasons
(2) Marks are given to the right of every question
(3) Draw neat diagrams wherever necessary

1 Solve (D2-4)y=1
2 M

2 Convert (3x2D2+5xD+7)y=2/x log x into an equation with constant coefficients.
2 M

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 M

4 State Green's theorem.
2 M

5 Find the constants a,b if f(z)=x+2ay+i(3x+by) is analytic.
2 M

6 Find the critical points of the transformation $w=1+\dfrac {2}{z}$
2 M

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 M

8 Find the residue of $f(z)= \dfrac {1-e^{-z}}{z^3} \ at \ z=0$
2 M

9 Find the Laplace transform of $f(t)=\dfrac {1-e^{-t}}{t}$
2 M

10 Find the Laplace transform of the function $f(t)\left\{\begin{matrix} 1,&t=0 \\0, &t \ne 0 \end{matrix}\right.$
2 M

11 (a) (ii Solve by the method of variation of parameters
$2 \dfrac {d^2y}{dx^2}+ 8y=\tan 2x$
8 M
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 M
11 (b) (i) $Solve \ x^2 \dfrac {d^2y}{dx^2}+ 4x\dfrac {dy}{dx}+ y=e^{\log x}$
8 M
11 (b) (ii) $Solve \ \dfrac {dx}{dt}+4x+3y=t; \ \dfrac {dy}{dt}+2x+5y=e^{2t}$
8 M

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 M
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 M
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 M
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 M

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 M
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 M
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 M
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 M

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 M
14 (a) (ii) Evaluate $\int^{2\pi}_0 \dfrac{\cos 2 \theta} {5+4 \cos \theta}d \theta$ using contour integration
8 M
14 (b) (i) show that $\int^{\infty}_{-\infty}\dfrac {x^2-x+2}{x^4+10x^2+9}dx = \dfrac {5\pi}{2}$
8 M
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 M

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 M
15 (a) (ii) Find the inverse Laplace transform of the following: $\log \left ( \dfrac {s+1}{s-1} \right )$
8 M
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 M
15 (b) (ii) Using Laplace transform, solve y''+y'=t2+2t, y(0)=4 and y(0)=-2
8 M

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