AU First Year Engineering (Semester 2)
Mathematics 2
December 2014
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 Find a particular integral of the differential equation (D3+6D+5)y=e.
2 M

2 Transform the differential equation x2y''-xy'+2y=0 with constant coefficients.
2 M

3 Find $\nabla (\nabla . ((x^2 - yz) \bar{i} + (y^2 -xz)\bar{j} + (z^2 -xy) \bar{k}))$ at the point (1, -1, 2).
2 M

4 State Green's theorem in the plane.
2 M

5 Give an example of a complex-valued function which is differentiable at a point but not analytic at that point.
2 M

6 $If \ u(x,y)= 3x^2 y + 2x^2 - y^3 - 2y^2,$ verify whether u is harmonic.
2 M

7 State Cauchy's integral formula.
2 M

8 Find the residue of $\left \{ \dfrac {\sin 3z}{z^6} \right \} \ at \ z=0$
2 M

9 State sufficient conditions for the existence of Laplace transform.
2 M

10 State final value theorem.
2 M

Answer any one question from Q11 (a) & Q11 (b)
11 (a) (i) Solve the differential equation y''+a2y=tan ax by variation of parameters method.
8 M
11 (a) (ii) Solve the following simultaneous differential equations. $\dfrac {dx}{dt} + 2x -3y = t \ and \ \dfrac {dy}{dt}- 3x + 2y =e^{2t}$
8 M
11 (b) (i) Solve ((x+1)2D2+(x+1)D+1)y=4 cos log (x+1).
8 M
11 (b) (ii) Solve (D2-7D-6)y=(1+x)e2x.
8 M

Answer any one question from Q12 (a) & Q12 (b)
12 (a) (i) Prove that $(\phi \bar{F})= \phi div \bar{F}+ \nabla \phi\cdot \bar{F}$ Also determine the value of n for which $r^{n}\bar{R}\ is \ solenoidal \ where \ \bar{R} = x\bar{i}+ y\bar{j}+ z\bar{k} \ and \ r=|\bar{R}|.$
8 M
12 (a) (ii) Verify Gauss divergence theorem for $\bar{F}= x^2 \overrightarrow{i}+ y^2 \overrightarrow{j} + z^2 \overrightarrow{k}$ over the volume of the cuboid formed by the planes x=0, x=a, y=0, y=b, z=0 and z=c.
8 M
12 (b) (i) Prove that $\bar{F} = (y^2 + 2xz^2 ) \bar{i} + (2xy - z) \bar{j}+ (2x^2 z - y+2z) \bar{k}$ irrotational and hence find its scalar potential.
8 M
12 (b) (ii) Verify Stokes' theorem for $\bar{F}= (x^2 +y^2) \bar{i} + 2xy \bar{j}$ where S is the rectangle in the xy-plane formed by the lines x=0, x=a, y=0 and y=b.
8 M

Answer any one question from Q13 (a) & Q13 (b)
13 (a) (i) If f=u+iv is an analytic function, prove that $\left ( \dfrac {\partial ^2 } {\partial x^2 }+ \dfrac {\partial^2}{\partial y^2} \right )| f(z)|^2=4 |f'(z)|^2$
8 M
13 (a) (ii) Find the bilinear transformation which maps the points ∞, 2, -1 to 1, ∞ and 0 respectively.
8 M
13 (b) (i) Find the analytic function f=u+iv given that. u(x,y)=e2x (x sin 2y+ y cos 2 y).
8 M
13 (b) (ii) If f=u+iv is analytic on a domain D and |f| is constant on D, prove that f must be a constant on D.
8 M

Answer any one question from Q14 (a) & Q14 (b)
14 (a) (i) $If \ F(a)= \oint_c \dfrac {3z^2+7z+1}{z-a}dz$ where C:|z|=2 and |a|≠2, find F and F''(1-i).
8 M
14 (a) (ii) Evaluate $\int^\infty_0 \dfrac {x^2}{(x^2+a^2) (x^2 +b^2)}dx \$ by the method of contour integration, if a and b are positive.
8 M
14 (b) (i) Find the Laurent's series of $f(z)= \dfrac {3z-2}{z(z^2-4)}$ valid in the region 2<|z+2|<4.
8 M
14 (b) (ii) Using contour integration method show that $\int^{2x}_0 \dfrac {d\theta}{a+b \cos \theta}= \dfrac {2\pi}{\sqrt{a^2 -b^2}}$ if a>b>0.
8 M

Answer any one question from Q15 (a) & Q15 (b)
15 (a) (i) Find the Laplace transform of $f(t) = \dfrac {\sin^2 t}{t}$
4 M
15 (a) (i) Find the value of $\int^\infty_0 te^{3t} \cos 2tdt$
4 M
15 (a) (ii) Solve y''+9y=cos 2t given that y(0)=1 and y(π/2)=-1, by given by the method of Laplace transform.
8 M
15 (b) (i) Find $L^{-1} \left ( \log \dfrac {s^2 +1} { s (s+1)} \right )$
4 M
15 (b) (i) Using convolution theorem, find y if $L(y)= \dfrac {s}{(s^2 +a^2)^2}$
4 M
15 (b) (ii) Find L(f(t)) if $f(t) = \begin{cases}t & 0\le t\le a\\2a-t & a\le t\le 2a\end{cases}$ and $f(t+2a)= f(t)$
8 M

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