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 M

2
Find the particular integral of (D-1)

^{2}y=e^{x}sin x.
2 M

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 M

4
State Gauss divergence theorem.

2 M

5
State the basic difference between the limit of a function of a real variable and that of a complex variable.

2 M

6
Prove that a bilinear transformation has atmost two fixed points.

2 M

7
Define singular point.

2 M

8
Find the residue of the function \[ f(z)= \dfrac {4}{z^3 (Z-2)} \] at a simple pole.

2 M

9
State the first shifting theorem on Laplace transforms.

2 M

10
verify initial value theorem for f(t)=1+e

^{t}(sin t+ cos t).
2 M

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

11 (a) (i)
Solve (D

^{2}+a^{2})y=sec ax using the method of variation of parameters.
8 M

11 (a) (ii)
Solve: (D

^{2}-4D+3)y=e^{x}cos 2x.
8 M

11 (b) (i)
Solve the differential equation \[ \left ( x^2 D^2 -xD +4 \right )y=x^2 \sin (\log x) \]

8 M

11 (b) (ii)
Solve the simultaneous differential equations \[ \dfrac {dx}{dt}+2y = \sin 2t, \ \dfrac {dy}{dt}- 2x = \cos 2t. \]

8 M

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 M

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 M

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 M

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 M

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 M

13 (a) (ii)
Find the bilinear transformation which maps the points z=0,1,? into w=i,1, -1 respectively.

8 M

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 M

13 (b) (ii)
Show that the image of the hyperbola x

^{2}-y^{2}=1 under the transformation ω=1/z is the lemniscate r^{2}= cos 2θ
8 M

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 M

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 M

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 M

14 (b) (ii)
Evaluate \[ \int^{\infty}_{0}\dfrac {\cos mx}{x^2+a^2}dx \] using contour integration.

8 M

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 M

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 M

15 (b) (i)
Find the Laplace transform of \[ \dfrac {e^{at}-e^{-bt}}{t} \]

4 M

15 (b) (ii)
Evaluate \[ \int^{\infty}_0 te^{-2t} \cos t \ dt \] using Laplace transform.

4 M

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 M

More question papers from Mathematics 2