AU Mathematics 2 - May 2012 Exam Question Paper

AU First Year Engineering (Semester 2)
Mathematics 2
May 2012

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 Transform the equation

(2 x + 3)^{2} \frac{d^{2} y}{d x^{2}} - 2 (2 x + 3) \frac{d y}{d x} - 12 y = 6 x

$(2x+3)^2 \dfrac {d^2y}{dx^2}-2 (2x+3)\dfrac {dy}{dx}-12 y =6x$ 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} = (3 x + 2 y + z) \bar{z} + (4 x + λ y - z) \bar{j} + (x - y + 2 z) \bar{k}

$\bar{F}= (3x+2y+z)\bar{z}+ (4x+\lambda y-z)\bar{j}+ (x-y+2z)\bar{k}$ 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) = \frac{4}{z^{3} (Z - 2)}

$f(z)= \dfrac {4}{z^3 (Z-2)}$ 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 (D²+a²)y=sec ax using the method of variation of parameters.

8 M

11 (a) (ii) Solve: (D²-4D+3)y=e^x cos 2x.

8 M

11 (b) (i) Solve the differential equation

(x^{2} D^{2} - x D + 4) y = x^{2} \sin (\log x)

$\left ( x^2 D^2 -xD +4 \right )y=x^2 \sin (\log x)$

8 M

11 (b) (ii) Solve the simultaneous differential equations

\frac{d x}{d t} + 2 y = \sin 2 t, \frac{d y}{d t} - 2 x = \cos 2 t .

$\dfrac {dx}{dt}+2y = \sin 2t, \ \dfrac {dy}{dt}- 2x = \cos 2t.$

8 M

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

12 (a) (i) Show that

\vec{F} = (y^{2} + 2 x z^{2}) \bar{i} (2 x y - z) \vec{j} + (2 x^{2} z - y + 2 z) \vec{k}

$\overrightarrow{F}= \left ( y^2 + 2xz^2 \right )\bar {i} \left (2xy -z \right )\overrightarrow{j}+ \left ( 2x^2z-y+2z \right )\overrightarrow{k}$ is irrotational and hence find its scalar potential.

8 M

12 (a) (ii) Verify Green's theorem in a plane for

\int_{c} [(3 x^{2} - 8 y^{2}) d x + (4 y - 6 x y) d y],

$\displaystyle \int_c \left [ \left (3x^2 -8y^2 \right )dx+(4y-6xy)dy \right ],$ 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

\int_{c} \vec{F} \cdot d \vec{r} w h e r e \vec{F} = y^{2} \vec{i} + x^{2} \vec{j} - (x + z) \vec{k}

$\displaystyle \int_c \overrightarrow{F}\cdot d\overrightarrow{r} \ where \ \overrightarrow{F}= y^2 \overrightarrow{i}+ x^2 \overrightarrow{j}- (x+z) \overrightarrow{k}$ 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

\vec{F} = 3 x^{2} \vec{i} + (2 x z - y) \vec{j} + z \vec{k}

$\overrightarrow{F}=3x^2\overrightarrow{i}+ (2xz-y)\overrightarrow{j}+ z\overrightarrow{k}$ 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

(\frac{\partial^{2}}{\partial x^{2}} + \frac{\partial^{2}}{\partial y^{2}}) \log | f (z) | = 0

$\left (\dfrac{\partial^2}{\partial x^2}+ \dfrac {\partial^2}{\partial y^2} \right )\log |f(z)|=0$

8 M

13 (b) (ii) Show that the image of the hyperbola x²-y²=1 under the transformation ω=1/z is the lemniscate r² = cos 2θ

8 M

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

14 (a) (i) Evaluate

\int_{c} \frac{z d z}{(z - 1) (z - 2)^{2}} w h e r e C i s | z - 2 | = \frac{1}{2}

$\displaystyle \int_c \dfrac {zdz}{(z-1)(z-2)^2} \ where \ C \ is \ |z-2| = \dfrac {1}{2}$ by using Cauchy's integral formula.

8 M

14 (a) (ii) Evaluate

f (z) = \frac{1}{(z + 1) (z + 3)}

$f(z)= \dfrac {1}{(z+1)(z+3)}$ in Laurent series valid for the regions |z|>3 and 1<|z|<3.

8 M

14 (b) (i) Evaluate

\int_{c} \frac{z - 1}{(z + 1)^{2} (z - 2)} d z

$\int_c \dfrac {z-1}{(z+1)^2(z-2)}dz$ where C is the circle |z=i|=2 using Cauchy's residue theorem.

8 M

14 (b) (ii) Evaluate

\int_{0}^{\infty} \frac{\cos m x}{x^{2} + a^{2}} d x

$\int^{\infty}_{0}\dfrac {\cos mx}{x^2+a^2}dx$ using contour integration.

8 M

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

15 (a) (i) Apply convolution theorem to evaluate

L^{- 1} [\frac{s}{{(s^{2} + a^{2})}^{2}}]

$L^{-1} \left [ \dfrac {s}{\left ( s^2+a^2 \right )^2} \right ]$

8 M

15 (a) (ii) Find the Laplace transform of the following triangular wave function given by

f (t) {\begin{matrix} t, & 0 \leq t \leq π \\ 2 π - t, & π \leq t \leq 2 π \end{matrix} a n d f (t + 2 π) = f (t)

$f(t)\left\{\begin{matrix}t, & 0\le t \le \pi \\ 2\pi-t, & \pi \le t \le 2\pi\end{matrix}\right. \ and \ f(t+2\pi)=f(t)$

8 M

15 (b) (i) Find the Laplace transform of

\frac{e^{a t} - e^{- b t}}{t}

$\dfrac {e^{at}-e^{-bt}}{t}$

4 M

15 (b) (ii) Evaluate

\int_{0}^{\infty} t e^{- 2 t} \cos t d t

$\int^{\infty}_0 te^{-2t} \cos t \ dt$ using Laplace transform.

4 M

15 (b) (iii Solve the differential equation

\frac{d^{2} y}{d t^{2}} - 3 \frac{d y}{d t} + 2 y = e^{- t}

$\dfrac {d^2 y}{dt^2}-3 \dfrac {dy}{dt}+2y=e^{-t}$ with y(0)=1 and y'(0)=0 using Laplace transform.

8 M

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