**BE (1st Semester)**

**Examination May-June, 2007**

**Applied Mathematics-II**

**UNIT-I**

1. (a) If x + 1 = 2 cos *ө *and y + 1= 2cos *ө*

x x

Show that one of the value of

*xmyn + 1 is 2cos(m ө + n*ф)* *

* xm + yn *

(b) If (*ө *+ *i*ф) *e ^{a}* show that

*ө = (n + 1/2)**∏**/2 and*

* * ф=1 log tan (*∏** * + *a**)*

2 4 2

(c) Sum the series by C+ is method:

xsin* ө *– 1 x^{2} sin2* ө* +1x^{3} sin 3*ө* ………∞

2 3

**UNIT- II**

2. (a) Solve the equation:

d^{2}y – 2dy + y= xe^{x} sin x.

dx^{2} dx

(b) Solve by the method of variation of parameter:

d^{2}y + y= x sin x

dx^{2}

(c) Solve the equation

x^{2}d^{2}y – 3xdy + 4y= (1+x)^{2}

dx^{2} dx

**UNIT- III**

3. (a) Change the order of the integration in

2-x

* ^{l}*ʃ

_{0}ʃ xy dx dy

and hence evaluate same.

(b) Find the volume bounded by the cylinder x^{2} + y^{2} = 4 and the planes

y + z = 4 and z = 0

(c) Solve ʃ ʃxy(x^{2} + y^{2})^{n/2} dxxy over the positive quadrant of

x^{2} + y^{2} = 4 sup *po*sin g n + 3> 0

**UNIT- IV**

4. (a) find the work done in moving a particle in the force field:

F= 3x^{3}*I* + (2xz – y)*j *+ zk along:

(i) the straight line form (0,0,0) to (2,1,3)

(ii) the curve defined by x^{2} =4y, 3x^{2} =8z from x = 0 x = 2

(b) Evaluate divergence & curl of

F = 3x^{2} *I* + 5xy^{2} *j* +xyz^{3} K

At the point (1, 2, 3)

(c) Use divergence theorem to evaluate ʃF *ds* where

F = 4x*l* – 2y^{2} *j** *+ z^{2} K and S is the surface bounding the region

x^{2} + y^{2} = 4, z = 0, z = 3

**UNIT-V**

5. (a) Solve the equation by Ferrari’s method:

x^{4 }– 12x^{2} + 41x^{2} – 18x- 72 = 0

(b) Solve the equation by Cardin’s method:

x^{3} – 15x – 126 = 0

(c) Solve x^{3} – 4x^{2} – 20x + 48 = 0 given that the roots *a* and *β* are connected by the relation *a* + 2*β* = 0