# CSVTU Exams Questions Papers – Ist Year – Applied Mathematics-I – April-May- 2009

BE (2nd Semester)

Examination April/May, 2009

Applied Mathematics-II

UNIT-I

1. (a) State the De-Moiver’s theorem.

(b) Find all the roots of the equation

(i) cos z=2

(ii) tanh z=2

(c) Separate sin-1 (cos? + I sin?) into real and imaginary parts, where is a positive acute angle.

(d) Sum the series:

sin +n (n+1) sin 2an (n+1) (n+2) sin 3a+………….?

1.2                      1.2.3

UNIT-II

2. (a) Explain briefly the method of variation of parameters.

(b) Solve:

(D2+2) y=x2e3X+eX cos2x.

(c) Solve the equation:

(1+x)2d2y+(1+x)dy+y=sin[2log(1+x)]

dx2            dx

(d) Solve the simultaneous equation:

dx +2y=et and dy– 2x=e-t

dt                      dt

UNIT- III

3. (a) Write the relation between Beta and  Gamma function.

(b) Evaluate the integral by changing the order of integration.

a  a

?  ?             y2         dxdy

0 ?ax    ?y4– a2x2

(c) Evaluate:

?  ?

?     ?  e(x2+y2) dxdy

0      0

by changing iro polar co-ordinates.

Hence show that:

?

?  e—x2 dx= ??

0                       2

(d) Find by double integration, the area lying between the parabola

Y=4x-x2 and the line y = x.

UNIT- IV

4. (a) State the Green’s theorem in the plane.

(b) Prove that:

A.V (B.V 1)=3(A.R)(B.R.)  _ A.B

r         r5                r3

Where A and B is constant vectors.

R=xI + yJ + K and r = ?x+ y2 +z2

(c) A vector field is given by:

F=(x– y2 + x)I – (2xy +y)J

Show that the Field is irrotational and find it’s scalar potential. Hence evaluate the line integral

from (1,2) to (2,1).

(d) Verify stokes theorem for the vector field

F=(2x – y) I – yz2J – y2zK on the upper half surface of x2 + y1 + z2= 1 bounded by its projection on the x y palne.

UNIT- V

5. (a) Write the relation between roots and coefficient of the equation.

a0xn + a1xn-1 + a2xn-2 +……….+ an-1x + an = 0

(b) If a?? be the roots of x3 + px + q = 0

Show that:

(i) a3 + ?3 +?3 = 5 a?? a?

(ii) 3?a2?a55? a? a4

(c) Show that the equation x4 – 10x3 + 23x2 – 6x ?15= 0 can be transformed into reciprocal equation by diminishing the roots by 2. Hence solve the equation.

(d) Solve the cardon’s method, the equation:

27x3 + 54x2 + 198x – 73= 0