# CSVTU Exam Papers – BE I Year – Applied Mathematics-Ii – Nov-Dec- 2006

BE (1st Semester)

Examination Nov-Dec- 2006

Applied Mathematics-II

UNIT-I

1. (a) If i………..∞ = A + iB∞ Prove that

tan A = B and A2 + B2= e-xB

2     A

(b) Show that

Sin2ө1 sin2өsin2 ө + 1 sin 3ө sin3 ө1  sin 4ө

2                         3                          4

Sin4ө +……….∞=tan-1(sin2 ө /(1 + cos ө sin ө)

(c) If (ө + iф) = reiu prove that

ф= 1 log  sin(ө – a)

2         sin(ө + a)

UNIT- II

2. (a) Solve the following:

dx + y= sin t. dx + x= cos t.

dt                    dt

Given that, x=2, y=0 when t=0

(b) Solve the differential equation

x3 d3y  +  2x2 d2y  + 2y = 10( x + 1)

dx3              dx2                           X

(c) Using method of variation of parameters, Solve:

d2y + 4y = tan 2x

dx2

UNIT- III

3. (a) Evaluate the following integral by changing of order integration:

ʃ0u   ʃaax    y2dxdy

y4 – a2 x2

(b) Find the volume common to gas cylinders x2 + y2 = a3

x2 + z2 = a2

(c) Define β function and show that:

β(m,n) √m√n

√m+n

UNIT- IV

4. (a) Verify Green’s theorem for:

ʃ[(xy + y2)dx + x2 dy]

Where C is bounded by line y=x and curve y=x2

(b) Evaluate ʃf, ds where

F=4xi – 2y2j + z2k and s is

The surface bounding the region x2 + y2 =4, z = 0 and z=3

(c) Evaluate  ʃf * dR where F= 2yi- zj + xk and the curve X= Cost, Y= sin t, Z= 2 cost from t=0 to t/2

c

UNIT- V

5. (a) Solve by Cardon’s method:

x3 + 3x2 + 3=0

(b) Solve the equation:

6x5 + x4 – 43x3– 43x2 + + 6=0

(c) If a, β,ϒ are the equation x3 + qx + r=0 find the equation whose roots are:

β + ϒ , ϒ + a , a + β

ϒ    β   a    ϒ   β     a