# 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 *A ^{2} + B*

^{2= }

*e*

^{-xB}2 *A*

(b) Show that

Sin^{2}*ө* – 1 sin2*ө*sin^{2} *ө* + 1 sin 3*ө* sin^{3}* ө* – 1 sin 4*ө*

2 3 4

Sin^{4}*ө *+……….∞=tan^{-1}(sin^{2}* ө* /(1 + cos *ө *sin *ө*)

(c) If (*ө *+ *i*ф*) *= *re** ^{iu}* 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

x^{3} d^{3}y + 2x^{2} d^{2}y + 2y = 10( x + 1)

dx^{3 } dx^{2 } X

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

d^{2}y + 4y = tan 2x

dx^{2}

**UNIT- III**

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

ʃ_{0}* ^{u}* ʃ

^{a}_{√}

*y*

_{ax}^{2}dxdy

√*y ^{4} – a^{2} x^{2}*

(b) Find the volume common to gas cylinders x^{2} + y^{2} = a^{3}

x^{2} + z^{2 }= a^{2}

(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 + y^{2})dx + x^{2} dy]

Where C is bounded by line y=x and curve y=x^{2}

(b) Evaluate ʃ*f*, *ds* where

F=4xi – 2y^{2}*j *+ z^{2}k and s is

The surface bounding the region x^{2} + y^{2} =4, z = 0 and z=3

(c) Evaluate ʃ*f* * 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:

x^{3} + 3x^{2} + 3=0

(b) Solve the equation:

6x^{5} + x^{4} – 43x^{3}– 43x^{2} + + 6=0

(c) If *a, β,**ϒ** *are the equation x^{3} + qx + r=0 find the equation whose roots are:

*β** + **ϒ** , ϒ + **a** ,** a** + β*

* ** ϒ **β a **ϒ **β a *

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