# 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:

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

* * 1.2 1.2.3

**UNIT-II**

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

(b) Solve:

(D^{2}+2)* y*=x^{2}*e*^{3}*X*+*e**X* cos2x.

(c) Solve the equation:

(1+x)^{2}*d ^{2}y*+(1+x)

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

*dx2 dx*

* *(d) Solve the simultaneous equation:

*dx +2y=e ^{t} 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*

* * ? ? y^{2} *dxdy*

0 ?*ax * ?*y**4**– a*^{2}x^{2}

(c) Evaluate:

?* *?

* * ? ? *e ^{—}*

^{(}x

^{2}+y

^{2)}

*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-x^{2} 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 r^{5} r^{3}

^{ } Where A and B is constant vectors.

R=*xI + yJ + K and r = *?x^{2 }+ y^{2} +z^{2}

(c) A vector field is given by:

*F=(x ^{2 }– y^{2} + 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* – yz^{2}*J** – y*^{2}zK on the upper half surface of x^{2} + y^{1} + z^{2}= 1 bounded by its projection on the x y palne.

**UNIT- V**** **

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

*a _{0}x^{n} + a_{1}x^{n-1} + a_{2}x^{n-2} +……….+ a_{n-1}x + a_{n} = 0*

(b) If *a??* be the roots of x^{3} + px + q = 0

Show that:

(i) *a ^{3} + ?^{3} +?^{3} = 5 a?? *?

*a?*

(ii) 3?*a ^{2}*?

*a*5?

^{5}=*a*?

*a*

^{4}

(c) Show that the equation x^{4} – 10x^{3} + 23x^{2} – 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:

27x^{3} + 54x^{2} + 198x – 73= 0

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