Answer any one question from Q1 & Q2

1 (a)
Solve the following differential equations. \[ (i) \ (x^4e^x-2mxy^2)dx+2mx^2 ydy=0 \\ (ii) \ \left ( \tan \dfrac {y}{x}- \dfrac {y}{x} \sec^2 \dfrac {y}{x} \right )dx+\sec^2 \dfrac {y}{x} dy =0 \]

8 M

1 (b)
A constant electromotive force E volts is applied to a circuit containing a constant resistance R ohms in series and a constant inductance L henries. If the initial current is zero, show that the current builds up to half its theoretical maximum in \[ \dfrac {L \log 2}{R} \ seconds. \]

4 M

2 (a)
Solve \[ \left [ \log (x^2+y^2) + \dfrac {2x^2}{x^2+y^2} \right ]dx+ \dfrac {2xy}{x^2+y^2}dy=0 \]

4 M

2 (b)
Solve the following :-

(i) A particle is moving in a straight line with an acceleration \[ k \left [ x+ \dfrac {a^4}{x^3} \right ] \] directed towards origin. If it starts from rest at a distance 'a' from the origin, prove that it will arrive at origin at the end of time \[ \dfrac {\pi}{4 \sqrt{k}} . \]

(ii) A pipe 10cm in diameter contains steam at 100° C. It is covered with asbestos,5cm thick,for which k=0.0006 and the outside surface is at 30°C .Find the amount of heat lost per hour from a meter long pipe.

(i) A particle is moving in a straight line with an acceleration \[ k \left [ x+ \dfrac {a^4}{x^3} \right ] \] directed towards origin. If it starts from rest at a distance 'a' from the origin, prove that it will arrive at origin at the end of time \[ \dfrac {\pi}{4 \sqrt{k}} . \]

(ii) A pipe 10cm in diameter contains steam at 100° C. It is covered with asbestos,5cm thick,for which k=0.0006 and the outside surface is at 30°C .Find the amount of heat lost per hour from a meter long pipe.

8 M

Answer any one question from Q3 & Q4

3 (a)
Express f(x)=π

^{2}-x^{2}, -π<x>π as a fourier series, where f(x)=f(x+2π)
5 M

3 (b)
Evaluate: [ int^{infty}_0 dfrac {x^8 - x^{14}}{(1+x)^{24}}dx ]

3 M

3 (c)
Trace the curve (Any one)

(i) y

(ii) r=2sin 3?

(i) y

^{2}=x^{2}(1-x)(ii) r=2sin 3?

4 M

4 (a)
show that the length of an arc of the curve

x=log (sec ? + \tan ?)-sin ?, y=cos ? from ?=0 to ?=t is log(sec t).

x=log (sec ? + \tan ?)-sin ?, y=cos ? from ?=0 to ?=t is log(sec t).

4 M

4 (b)
Evaluate: \[ \int^{\pi}_0 x \sin^5 x \cos^2 x \ dx \]

4 M

4 (c)
Evaluate: \[ \int^{1}_0 \left [ \dfrac {x^m-1}{\log x} \right ]dx \]

4 M

Answer any one question from Q5 & Q6

5 (a)
Find the equation of the sphere, having its center on the plane 4x - 5y - z = 3 and passing through the circle. x

^{2}+y^{2}+z^{2}-2x-3y+4z+8=0, x-2y+z=8.
5 M

5 (b)
Find the equation of a right cicular cone, having vertex at the point (0,0,3) and passing through the circle x

^{2}+y^{2}=16, z=0.
4 M

5 (c)
Find the equation of a right circular cylinder of radius 2,whose axis passes through the point (1,1,-2) and has direction cosines proportional to 2,1,2.

4 M

6 (a)
Find the equation of the sphere which is tangential to the plane 4x - 3y + 6z - 35 = 0 at (2,-1,4) and passing through the point (2,-1,-2).

5 M

6 (b)
Find the equation of a right circular cone with vertex at origin,the line x = y = 2z as the axis and semi-vertical angle 30°.

4 M

6 (c)
Find the equation of a right circular cylinder whose axis is 2(x-1) = y+2 = z and radius is 4.

4 M

Answer any one question from Q7 & Q8

7 (a)
Solve any two:

(a) Evaluate \[ \int^{a}_0 \int^{y}_{y^2/a}\dfrac {ydxdy}{(a-x)\sqrt{ax-y^2}} \]

(a) Evaluate \[ \int^{a}_0 \int^{y}_{y^2/a}\dfrac {ydxdy}{(a-x)\sqrt{ax-y^2}} \]

7 M

7 (b)
Evaluate \[ \int \int_y \int \sqrt{x^2+y^2}dxdydz, \] where V is bounded by the surface x

^{2}+y^{2}=z^{2}, z ? 0 and the plane z=1.
6 M

7 (c)
Find the moment of inertia (M.I.) about the line ? =?/2 of the area enclosed by r=a(1-cos ?).

6 M

8 (a)
Solve any two:

(a) Find by double integration the area between the curve y

(a) Find by double integration the area between the curve y

^{2}x=4a^{2}(2a-x) and its asymptote.
7 M

8 (b)
Find the volume of the cylinder x

^{2}+y^{2}=2ax intercepted between the paraboloid x^{2}+y^{2}=2az and xoy-plane.
6 M

8 (c)
Find the centre of gravity (C.G.)of one loop of the curve r = a sin?

6 M

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