Solve any one question from Q1 & Q2

1 (a)
Solve the following \[i) \ \dfrac {dy}{dx} = \dfrac {x^2 + y^2 +1}{2xy} \\ ii) \ (1-x^2) \dfrac {dy} {dx} = 1+ xy \]

8 M

1 (b)
An electric circuit contains an inductance of 0.5 henries and a resistance of 100 ohms in series with an e.m.f. of 20 volts. Find the current at any time t, if it, is zero at t=0.

4 M

2 (a)
Solve: (2x+3y-1)dx=(6x+9y+6)dy

4 M

2 (b)
Solve the following:

i) A bullet is fired sand tank, its retardation is proportinal to square root of its velocity. Show that the bullet will come to rest in time \[ \dfrac {2\sqrt{v}} {k} \] where V is initial velocity.

ii) A pipe 20 cm in diameter steam at 150°C and is protected with a covering 5cm thick for which k=0.0025. If the temperature of the outer surface of the covering is 40°C, find the temperature half-way through the covering under steady state conditions.

i) A bullet is fired sand tank, its retardation is proportinal to square root of its velocity. Show that the bullet will come to rest in time \[ \dfrac {2\sqrt{v}} {k} \] where V is initial velocity.

ii) A pipe 20 cm in diameter steam at 150°C and is protected with a covering 5cm thick for which k=0.0025. If the temperature of the outer surface of the covering is 40°C, find the temperature half-way through the covering under steady state conditions.

8 M

Solve any one question from Q3 & Q4

3 (a)
Find the half range cosine series for the function F(x)=x-x

^{3}, 0?x?1
5 M

3 (b)
\[ Evaluate \ int^5_2 (x-2)^2 (5-x)^2 dx \]

3 M

3 (c)
Trace the curve (any one)

i) x=a(t+sin t), y=a(l-cos t)

ii) x

i) x=a(t+sin t), y=a(l-cos t)

ii) x

^{2}y^{2}=a^{2}(y^{2}-x)
4 M

4 (a)
\[ If \ I_n = \int^\infty_0 e^{-x} \sin^n x \ dx \] obtain the relation between I

_{n}and I_{n2}.
4 M

4 (b)
Show that \[ \int^b_a e^{-x^2} dx = \dfrac {\sqrt{\pi}}{2} [erf (b)- erf (a)] \]

4 M

4 (c)
Find the arclength of one loop of Lemniscate r

^{2}=a^{2}= cos 2?
4 M

Solve any one question from Q5 & Q6

5 (a)
Find the equation of right circular cylinder of radius a, whose axis passes through the origin and makes equal angles with the coordinate axes.

4 M

5 (b)
Lines are drawn from the origin with direction co-sines proportional to (1,2,2), (2,3,6)(3,4,12). Find direction co-sines of the axis of right circular cone through them, and prove that the semivertical angle of cone is \[ \cos^{-1} \dfrac {1}{\sqrt{3}} \]

4 M

5 (c)
Find the equation of the sphere which passes through the points (1,-4,3)(1,-5,2)(1,-3,0) and whose centre lies on the plane x+y+z=0

5 M

6 (a)
A sphere of constant radius K passes through the origin and meets the axes in A,B,C prove that the centroid of the triangle ABC lies on the sphere g(x

^{2}+y^{2}+z^{2}) =4K^{2}
5 M

6 (b)
Find the equation of the right circular cone which has its vertex at the point (0,0,10) and whose intersection with the plane XOY is a circle of diameter 10.

4 M

6 (c)
Find the equation of the right circular cylinder of radius 3 and axis \[ \dfrac {x-1}{2} = \dfrac {y-3}{2} = \dfrac {z-5}{-1} \]

4 M

Answer any two from Q7

7 (a)
\[ Evaluate \ \iint (x+y)^2 dxdy \ over \ the \ area \ bounded \ by \ an ellipse \ \dfrac {x^2}{a^2} + \dfrac {y^2}{b^2} =1 \]

7 M

Solve any one question from Q7 & Q8

7 (b)
Find the volume of the tetrahydron bounded by the co-ordinate planes and the plane x+y+z=1

6 M

7 (c)
Find the moment of inertia about the initial line of the cardiode r=a(1+cos ?)

6 M

Answer any two from Q8

8 (a)
Find the area bounded by the parabola y=x

^{2}& the Line y=2x+3.
7 M

8 (b)
\[ Evaluate \ \iiint z(x^2 + y^2)dxdydz \] over the volume of the cylinder x

^{2}+y^{2}=1 intercepted by the planes z=2 and z=3.
6 M

8 (c)
Find the x-co-ordinate of centre of gravity of an area bounded by the parabola y

^{2}=x and the line x+y=2.
6 M

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