Answer any one question from Q1 and Q2

1 (a)
Solve the following differential equations: \[ i) \ \ \dfrac {dy}{dx} = \cos x \cos y + \sin x \sin y \\ ii) \ (x^2 + y^2 +1) dx - 2xy \ dy =0 \]

8 M

1 (b)
In a circuit containing inductance L, resistance R and voltage E, the current I is given by: \[ E=RI + L \dfrac {dI}{dt} \] Given:

L=640H, R=250 Ω, E=500 Volts. I being zero when t=0. Find the time that elapses before it reaches 80% of its maximum value.

L=640H, R=250 Ω, E=500 Volts. I being zero when t=0. Find the time that elapses before it reaches 80% of its maximum value.

4 M

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

4 M

2 (b)
Solve the following: i) A body at temperature 100°C is placed in a room whose temperature is 20°C and cools to 60°C in 5 minutes. Find its temperature after a further interval of 3 minutes.

(ii) A steam pipe 20 cm in diameter is protected with a covering 6 cm thick for which the coefficient of thermal conductivity is k = 0.003 cal/cm deg. sec in steady state. Find the heat lost per hour through a meter length of the pipe, if the surface of pipe is at 200°C and outer surface of the covering is at 30°C.

(ii) A steam pipe 20 cm in diameter is protected with a covering 6 cm thick for which the coefficient of thermal conductivity is k = 0.003 cal/cm deg. sec in steady state. Find the heat lost per hour through a meter length of the pipe, if the surface of pipe is at 200°C and outer surface of the covering is at 30°C.

8 M

Answer any one question from Q3 and Q4

3 (a)
Find a half range cosine series of f(x) =πx-x

^{2}in the interval 0
5 M

3 (b)
Evaluate: \[ \int^\infty_0 \dfrac {x^3}{3^x} dx \]

3 M

3 (c)
Trace the following curve (any one):

i) y

ii) r=a sin 2θ

i) y

^{2}=x^{5}(2a-x)ii) r=a sin 2θ

4 M

4 (a)
\[ If \ I_n = \int^{\pi /2}_{\pi /4} \cot^n \theta d \theta \\ prove \ that \ I_n=\dfrac{1}{n-1}- I_{n-2}. Hence \ evaluate \ I_3. \]

4 M

4 (b)
Using differentiation under Integral sign prove that: \[ \int^\infty _{0} \dfrac {e^{-x}-e^{-ax}}{x \sec x}dx = \dfrac {1}{2} \log \left ( \dfrac {a^2+1}{2} \right ) \] for a>0.

4 M

4 (c)
Find the length of the curve

x=a(θ- sin θ), y=a (1-cos θ) between θ=0 to θ=2 π.

x=a(θ- sin θ), y=a (1-cos θ) between θ=0 to θ=2 π.

4 M

Answer any one question from Q5 and Q6

5 (a)
Show that the plane 4x-3y+6z-35=0 is tangential to the sphere x

^{2}+y^{2}+z^{2}-z-2z-14=0 and find the point of contact.
5 M

5 (b)
Find the equation of the right circular cone whose vertex is given by (-1, -1, 2) and axis is the line \[ \dfrac {x-1}{2} = \dfrac {y+1}{1} = \dfrac {z-2}{-2} \] and semi-vertical angle is 45°.

4 M

5 (c)
Find the equation of right circular cylinder of radius 2 and axis is given by:

\[ \dfrac {x-1}{2} = \dfrac {y-2}{-3}= \dfrac {z-3}{6} \]

\[ \dfrac {x-1}{2} = \dfrac {y-2}{-3}= \dfrac {z-3}{6} \]

4 M

6 (a)
Find the equation at the sphere through the circle x

^{2}+y^{2}+z^{2}=1, 2x+3y+4z=5 and which intersects the sphere x^{2}+y^{2}+z^{2}+3 (x-y+z)-56=0 orthogonally.
5 M

6 (b)
Find the equation of right circular cone with vertex at origin
making equal angles with the co-ordinate axes and having generator with direction cosines proportional to 1, ?2, 2.

4 M

6 (c)
Obtain the equation of the right circular cylinder of radius 5
where axis is: \[ \dfrac {x-2}{3}= \dfrac {y-3}{1}= \dfrac {z+1}{1} \]

4 M

Attempt any two of the following:

7 (a)
Change the order of integration in the double integral: \[ \int^5_0 \int^{2+x}_{2-x} f(x,y) dy \ dx \]

6 M

Answer any one question from Q7 and Q8

7 (b)
Evaluate: \[ \int^2_0 \int^x_0 \int^{2x+2y}_0 e^{x+y+z}dx \ dy \ dz \]

7 M

7 (c)
Find the centroid of the loop of the curve: r

^{2}=a^{2}cos 2 θ.
6 M

Attempt any two of the following:

8 (a)
Evaluate: \[ \int^a_0 \int^{\sqrt{a^2-x^2}}_0 e^{-x^2 - y^2}dx dy. \]

6 M

8 (b)
Evaluate: \[ \iiint \sqrt{1- \dfrac {x^2}{a^2} - \dfrac {y^2}{b^2} - \dfrac {z^2}{c^2}}dx \ dy \ dz \] through the volume of ellipsoid \[ \dfrac {x^2}{a^2} + \dfrac {y^2}{b^2}+ \dfrac {z^2}{c^2}=1 \]

6 M

8 (c)
Prove that the moment of inertia of the area included between the curves y

^{2}=a ax and x^{2}=4ay about x-axis is 144/35 Ma^{2}where M is the mass of the area included between the curves.
7 M

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