Solve the following differential equation, Solve any one question from Q.1(a, b) & Q.2 (a, b)

1(a)(i)
\( \dfrac{dx}{dy}=\dfrac{x}{y}+ \cot\left ( \dfrac{x}{y} \right ) \)/

4 M

1(a)(ii)
\( \dfrac{dx}{dy}=e^{x-y}=4x^3e^{-y} \)

4 M

1(b)
A voltage e

^{-at}is applied at t = 0 to a circuit containing inductance L, and resistance R. Show that the current at any time t is given by : Provided i = 0 at t = 0 \[i=\dfrac{1}{R-aL}\left [ e^{-at} -e^{-\dfrac{Rt}{L}}\right ]\]
4 M

2(a)
Obtain a differential equation from its general solution

y = c

where c

y = c

_{1}e^{4x}+ c_{2}e^{-3x},where c

_{1}, c_{2}are arbitrary constants.
4 M

2(b)(ii)
The temperature of air is 30°C. The substance kept in air cools from 100°C to 70°C in 15 minutes. Find the time required to reduce the temperature of the substance upto 40°C.

4 M

Solve

2(b))(i)
A body of mass m, falling from rest, is subject to the force of gravity and an air resistance proportional to the square of velocity i.e. kv

^{2}, where k is a constant of proportionality. If it falls through a distance x and possesses a velocity v at that instant, show that : where mg = ka^{2}\[x=\dfrac{m}{2k}\log\left [ \dfrac{a^2}{a^2-v^2} \right ]\]
4 M

Solve any one question from Q.3(a, b, c) & Q.4 (a, b, c)

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 : \( \int ^1_0 x^m(1-x^n)^p dx. \)

3 M

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

(i) x = a(t + sint), y = a(1 cost)

(ii) y

(i) x = a(t + sint), y = a(1 cost)

(ii) y

^{2}= x^{2}(1 x).
4 M

4(a)
Find the perimeter of cardioid r = a(1 + cosθ).

4 M

4(b)
If \( I_n=\int ^{\pi/4}_0 \cos^{2n}x\ dx
\) Prove that : \( I_n=\dfrac{1}{n \cdot 2^{n+1}}+\dfrac{2n-1}{2n}I_{n-1}. \)

4 M

4(c)
Evaluate : \( \int ^{\infty}_0 \dfrac{x^4}{4^x}dx. \)

4 M

Solve any one question from Q.5(a, b, c) & Q.6 (a, b, c)

5(a)
Find the equation of the sphere which touches the coordinate axes, whose centre is in the positive octant and has radius 4.

5 M

5(b)
Find the equation of the cone with vertex at (1, 2, -3), semivertical angle \( \cos^{-1}\left ( \dfrac{1}{\sqrt{3}} \right ) \) and the line : \( \dfrac{x-1}{1}=\dfrac{y-2}{2}=\dfrac{z+1}{-1} \) as axis of the cone.

4 M

5(c)
Find the equation of the right circular cylinder whose guiding curve is :

x

x y + z = 3.

x

^{2}+ y^{2}+ z^{2}= 9,x y + z = 3.

4 M

6(a)
Find the centre and radius of the circle of intersection of the sphere x

^{2}+ y^{2}+ z^{2}- 2y - 4z - 11 = 0 by the plane x + 2y + 2z = 15.
5 M

6(b)
Obtain the equation of a right circular cone which passes through the point (2, 1, 3) with vertex (2, 1, 1) and axis parallel to the line : \[\dfrac{x-2}{2}=\dfrac{y-1}{1}=\dfrac{z+2}{2}\]

4 M

6(c)
Find the equation of the right circular cylinder whose axis is : \( \dfrac{x-2}{2}=\dfrac{y-1}{1}=\dfrac{z}{3} \) and which passes through the point (0, 0, 3).

4 M

Attempt any two of the following, Solve any one question from Q.7(a, b, c) & Q.8 (a, b, c)

7(a)
Evaluate by changing the order of integration : \[\int^{\infty}_0 \int ^x_0xe^{-x^2/y}\ dy\ dx.\]

7 M

7(b)
Find the volume of solid common to the cylinders :

x

x

x

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

^{2}+ z^{2}= a^{2}.
6 M

7(c)
Find the moment of inertia of the circular plate r = 2a cosθ about θ = π/2 line.

6 M

8(a)
Find the total area of the Astroid :

x

x

^{2/3}+ y^{2/3}= a^{2/3}.
7 M

8(b)
Evaluate: \[\iiint _v \sqrt{x^2+y^2}\ dx\ dy\ dz,\]

whereV is the volume of the cone x

whereV is the volume of the cone x

^{2}+ y^{2}= z^{2}, z > 0 bounded by z = 0 and z = 1 plane.
6 M

8(c)
Find centre of gravity of area of the cardioid :

r = a(1 + cosθ).

r = a(1 + cosθ).

6 M

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