Time: 2 hours

Maximum marks: 50

Notes :

1. Attempt Q. No. 1 or Q. No. 2, Q. No. 3 or Q. No. 4 and Q. No. 5 or Q. No. 6.

2. Neat diagram must be drawn wherever necessary.

3. Figures to the right indicate full marks.

4. Assume suitable data, if necessary and clearly state.

5. Use of cell phone is prohibited in the examination hall.

6. Use of electronic pocket calculator is allowed.

1. (a) Solve the following differential equations : [8M]

(i) $latex \frac{dx}{dy}=\frac xy+\cot\left(\frac xy\right)$

(ii) $latex \frac{dx}{dy}=e^{x-y}=4x^3e^{-y}$

(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 :

$latex i=\frac1{R-aL}\left[e^{-at}-e^{-\frac{Rt}L}\right]$

provided i = 0 at t = 0. [4M]

**OR**

2. (a) obtain a differential equation from its general solution :

y = c_{1}e^{4x} + c_{2}e^{-3x}, [4M]

Where c_{1}, c_{2} are arbitrary constants. [4M]

(b) Solve : [8M]

(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 :

$latex x=\frac m{2k}\log\left[\frac{a^2}{a^2-v^2}\right]$,

where mg = ka^{2}.

(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.

3. (a) Express f(x) = π^{2} – x^{2}, -π ≤ x ≤ π as a Fourier series where f(x) = f(x + 2π). [5M]

(b) Evaluate : $latex \int_0^1x^m(1-x^n)^pdx.$ [3M]

(c) Trace the curve (any one) : [4M]

(i) x = a(t + sint), y = a (1 – cost)

(ii) y^{2} = x^{2}(1 – x).

**OR**

4. (a) Find the perimeter of cardioid r = a(1 – cosθ). [4M]

(b) If $latex I_n=\int_0^{\pi/4}\cos^{2n}x\;dx$

prove that : $latex I_n=\frac1{n\cdot2^{n+1}}+\frac{2n-1}{2n}I_{n-1}.$ [4M]

(c) Evaluate : $latex \int_0^\infty\frac{x^4}{4^x}dx.$ [4M]

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

(b) Find the equation of the cone with vertex at (1, 2, 3), semivertical angle $latex \cos^{-1}\left(\frac1{\sqrt3}\right)$ and the line : $latex \frac{x-1}1=\frac{y-2}2=\frac{z+1}{-1}$ as axis of the cone. [4M]

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

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

x – y + z = 3. [4M]

**OR**

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. [15M]

(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 linen :

$latex \frac{x-2}2=\frac{y-1}1=\frac{z+2}2$ [4M]

(c) Find the equation of the right circular cylinder whose axis is :

$latex \frac{x-2}2=\frac{y-1}1=\frac z3$

and which passes through the point (0, 0, 3) [4M]

7. Attempt any two of the following :

(a) Evaluate by changing the order of integration : [7M]

$latex \int_0^\infty\int_0^xxe^{-x^2/y}\;dy\;dx.$

(b) Find the volume of solid common to the cylinders : [6M]

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

x^{2} + z^{2} = a^{2}.

(c) Find the moment of inertia of the circular plate r = 2a cos θ about θ = π/2 line. [6M]

**OR**

8. Attempt any two of the following :

(a) Find the total area of the astroid : [7M]

x^{2/3} + y^{2/3} = a^{2/3}.

(b) Evaluate :

$latex \iiint_v\sqrt{x^2+y^2}\;dx\;dy\;dz,$

where v is the volume of the cone x^{2} + y^{2} = z^{2}, z > 0 bounded by z = 0 and z = 1 plane. [6M]

(c) Find centre of gravity of area of the cardioid : [6M]

r = a(1 + cosθ).

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