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) $latex x^4\frac{dx}{dy}+x^3y=\;sec(xy)$

(ii) $latex \frac{dx}{dy}=\frac{1+y^2+3x^2y}{1-2xy-x^3}$

(b) A body start moving from rest is opposed by a force per unit mass of value cx and resistance per unit mass of a value bv^{2} where x and v are the displacement and velocity of the particle at that instant. Show that the velocity of the particle is given by :

$latex v^{2}=\frac{c}{2b^{2}}\left ( 1-e^{-2bx} \right )-\frac{cx}{b}$ [4M]

**OR**

2. (a) Solve :

$latex \frac{dy}{dx}+x\sin 2y=x^{3} \cos^{2} y.$ [4M]

(b) Solve the following : [8M]

(i) Water at temperature 100°C cools in 10-minutes to 88°C in a room of temperature 25°C. Find the temperature of water after 20 minutes.

(ii) A resistance of 100 Ω, an inductance of 0.5 henry are connected in a series with battery of 20 volts. Fing the current in a circuit as a function of time t.

3. (a) Find Fourier series to represent the function f(x) = x in – π < x < π and f(x) = f(x + 2π). [5M]

(b) Evaluate :

$latex \int_{0}^{\infty \sqrt{y}}.e^{-\sqrt{y}}dy$[3M]

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

(i) $latex y^{2}\left ( x^{2}-1 \right )=x$

(ii) $latex r = a(1+\cos \theta ).$

**OR**

4. (a) If :

$latex I_n=\int_{\mathrm\pi/4}^{\mathrm\pi/2}cot^n\theta\;d\theta,$

Prove that :

$latex I_n=\frac1{n-1}-I_{n-2}.$ [4M]

(b) Prove that :

$latex \int_0^1\frac{x^a-x^b}{\log x}dx=\log\left(\frac{a+1}{b+1}\right),\;a>0,\;b>0$ [4M]

(c) Find the length of the are of cycloid $latex x=a\left(\theta+\sin\theta\right),y=a\left(1-\cos\theta\right)$ between two consecutive cusps. [4M]

5. (a) Find the centre and radius of the circle which is an intersection of the sphere $latex x^2+y^2+z^2-2y-4z-11=0$ by the plane $latex x+2y+2z=15.$ [5M]

(b) Find the equation of the right circular cone which passes through the point (1, 1, 2) and has its axis along the line 6x = – 3y = 4z and vertex at (0, 0, 0). [4M]

(c) Find the equation of a right circular cylinder of radius 2 whose axis passes through (1, 2, 3) and has direction cosines proportional to 2, -3, 6. [4M]

**OR**

6. (a) Show that the plane 4x – 3y + 6z – 35 = 0 is tangential to the sphere $latex x^2+y^2+z^2-y-2z-14=0.$ [5M]

(b) Find the equation of a right circular cone whose vertex is at (1, 2, 3) and axis has direction ratios (2, -1, 4) and semivertical angle 60°. [4M]

(c) Find the equation of the right circular cylinder of radius 3 whose axis is the line

$latex \frac{x-1}2=\frac{y-3}2=\frac{z-5}{-1}.$ [4M]

7. Attempt any two of the following :

(a) Evaluate : $latex \iint\frac{x^2y^2dxdy}{x^2y^2}$

where R is annulus between $latex x^2+y^2=4,x^2+y^2=9.$ [6M]

(b) Evaluate : $latex \iiint\left(x^2y^2+y^2z^2+z^2x^2\right)dxdydz$

throughout the volume of sphere $latex x^2+y^2+z^2=a^2.$ [7M]

(c) Find the moment of inertia of one loop of lemniscate $latex r^2=a^2\cos2\theta$ about initial line. [6M]

**OR**

8. Attempt any two of the following

(a) Find the total area included between the two cardioids.

$latex r=a\left(1+\cos\theta\right)\;\;\text{and}\;r=a\left(1-\cos\theta\right).$ [6M]

(b) Find the volume cut-off from the paraboloid $latex x^2+\frac{y^2}4+z=1$ by the plane z = 0. [7M]

(c) Find the C.G. of an area of the cardioid

$latex r=a\left(1+\cos\theta\right).$ [6M]

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