Solve any one question from Q1 & Q2

1 (a)
Solve the following:

i) \[ i) \ \dfrac {dy}{dx} = \dfrac {x+y-2}{y-x-4} \\ ii) \ \dfrac {dy}{dx} =x^2 \cos^2 y -x \sin 2y \]

i) \[ i) \ \dfrac {dy}{dx} = \dfrac {x+y-2}{y-x-4} \\ ii) \ \dfrac {dy}{dx} =x^2 \cos^2 y -x \sin 2y \]

8 M

1 (b)
An e.m.f. 200 e

^{-5t}is applied to a series circuit consisting 20? resistor and 0.01 F capacitor. Find the charge and current at any time, assuming that there is no initial change on capacitor.
4 M

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

4 M

2 (b)
Solve the following

i) Find the orthogonal trajectory to of x

A body originally at 85°C cools to 65°C in 25 minutes, the temperature of air being 40°, what will be the temperature of the body after 40 minutes.

i) Find the orthogonal trajectory to of x

^{2}+cy^{2}=1.A body originally at 85°C cools to 65°C in 25 minutes, the temperature of air being 40°, what will be the temperature of the body after 40 minutes.

8 M

Solve any one question from Q3 & Q4

3 (a)
Find the fourier expansion for y in terms x upto first harmonic as given in following table.

x° | 0 | 30 | 60 | 90 | 120 | 150 | 180 | 210 | 240 | 270 | 300 | 330 |

y | 10.5 | 20.2 | 26.4 | 29.3 | 27 | 21.5 | 12.5 | 1.6 | -19.2 | -18.0 | -15.8 | -0.4 |

5 M

3 (b)
\[ Evaluate: \ \int^\infty_0 \sqrt[4]{x}e^{\sqrt[-]{x}}dx \]

3 M

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

i) x=a(t-sin t), y=a(1-cos t)

ii) r=a sin 3?

i) x=a(t-sin t), y=a(1-cos t)

ii) r=a sin 3?

4 M

4 (a)
\[ If \ l_n = \int^{\pi /4}_0 \cos^{2n}x dx, \ prove \ that l_n = \dfrac {1}{2^{n+1}}+ \dfrac {2n-1}{2n}l_n \]

4 M

4 (b)
Prove that \[ \phi (a) = \int^{\pi/2a}_{\pi/6a} \dfrac{\sin ax}{x} dx \] is independent of 'a'.

4 M

4 (c)
Find the length of the arc of cardioide r=a (1-cos ?) which lies outside the circle r=a cos ?.

4 M

Solve any one question from Q5 & Q6

5 (a)
Find the equation of the sphere tangential to the plane x-2y-2z=7 at (3, -1, -1) and passing through the point (1, 1, -3).

5 M

5 (b)
Find the equation of the right circular cone which passes through the point (1, 1, 2) has its axis at the line ?x= -3y=4z and vertex at origin.

4 M

5 (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

6 (a)
A sphere s has points (1, -2, 3) and (4, 0, 6) as opposite ends of a diameter. Find the equation of the sphere having the intersection of s with the plane x+y-2z=6=0 as its great circle.

5 M

6 (b)
Find the equation of right circular cone whose vertex is (1, 2, 3) and the axis is given by \[ \dfrac {x-1}{2} = \dfrac {y-2}{-1}= \dfrac {z-3}{4} \] and semi-vertical angle is 60°.

4 M

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

4 M

Answer any two from Q7

7 (a)
Change the order of integration and evaluate: \[ \int^\infty_0 \int^{\infty}_x \dfrac {e^{-y}}{y}dx \ dy \]

6 M

7 (b)
Find the volume of the tetrahydron bounded by the co-ordinate planes and the plane \[ \dfrac {x}{a}+ \dfrac {y}{b}+ \dfrac {z}{c} =1 \]

6 M

7 (c)
Find the centre of gravity of one loop of the curve r=a sin 2?.

7 M

8 (a)
\[ Evaluate: \iint_R \sin (x^2+y^2)dx \ dy \] where R is circle x

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

8 (b)
Find the total area included between the two cardiodes r=ac (1+ cos ?) and r=a (1-cos ?).

7 M

8 (c)
Find the moment of inertia about x-axis of the area enclosed by the lines x=0, y=0 \[ \dfrac {x}{a} + \dfrac {y}{b}=1 \]

6 M

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