Solve any one question from Q.1(a,b) & Q.2(a,b)

1(a)
Solve the following differential equations:\[\begin{align*}& i)x^{4}\frac{dy}{dx}+x^{3}y=sec(xy)\\
&ii)\frac{dy}{dx}=\frac{1+y^{2}+3x^{2}y}{1-2xy-x^{3}}\end{align*}\]

8 M

1(b)
A body starts 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:\[v^{2}=\frac{c}{2b^{2}}\left ( 1-e^{-2bx} \right )-\frac{cx}{b}\]
4 M

2(a)
Solve: \[\frac{dy}{dx}+x\sin 2y=x^{3} \cos^{2} y.\]

4 M

2(b)
Solve the following :

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 100Ω, an inductance of 0.5 henry are connected in a series with battery of 20 volts. Find the current in a circuit as a function of time t .

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 100Ω, an inductance of 0.5 henry are connected in a series with battery of 20 volts. Find the current in a circuit as a function of time t .

8 M

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

3(a)
Find Fourier series to represent the function f(x) = x in -π < x< π and f(x) = f(x+2π)

5 M

3(b)
Evaluate:\[\int_{0}^{\infty }\sqrt{y.}e^\sqrt{y}\ dy\]

3 M

Solve any one question from Q.3(c) (i,ii)

3(c)(i)
Trace the curve:\[ y^{2}\left ( x^{2}-1 \right )=x\]

4 M

3(c)(ii)
\[r = a(1+\cos \theta ).\]

4 M

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

4(a)
If : \(\begin{align*}I_{n}=\int_{\pi /4}^{\pi /2}\cot ^{n}\ \theta\ d\theta ,\\
\text{prove that}:\\
I_{n}=\frac{1}{n-1}-I_{n-2}. \end{align*} \)/

4 M

4(b)
\( \text{Prove that}:\\
\\
\int_{0}^{1}\frac{x^{a}x^{b}}{\log x}dx=\log \left ( \frac{a+1}{b+1} \right ),a>0, b>0\)/

4 M

4(c)
Find the legth of the arc of cycloid \(x = a \left ( \theta +\sin \theta \right ), y=a\left ( 1-\cos \theta \right ) \)/ between two consecutive cups.

4 M

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

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

5 M

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

4 M

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

4 M

Solve any two question from Q.6 (a, b, c)

6(a)
Show that the plane 4x-3y+6z-35 = 0 is tangential to the sphere \( x^{2}+y^{2}+z^{2}-y-2z-14=0. \)/

5 M

6(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°.

4 M

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

4 M

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

7(a)
Evaluate \( \iint \frac{x^{2}y^{2}dxdy}{x^{2}y^{2}} \)/ where R is annulus between \( x^{2}+y^{2}=4, x^{2}+y^{2}=9. \)/

6 M

7(b)
Evaluate \( \iiint \left ( x^{2}y^{2}+y^{2}z^{2}+z^{2}x^{2} \right )dxdydz\)/ throughout the volume of sphere\( x^{2}+y^{2}+z^{2}= a^{2}. \)/

7 M

7(c)
Find the moment of inertia of one loop of lemniscate \( r^{2}=a^{2}\cos 2\theta \)/

6 M

Solve any two question from Q.8(a, b, c)

8(a)
Find the total area included between the two cardioids \[r = a\left ( 1+\cos \theta \right )\ \ \text{and}\ r =a\left ( 1-\cos \theta \right ).\]

6 M

8(b)
Find the volume cut-off from the paraboloid \( x^{2}+\frac{y^{2}}{4}+z=1 \)/ by the plane z=0.

7 M

8(c)
Find the C.G. of an area of the cardioid:\[r =a\left ( 1+\cos \theta \right ).\]

6 M

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