STUPIDSID
Solve any one question from Q1 & Q2
1 (a)
Solve the following \[i) \ \dfrac {dy}{dx} = \dfrac {x^2 + y^2 +1}{2xy} \\ ii) \ (1-x^2) \dfrac {dy} {dx} = 1+ xy \]
8 M
1 (b)
An electric circuit contains an inductance of 0.5 henries and a resistance of 100 ohms in series with an e.m.f. of 20 volts. Find the current at any time t, if it, is zero at t=0.
4 M
2 (a)
Solve: (2x+3y-1)dx=(6x+9y+6)dy
4 M
2 (b)
Solve the following:
i) A bullet is fired sand tank, its retardation is proportinal to square root of its velocity. Show that the bullet will come to rest in time \[ \dfrac {2\sqrt{v}} {k} \] where V is initial velocity.
ii) A pipe 20 cm in diameter steam at 150°C and is protected with a covering 5cm thick for which k=0.0025. If the temperature of the outer surface of the covering is 40°C, find the temperature half-way through the covering under steady state conditions.
i) A bullet is fired sand tank, its retardation is proportinal to square root of its velocity. Show that the bullet will come to rest in time \[ \dfrac {2\sqrt{v}} {k} \] where V is initial velocity.
ii) A pipe 20 cm in diameter steam at 150°C and is protected with a covering 5cm thick for which k=0.0025. If the temperature of the outer surface of the covering is 40°C, find the temperature half-way through the covering under steady state conditions.
8 M
Solve any one question from Q3 & Q4
3 (a)
Find the half range cosine series for the function F(x)=x-x3, 0?x?1
5 M
3 (b)
\[ Evaluate \ int^5_2 (x-2)^2 (5-x)^2 dx \]
3 M
3 (c)
Trace the curve (any one)
i) x=a(t+sin t), y=a(l-cos t)
ii) x2y2=a2 (y2-x)
i) x=a(t+sin t), y=a(l-cos t)
ii) x2y2=a2 (y2-x)
4 M
4 (a)
\[ If \ I_n = \int^\infty_0 e^{-x} \sin^n x \ dx \] obtain the relation between In and In2.
4 M
4 (b)
Show that \[ \int^b_a e^{-x^2} dx = \dfrac {\sqrt{\pi}}{2} [erf (b)- erf (a)] \]
4 M
4 (c)
Find the arclength of one loop of Lemniscate r2=a2= cos 2?
4 M
Solve any one question from Q5 & Q6
5 (a)
Find the equation of right circular cylinder of radius a, whose axis passes through the origin and makes equal angles with the coordinate axes.
4 M
5 (b)
Lines are drawn from the origin with direction co-sines proportional to (1,2,2), (2,3,6)(3,4,12). Find direction co-sines of the axis of right circular cone through them, and prove that the semivertical angle of cone is \[ \cos^{-1} \dfrac {1}{\sqrt{3}} \]
4 M
5 (c)
Find the equation of the sphere which passes through the points (1,-4,3)(1,-5,2)(1,-3,0) and whose centre lies on the plane x+y+z=0
5 M
6 (a)
A sphere of constant radius K passes through the origin and meets the axes in A,B,C prove that the centroid of the triangle ABC lies on the sphere g(x2+y2+z2) =4K2
5 M
6 (b)
Find the equation of the right circular cone which has its vertex at the point (0,0,10) and whose intersection with the plane XOY is a circle of diameter 10.
4 M
6 (c)
Find the equation of the right circular cylinder of radius 3 and axis \[ \dfrac {x-1}{2} = \dfrac {y-3}{2} = \dfrac {z-5}{-1} \]
4 M
Answer any two from Q7
7 (a)
\[ Evaluate \ \iint (x+y)^2 dxdy \ over \ the \ area \ bounded \ by \ an ellipse \ \dfrac {x^2}{a^2} + \dfrac {y^2}{b^2} =1 \]
7 M
Solve any one question from Q7 & Q8
7 (b)
Find the volume of the tetrahydron bounded by the co-ordinate planes and the plane x+y+z=1
6 M
7 (c)
Find the moment of inertia about the initial line of the cardiode r=a(1+cos ?)
6 M
Answer any two from Q8
8 (a)
Find the area bounded by the parabola y=x2 & the Line y=2x+3.
7 M
8 (b)
\[ Evaluate \ \iiint z(x^2 + y^2)dxdydz \] over the volume of the cylinder x2+y2=1 intercepted by the planes z=2 and z=3.
6 M
8 (c)
Find the x-co-ordinate of centre of gravity of an area bounded by the parabola y2=x and the line x+y=2.
6 M
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