List of practice Questions

Two spherical black bodies A and B of equal radii are at temperatures \(2T\) and \(3T\). If surrounding temperature is \(T\), then ratio of radiant powers emitted by A and B is

A block slides down a \(30^\circ\) inclined plane. Coefficient of friction on upper half is \[ \mu_1=\frac{1}{2\sqrt3} \] and on lower half is \[ \mu_2=\frac{1}{4\sqrt3} \] Find ratio of velocities at midpoint and bottom.

The power \(P\) (in watt) acting on a body of mass \(2\text{ kg}\) is given by \[ 4.5P=8t^2+14t+9 \] where \(t\) is time in second. If the body starts from rest at \(t=0\), then the velocity of the body at time \(t=3\text{ s}\) is

A bomb of mass \(m\) at rest explodes into three parts. If these three parts move horizontally with equal speeds in different directions, then the masses of the three parts can be

A body P of mass \(1.5\text{ kg}\) moving with velocity \(10\text{ ms}^{-1}\) makes a one dimensional elastic collision with another body Q at rest. If ratio of velocities after collision is \(1:3\), then velocity of centre of mass is

A solid sphere rolls down without slipping on an inclined plane of angle \(30^\circ\). If angle increases to \(45^\circ\), percentage increase in acceleration is nearly

For a particle executing simple harmonic motion, if the velocities at distances \(6\text{ cm}\) and \(8\text{ cm}\) from mean position are \(16\pi\text{ cms}^{-1}\) and \(12\pi\text{ cms}^{-1}\) respectively, then maximum acceleration is

Evaluate \[ \int_{\pi/6}^{\pi/3} \frac{dx}{\sin2x(\tan^4x-\cot^4x)} \]

If \(a,b\) are arbitrary constants, then the differential equation corresponding to family \[ y=ax^2-2abx+ab^2 \] is

The general solution of the differential equation \[ \frac{dy}{dx} = x^2y^2+3x^2y-2xy^2-6xy \] is

The unification of electromagnetism and optics is based on the discovery that

If \(B\) is magnetic induction, \(e\) is charge of electron, \(m\) is mass and \(c\) is speed of light in vacuum, then the physical quantity having dimensions of \[ \frac{4\pi mc}{Be} \] is

A body is thrown vertically upwards from earth with velocity \(60\text{ ms}^{-1}\). The ratio of displacements during first, second and third seconds is (\(g=10\text{ ms}^{-2}\))

If minimum velocity of a projectile is \(15\text{ ms}^{-1}\) and maximum height reached is \(20\text{ m}\), then velocity of projection is (\(g=10\text{ ms}^{-2}\))

Evaluate \[ \int\frac{\cos x}{\sqrt{16\cos^2x+9}}dx \]

If \[ \int(e^{2x}+2e^x)\sqrt{e^{2x}-4e^x+5}\,dx = \frac13[f(x)]^{3/2} + 4\left[ \frac{e^x-2}{2}\sqrt{f(x)} + \frac12g(x) \right]+c \] then \(f(0)=\)

Evaluate \[ \int\frac1{2\cot x-3\tan x}\,dx \]

Evaluate \[ \int_{\frac{\pi}{4}}^{\frac{3\pi}{4}} \frac{x\sin x}{1+\cos2x}\,dx \]

The area of the region bounded by the curves \[ y=x^2-3x+3 \] \[ y=2x^2-1 \] is

If \(\theta\) is angle between parabolas \[ y^2=108x \] and \[ x^2=32y \] then \[ \cos2\theta= \]

If the real valued function \[ f(x)=\log(2x-3)-2x^2+6x-4 \] then the interval in which \(f(x)\) is increasing is

If \(m\) and \(M\) are respectively the absolute minimum and absolute maximum values of the function \[ f(x)=|2x^2-x-6|+2x-3 \] in the interval \([-2,4]\), then \(2M+8m=\)

Evaluate \[ \int \frac{\cos x+\sin2x}{1-\sin x-2\sin^2x}\,dx \]

The domain of derivative of real valued function \[ f(x)=(x^2-x-2)|x^2+x-6| \] is

If \[ f(x)=\pi-\cos^{-1}\left(\frac{x^2+4x+3}{x^2+4x+5}\right) \] then \(f'(1)=\)