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A book with many printing errors contains four different formulas
for the displacement y of a particle undergoing a certain
periodic motion:

(a)

(b) y = a sin vt

(c)

(d)

(a = maximum displacement of the particle, v = speed
of the particle. T = time-period of motion). Rule out the
wrong formulas on dimensional grounds.

An ideal gas expands isothermally from a volume $V_1$ to $V_2$ and then compressed to original volume $V_1$ adiabatically. Initial pressure is $P_1$ and final pressure is $P_3$. The total work done is $W$. Then

Water is allowed to come out of a hole $P$ in one of the walls at a depth $h_1$ below the surface of water. Express the horizontal distance covered by water jet in terms of $h_1$ and $h_2$.

Two thin metallic spherical shells of radius $a$ and $2a$ are placed with their centers coinciding. A material of thermal conductivity $k$ is filled in the space between them. Determine the expression for thermal resistance of the system.

Find equivalent capacitance across $\text{A}$ and $\text{B}$

A projectile is thrown with a velocity of $50{\text{ms}}^{-1}$ at an angle of ${53}^{\circ }$ with the horizontal. The equation of the trajectory is given

(Take $g=10{\text{m/s}}^2$)

Two projectiles are projected with same speed at angles $(\frac{\pi }{4}+\theta )$ and $(\frac{\pi }{4}-\theta )$ with the horizontal, where $\theta <\frac{\pi }{4}$ . The ratio of horizontal ranges described by them is

A hollow sphere of mass $M$ and radius $R$ is rolling without slipping on a rough horizontal surface. Then the percentage of it's total $KE$ which is translational, will be

A spring when stretched by 2 mm its potential energy becomes 4 J. If it is stretched by 10 mm, its potential energy is equal to

The acceleration due to gravity on a planet $\text{A}$ is $9$ times the acceleration due to gravity on planet $\text{B}$. A man jumps to a height of $2\text{m}$ on the surface of $\text{A}$. What is the height of jump by the same person on the planet $\text{B}$ ?

If a current of $0.8\text{A}$ is flowing through a solenoid of $10000\text{turns}$ having a total length of $30\text{cm}$, then what is the magnitude of the magnetic field?

Body $A$ of mass $4m$moving with speed $u$ collides with another body $B$ of mass $2m$ at rest. The collision is head on and elastic in nature. After the collision, the fraction of kinetic energy lost by the colliding body $A$ is

The velocity at the maximum height of a projectile is $\frac{\sqrt{3}}{2}$ times its initial velocity of projection $\left(u\right)$. Its range on the horizontal plane is

The length of shorter diagonal of parallelogram constructed on $5\overrightarrow{a}-6\overrightarrow{b}$ and $4\overrightarrow{a}-3\overrightarrow{b}$ where $|\overrightarrow{a}|=4$ and $|\overrightarrow{b}|=3$ and $\overrightarrow{a}$ and $\overrightarrow{b}$ are anti parallel is

A uniform rod of length 'l' and mass 'm' is hung from two strings of equal length from a ceiling as shown in figure. The tensions in the strings are

A horizontal $90\text{kg}$ merry-go-round is a solid disc of radius $1.50\text{m}$ is started from rest by a constant horizontal force of $50\text{N}$ applied tangentially to the edge of the disc. The kinetic energy of the disc after $3\text{s}$ is

The phase difference between the alternating current and voltage represented by the following equation $I=I_0\sin \omega t,E=E_0\cos (\omega t+\frac{\pi }{3}),$ will be

The number of molecules in one litre of an ideal gas at $300\text{K}$ and $2$ atmospheric pressure with mean kinetic energy $2\times {10}^{–9}\text{J}$ per molecule is :

A wire is wound on a long rod of material of relative permeability ${\mu }_r=4000$ to make a solenoid. If the current through the wire is $5A$ and number of turns per unit length is $1000$ per metre, then the magnetic field inside the solenoid is:

Two coils have a mutual inductance 0.005 H. The current changes in the first coil according to equation $I=I_{0}sin\omega t$, where $I_0=10Aand\omega =100\pi radian/sec$. The maximum value of e.m.f. in the second coil is

The energy stored in the system of capacitors shown in the circuit is (Capacitance of each capacitor is C = 0.1 $\mu F)$

Six particles situated at the corners of a regular hexagon of side a move at a constant speed v. Each particle maintains a direction towards the particle at the next corner. Calculate the time the particles will take to meet each other.

A block of mass $1\text{kg}$ slides down on a rough inclined plane of inclination ${60}^{\circ }$ starting from its top. If the coefficient of kinetic friction is 0.5 and the length of the plane is $1\text{m}$, then work done against friction is (Take $g=9.8{\text{m/s}}^2$)

How many spectral line will be observed when an electron jumps from $n=5$ to $n=1$ in the visible line spectra of $H$ spectrum?

A convex lens of focal length $f$ is cut into two halves as shown. Light rays parallel to the principal axis are incident on one half of the lens (say $L_1$) This will form an image at

If the work function of a metal is ${}^{\prime }{\varphi }^{\prime }$ and the frequency of the incident light is 'v', there is no emission of photoelectron if