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A cyclist rides along the circumference of a circular horizontal plane of radius R, the frictional coefficient being dependent only on distance `r' from the centre of the plane as $\mu ={\mu }_0[1-\frac{r}{R}]$ , where ${\mu }_0$ is a constant. Find the radius of the circle with the centre at the point along which the cyclist can ride with the maximum velocity.

In the figure a ball of mass $m$ is tied with two strings of equal length as shown. If the rod is rotated with angular velocity $\omega$ and $T_1,T_2$ be tension in the strings as shown, then

In a Fraunhofer diffraction experiment, a single slit of width $0.5mm$ is illuminated by a monochromatic light of wavelength $600nm$. The diffraction pattern is observed on a screen at a distance of $50cm$ from the slit. What will be the linear separation of the first order minima?

A capacitor of capacitance $C_1$ is charged by connecting it to a battery. The battery is now removed and this capacitor is connected to a second uncharged capacitor of capacitance $C_2$ . If the charge gets distributed equally on the two capacitors after connection, the ratio of the total energy stored in the capacitors after connection to the total energy stored in them before connection is

A raft of wood $(\rho =600{\text{kg/m}}^3)$ of mass $120\text{kg}$ floats on water. The mass which can be put on the raft to make it just sink is

In a series $LCR$ circuit, the inductive reactance $\left(X_L\right)$ is $10\Omega$ and the capacitive reactance $\left(X_C\right)$ is $4\Omega$. The resistance $\left(R\right)$ in the circuit is $6\Omega$. Find the power factor of the circuit.

Two sinusoidal waves are defined by the functions :

$y_1=\left(2.00\text{cm}\right)\sin (20x-32t)$ and $y_2=\left(2.00\text{cm}\right)\sin (25x-40t)$ where $y_1,y_2$ and $x$ are in centimeters and $t$ is in seconds. What is the phase difference between these two waves at the point $x=5.00\text{cm}$ at $t=2\text{s}$?

A $50\text{cm}$ long wire having a mass of $20\text{g}$ is fixed at two ends and is vibrated in its fundamental mode. A $75\text{cm}$ long closed organ pipe, placed with its open end near the wire, is set up into resonance in its first overtone mode by the vibrating wire. Find the tension in the wire. Speed of sound in air is $340\text{m/s}$.

Let $\overrightarrow{x},\overrightarrow{y}$ and $\overrightarrow{z}$ be three vectors each of magnitude $\sqrt{2}$ and the angle between each pair of them is $\frac{\pi }{3}$. If $\overrightarrow{a}$ is a nonzero vector perpendicular to $\overrightarrow{x}$ and $\overrightarrow{y}\times \overrightarrow{z}$ and $\overrightarrow{b}$ is a nonzero vector perpendicular to $\overrightarrow{y}$ and $\overrightarrow{z}\times \overrightarrow{x}$, then

Equation of a stationary wave is given by $y=5\cos \left(\frac{\pi x}{25}\right)\sin \left(100\pi t\right)$. Here, $x$ is in $\text{centimeter}$ and $t$ in $\text{seconds}$. Node will not occur at distance

A cannon on a level plane is aimed at an angle $\theta$ above the horizontal and a shell is fired with a muzzle velocity $v_0$ towards a vertical cliff a distance $D$ away. Then the height from the bottom at which the shell strikes the side walls of the cliff is

A long straight wire carries a current $i=2\text{A}$. A particle having a positive charge $q=1\text{C}$ and mass $5\text{mg}$, kept at a distance $x_0=5\text{cm}$ from the wire is projected towards it with a speed $v=20\text{cm/s}$. Find the minimum separation between the wire and the particle.

There are two vectors $\overrightarrow{A}=3\hat{i}+\hat{j}$ and $\overrightarrow{B}=\hat{j}+2\hat{k}.$ Find the magnitude of component of $\overrightarrow{A}$ along $\overrightarrow{B}$

The density $\rho$ of water of bulk modulus B at a depth y in the ocean is related to the density at surface by the relation

The resistance of a galvanometer is $999\Omega$. It is shunted by $1\Omega$ resistance. If the current in the circuit is ${10}^{-2}\text{A},$ then the current through galvanometer is

A rope of length $10m$ and linear mass density $4kg/m$ is lying lengthwise on a horizontal smooth table. One end of the rope is pulled horizontally by a force of $40N$. The tension in the rope at a point $4m$ from point of application of force will be

Fourier series of any periodic signal x(t) can be obtained if

1. ${\int }_0^T|x\left(t\right)|dt<\infty$

2. Finite number of discontinuities within finite interal t

3. Infinite number of discontinuities



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A police jeep is chasing a culprit going on a motorbike. The motorbike crosses a turning at a speed of 72 km/h, The jeep follows it at a speed of 90 km/h, crossing the turning ten seconds later than the bike. Assuming that they travel at constant speeds, how for from the turning will the jeep catch up with the bike?

A time varying voltage is given by $E=240\sin 120t$, where $t$ is in second. This voltage is applied across a $120\Omega$ resistor. The average value of current for the half cycle is -

A harmonic wave propagates in a medium. Find the average energy density of the wave, if at any point, the energy density becomes equal to $W_0$ at an instant $t=t_0+T/6$, where $t_0$ is the instant when amplitude is maximum at this location and $T$ is the time period of oscillation.

The electrical resistance of depletion layer is large because

A particle executes S.H.M (a) What fraction of total energy is kinetic? What fraction is potential when displacement is one half of the amplitude?

Given that $\overrightarrow{C}=\overrightarrow{A}+\overrightarrow{B}$. Also, the magnitude of $\overrightarrow{A},\overrightarrow{B}$ and $\overrightarrow{C}$ are $12,5$ and $13$ units respectively. The angle between $\overrightarrow{A}$ and $\overrightarrow{B}$ is

A metal ball falls from a height of 32 m on a steel plate. If the coefficient of restitution is 0.5, to what height will the ball rise after the second bounce

Two blocks of mass $10\text{kg}$ and $2\text{kg}$ respectively are connected by an ideal string passing over a fixed smooth pulley as shown in figure. A monkey of mass $8\text{kg}$ starts climbing the string with a constant acceleration of $2{\text{ms}}^{-2}$ with respect to the string at $t=0$. Initially, the monkey is $2.4\text{m}$ from the pulley. Find the time taken by the monkey to reach the pulley.