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A projectile is projected from the base of an inclined plane of inclination '$\alpha$'. The angle of projection of the projectile with the ground is $\theta$. If it strikes the incline plane at it's maximum height from the ground, then, (assume the plane of motion contains the line of greatest slope on the incline plane)

The height and horizontal distance $x$ of a projectile are given by $y=(8t-5t^2)m$ and $x=6tm$ where $t$ is in seconds. Find the maximum height of a projectile.

(Take $g=10m/s^2$)

How many external forces would appear to act on $(m_2+m_3)$ if they are considered as a system and $m_1$ is considered as another system in the figure shown?

Two containers of sand are arranged like the block as shown in figure. The containers alone have negligible mass; the sand in these containers has a total mass $M_{tot}$; the sand in the hanging container $H$ has mass $m$.

To measure the magnitude of the acceleration of the system, a large number of experiments have been carried out where $m$ varies from experiment to experiment but $M_{tot}$ does not; that is, sand is shifted between the containers before each trial.





Which of the curves gives the tension in the connecting cord (the vertical axis is for tension)?

Two wires are made of the same material and have the same volume. However wire $1$ has cross-sectional area $A$ and wire $2$ has cross-sectional area $3$A. If the length of wire $1$ increases by $\Delta x$ on applying force $F$, how much force is needed to stretch wire $2$ by the same amount?

In the figure shown, all pulleys are massless and the strings are light and inextensible. Find the tension in the string attached to the block $m$ of mass $10kg$? (All surfaces are smooth and take $g=10m/s^2$ )

Determine the change in volume of $6\text{litres}$ of alcohol if the pressure is decreased from $200\text{cm}$ of $Hg$ to $75\text{cm}$ of $Hg$.

[Velocity of sound in alcohol is $1280\text{m/s}$ density of alcohol $=0.81\text{gm/cc}$, density of $Hg=13.6\text{gm/cc}$ and $g=981{\text{cm/s}}^2$]

On the superposition of the two waves represented by equation,

$y_1=A\sin (\omega t-kx)$

$y_2=A\cos (\omega t-kx+\frac{\pi }{6})$

The resultant angular frequency of the oscillation will be

Match the Van der Waal's for one $\text{mole}$ of gas given in column $2$ to the conditions in column $1$.

$$\begin{array}{cccc}& \text{Column 1}& & \text{Column 2}\\ A& \text{At low pressure}& i& P(V-b)=RT\\ B& \text{At high pressure}& ii& (P+\frac{a}{V^2})V=RT\\ C& \text{At low pressure and high temperature}& iii& PV=RT\\ D& \text{At room temperature and pressure}& iv& (P+\frac{a}{V^2})(V-b)=RT\end{array}$$

A cylinder of mass $M$, radius $R$ is resting on a horizontal platform (which is parallel to $X-Y$ plane) with its axis fixed along the $y-$ axis and free to rotate about its axis. The platform is given a motion in $X-$ direction given by: $x=A\cos \omega t$. There is no slipping between the cylinder and platform. The maximum torque acting on the cylinder during its motion is:

A man of mass $M=50\text{kg}$ stands at one end of the boat which is floating on still water. The man walks to the other end of the boat. Length of the boat is $L=10\text{m}$ and mass $M^{\prime }=100\text{kg}$. Then find the distance moved by the boat relative to the ground

If the surface shown in the figure is frictionless, and $T_2=6\sqrt{3}N$ , find $F$.

The temperature of a liquid drops from $365\text{K}$ to $361\text{K}$ in $2\text{min}$. Find the time during which temperature of the liquid drops from $344\text{K}$ to $342\text{K}$.

[Take the room temperature as $293\text{K}$]

When a rubber ball is taken to the bottom of a sea of depth $2000\text{m}$, the volume of the rubber ball decreases by $1.5\%$. Find the bulk modulus of the rubber ball.

Take the density of water $\rho =1000{\text{kg/m}}^3$ and $g=10{\text{m/s}}^2$.

Four spheres of radii $R$, $2R$, $2R$ and $3R$ respectively, are placed such that their respective centres lie at $x=2\text{m}$, $x=6\text{m}$, $x=10\text{m}$ and $x=20\text{m}$ respectively. If density of each sphere is the same, then find the $x$ coordinate of $\text{COM}$ of the system of spheres.

While you are travelling in a car on a straight road at $90\text{km/h}$, another car passes you in the same direction; its speedometer reads $120\text{km/h}$. What is your velocity relative to the other driver? (in km/h)

What will be the daughter nuclei after 4 alpha decays of ${}_{91}^{231}Pa$

The velocity $v$ and time $t$ graph of a body in a straight line motion is shown in the figure. The point $S$ is at $4.333\text{sec}$. The total distance covered by the body in $6\text{s}$ is

Two identical discs of the same radius R are rotating about their axes in opposite directions with the same constant angular speed $\omega .$ The discs are in the same horizontal plane. At time t = 0, the points P and Q are facing each other as shown in the figure. The relative speed between the two points P and Q is $v_r$. In one time period (T) of rotation of the discs, $v_r$ as a function of time is best represented by

A rope 1 cm in diameter breaks if the tension in it exceeds 500N. The maximum tension that may be given to a similar rope of diameter 2 cm is?

A ray of light undergoes deviation of ${30}^{\circ }$ when incident on an equilateral prism of refractive index $\sqrt{2}$. The angle made by the ray inside the prism with the base of the prism is

Three equal charges are placed on the three corners of a square. If the force between $q_1$ and $q_2$ is $F_{12}$ and that between $q_1$ and $q_3$ is $F_{13}$, the ratio of magnitudes $\frac{F_{12}}{F_{13}}$ is

A doubly ionized Lithium atom is like a hydrogen atom with atomic number 3. What is the wavelength of the radiation required to excite the electron from first to third Bohr's orbit?

A simple pendulum is made of a string of length $l$ and a bob of mass $m$, is released from a small angle ${\theta }_0$. It strikes a block of mass $M$, kept on a horizontal surface at its lowest point of oscillations, elastically. It bounces back and goes up to an angle ${\theta }_1$. Then, $M$ is given by

A block of mass m is placed on a suface with a vertical cross section given by $y=\frac{x^3}{6}.$ If the coefficient of friction is $0.5$, the maximum height above the ground at which the block can be placed without slipping is:

A gas at temperature $250\text{K}$ is contained in a closed vessel. If the gas is heated and the temperature rises by $1^{\circ }\text{C},$ the percentage increase in its pressure is nearly

Two thin dielectric slabs of dielectric constants $K_1$ and $K_2(K_1<K_2)$ are inserted between the plates of a parallel plate capacitor, as shown in the figure below. The variation of electric field $E$ between the plates with distance $d$ as measured from plate $P$ is correctly shown by -

If $C_P$ and $C_V$ denote the specific heats of nitrogen per unit mass at constant pressure and constant volume respectively, then

Which of the following graphs correctly represent adiabatic expansion?

Convection coefficient depends upon

A wooden plank of mass $30\text{kg}$ is resting on a smooth horizontal floor. A man of mass $70\text{kg}$ starts moving from one end of the plank to the other end. The length of plank is $20\text{m}$. Find the displacement of the plank over the floor when the man reaches the other end of the plank.

Stationary waves of frequency $300\text{Hz}$ are formed in a string, in which the velocity of wave is $200\text{m/s}$. The distance between a node and the consecutive antinode is

A particle of mass $m$ is located at point O. Find $I_A,I_B,I_{AB}$ and $I_{A{A}^{\prime }}$ respectively.

At Deoprayag (Garhwal, UP) river Alaknanda mixes with the river Bhagirathi and becomes river Ganga. Suppose Alaknanda has a width of 12 m, Bhagirathi has a width of 8 m and Ganga has a width of 16 m. Assume that the depth of water is same in the three rivers. Let the average speed of water in Alaknanda be 20 km h^-1 and in Bhagirathi be 16 km h^-1. Find the average speed of water in the river Ganga.

Select the correct statement from the following

A bar is subjected to equal and opposite forces as shown in the figure. PQRS is a plane making angle $\theta$ with the cross-section of the bar. If the area of cross-section be ‘A’, then when is the shearing stress maximum

A truck driver drives $20\text{km}$ down the road in $5\text{minutes}$ . He then reverses and drive $12\text{km}$ back down the road in $3\text{minutes}$. What is the average velocity?

Two trains $\text{A}$ and $\text{B}$ moving with speed of $36\text{km/hr}$ and $72\text{km/hr}$ respectively in opposite direction. A man moving in train $\text{A}$ with speed of $1.8\text{km/hr}$ opposite to the direction of train $\text{A}$. Find velocity of man as seen from train $\text{B}$ (in $\text{m/s}$).

An electric dipole of moment p is placed in the position of stable equilibrium in uniform electric field of intensity E. If it is rotated through an angle $\theta$ from the initial position, the potential energy of electric dipole in the final position is

One mole of a monoatomic gas of molar mass $M$ undergoes a cyclic process as shown in the figure. Here, $\rho$ is the density and $P$ is the pressure of the gas. Find the efficiency of the cycle.

A wooden cylinder of base area $50c{m}^2$ and height $1m$ has mass of $2.5kg$. It is carefully lowered in a cylindrical vessel of water. The vessel has inner area $100c{m}^2$ and height of $1m$. Which of the four figures correctly represents the equilibrium state?

Which of the following is the necessary and sufficient criterion for a body to be called rigid?

A block of mass $3\text{kg}$ produces an extension of $1\text{mm}$ in a wire of length $4\text{m}$ and diameter $4\text{mm}$ when suspended at one end. The Young's modulus of the material of the wire will be (Assume: $g=10{\text{m/s}}^2\text{and}\pi =3$)

Two rods of length $l_1$ and $l_2$ are made of materials whose coefficient of linear expansion are ${\alpha }_1$ and ${\alpha }_2$. If difference between two lengths is independent of temperature then

A solid uniform cylinder of mass $M$ performs small oscillations in the horizontal plane if slightly displaced from its mean position as shown in the figure. Initially, the springs are at their natural lengths and the cylinder does not slip on the ground during oscillations due to friction between the ground and cylinder. The force constant of each spring is $k$. What is the time period of this oscillation?

A ball is dropped from a height of $5\text{m}$. Find the velocity with which it hits the ground.