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Two rods, one of aluminum and the other made of steel, having initial lengths $l_1$ and $l_2$ are connected together to form a single rod of length $l_1+l_2$. The co-efficients of linear expansion for aluminum and steel are ${\alpha }_a$ and ${\alpha }_s$ respectively. If the length of each rod increases by the same amount when their temperature are raised by $t^{\circ }C$, then find the ratio $\frac{l_1}{(l_1+l_2)}$

At time $t=0$, a container has $N_0$ radioactive atoms with a decay constant $\lambda$. In addition, $C$ numbers of atoms of the same type are being added to the container per unit time. How many atoms of this type are there at $t=T$ ?

Calculate the ratio of intensity of wavetrain A to wavetrain B.

A child pushes a $2.2\text{kg}$ swing that is initially at rest. The net force on the swing over time is shown below.





What is the swing's speed at $t=40\text{ms}$? Round off the answer to two decimal places.

A spring of force constant k is cut into lengths of ratio $1:2:3.$ They are connected in series and the new force constant is $k^{\prime }$. Then they are connected in parallel and force constant is $k^{\prime \prime }$. The ratio $k^{\prime }:k^{\prime \prime }$ is

A ball falls from a height of $1\text{m}$ on the ground and jumps upto a height $H$. If the coefficient of restitution of the collision is $0.9$, then find the value of $H$.

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

When a ball is thrown vertically upwards with velocity $v_0$, it reaches a maximum height of $h$. If one wishes to triple the maximum height then the ball should be thrown with velocity.

Find the binding energy of ${}_{26}^{56}Fe$. Atomic mass of ${}^{56}Fe$ is 55.9349 u and that of ${}^{1}H$ is 1.00783 u. Mass of neutron = 1.00867 u.

In the figure shown, find the tension $T_1$ (in $N$) in the string. Assume the strings to be inextensible and pulley frictionless.

A wire having a linear mass density $5\times {10}^{-3}\text{kg/m}$ is stretched between two rigid supports with a tension of $450\text{N}$. The wire resonates at a frequency of $420\text{Hz}$. The next higher frequency at which the same wire resonates is $490\text{Hz}$. The length of the wire will be

At what angle should a body be projected with a velocity $24{\text{ms}}^{–1}$ just to pass over the obstacle $14\text{m}$ high at a distance of $24\text{m}$.

[Take $g=10{\text{ms}}^{–2}$]

The shortest wavelength of X-rays emitted from an X-ray tube depends on the

For a particle moving along a straight line, the position $x$ depends on time $t$ as $x=A{t}^3+B{t}^2+Ct+D$. The ratio of its initial velocity to its initial acceleration will be

Consider a two-particle system with the particles having masses $m_1$ and $m_2$. If the first particle is pushed towards the centre of mass by a distance 'd', by what distance should the second particle be moved so as to keep the centre of mass at the same position?

An amount $Q$ of heat is added to a monoatomic ideal gas in a process in which the gas performs a work of $\frac{Q}{2}$ on its surrounding. Find equation of the process that monoatomic gas is undergoing.

Two rigid boxes containing different ideal gases are placed on a table. Box $A$ contains one mole of nitrogen at temperature $T_0$, while box $B$ contains one mole of helium at temperature $\left(\frac{7}{3}\right)T_0$. The boxes are then put into thermal contact with each other, and heat flows between them until the gases reach a common final temperature (Ignore the heat capacity of boxes). Then, the final temperature of the gases, $T_f$ in terms of $T_0$ is -

Two blocks of equal masses are placed on a horizontal surface. The surface of $A$ is smooth but that of $B$ has a friction coefficient of $0.2$ with the floor. Block $A$ is given a speed $5\text{m/s}$, towards $B$ which is kept at rest. Find the distance travelled by $B$, if the collision is perfectly elastic.

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

A wire of length $16\text{m}$ and mass $8\text{kg}$ is bent in the form of a rectangle $\text{ABCD}$ with $\frac{\text{AB}}{\text{BC}}=2$. The moment of inertia of this wire frame about side $\text{BC}$ is

In a cricket match, Sachin hit a 1 kg ball thrown at him at 126 km/hr exactly in the opposite direction with the exact same speed. Find the impulse that his bat imparted on the ball in the short duration for which they were in contact.

Figure below shows a flat car of mass 'M' on a frictionless road. A small massless wedge is fitted on it as shown. A small ball of mass 'm' is released from the top of the wedge, it slides over it and falls in the hole at distance 'l' from the initial position of the ball. Find the distance the flat car moves till the ball gets into the hole.

A hoop of radius r and mass m rotating with an angular velocity ${\omega }_0$ is placed on a rough horizontal surface. The initial velocity of the centre of the hoop is zero. What will be the velocity of the centre of the hoop when it ceases to slip?

The amount of heat required to convert $5\text{g}$ of ice at $0^{\circ }\text{C}$ to $5\text{g}$ of steam at ${100}^{\circ }\text{C}$ is -

[Latent heat of vaporization and fusion are $L_v=540{\text{cal g}}^{-1}$ and $L_f=80{\text{cal g}}^{-1}$]

A platinum sphere floats in mercury. Find the percentage change in the fraction of volume of sphere immersed in mercury when the temperature is raised by ${80}^{\circ }C$ (Volume expansivity of mercury is $182\times {10}^{-6}{/}^{\circ }C$ and linear expansivity of platinum is $9\times {10}^{-6}{/}^{\circ }C$)

Water flows at a speed of 6 cm s^-1 through a tube of radius 1 cm. Coefficient of viscosity of water at room temperature is 0.01 poise. Calculate the Reynolds number. Is it a steady flow?

A two dimensional object lies in $x-z$ plane having moment of inertia $I_X=M{R}^2$ and $I_Y=4M{R}^2$. Then, the moment of inertia of object along $z-$axis is

The pressure at the bottom of a water tank is $6P$, where $P$ is the atmospheric pressure. If water is drawn out till the level decreases by $\frac{2}{5}th$, then the pressure at the bottom of the tank is

In an adiabatic process, $R=\frac{2}{3}{C}_v$. The pressure of the gas will be proportional to

Two bodies are projected vertically upwards from one point with the same initial velocities $v_{0}m/s$. The second body is thrown $\tau s$ after the first. The two bodies meet after time

The distance of neptune and saturn from sun are nearly ${10}^{13}$ and ${10}^{12}$ meters respectively. Assuming that they move in circular orbits, their periodic times will be in the ratio

What is the percentage increase in length of a wire having diameter $1.5\text{mm}$, stretched by a force of $50\text{kg-wt}$?

(Young's modulus of elasticity of wire $\left(Y\right)=15\times {10}^{11}{\text{dyne/cm}}^2\text{and}g=10{\text{m/s}}^2$)

Four bodies $A,B,C$ and $D$ of masses $[M_A=m,M_B=2m,M_C=m,M_D=2m]$ are initially at rest. They start moving under the influence of the internal force of attraction and at a particular instant, their velocities are $\overrightarrow{V_A}=2\hat{i},\overrightarrow{V_B}=-\hat{i}+2\hat{j}$ and $\overrightarrow{V_C}=-2\hat{i}+2\hat{j}$. Find the velocity of body $D\left(\overrightarrow{V_D}\right)$ at that instant.

Water and mercury are filled in two identical cylindrical vessels up to the same height. Both vessels have a tiny hole in the wall at the bottom. The velocity of water and mercury coming out of the holes are $v_1$ and $v_2$ respectively. Then:

[Density of mercury $=13.6\text{g/cc}$]

In the system shown in the figure, the string, springs and pulley are weightless. The force constants of the two springs are $k_1=k$ and $k_2=2k$. Block of mass $M$ is pulled vertically down from its equilibrium position and released. Calculate the angular frequency of oscillation.

[Assume the top surface of the block (represented by line $AB$) always remains horizontal]

Figure shown below represents, parallel combination of two metallic slabs of same thickness having area of cross sections $2{\text{m}}^2$ and $3{\text{m}}^2$ respectively, connected to the same temperature difference. If the thermal conductivities of each slab is $300{\text{W/m}}^{\circ }\text{C}$, then equivalent thermal conductivity is (in ${\text{W/m}}^{\circ }\text{C}$)

The magnetic field energy in an inductor changes from maximum value to minimum value in 5 ms when connected to an AC source. The frequency of the source is

Let a force $\overrightarrow{F}$ be acting on a body free to rotate about a point $O$ and let $\overrightarrow{r}$ be the position vector of any point $P$ on the line of action of the force. Then torque $\overrightarrow{\tau }$ of this force about point $O$ is defined as :

$\overrightarrow{\tau }=\overrightarrow{r}\times \overrightarrow{F}$

Given, $\overrightarrow{F}=(2\hat{i}+3\hat{j}-\hat{k})N$ and $\overrightarrow{r}=(\hat{i}-\hat{j}+6\hat{k})m$. Find the torque of this force.

A particle of mass 100 g is fired with a velocity $20mse{c}^{-1}$ making an angle of ${30}^{\circ }$ with the horizontal. When it rises to the highest point of its path then the change in its momentum is

A person sees his virtual image by holding a mirror very close to the face.

When he moves the mirror away from his face, the image becomes inverted.

What type of mirror he is using

The potential at a point is $20V$. What will be the work done to bring a charge of $0.5C$ from infinity to this point?

Find the differentiation of $y$ w.r.t $x$ ,if $y=\frac{\sin x}{x}$

Find the value of $M$ in terms of m if the system is in equilibrium.

The sum of all the angles in an octagon is

A player kicks a football at an angle of ${45}^{\circ }$ with a velocity of $30m{s}^{-1}$. A second player $120m$ away along the direction of kick starts running to receive the ball at that instant. Find the speed with which second player should run to reach the ball before it hits the ground $(g=10m{s}^{-2})$