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An enclosure of volume V contains a mixture of 8g of oxygen, 14g of nitrogen and 22g of carbon dioxide at absolute temperature T. The pressure of the mixture of gases is (R is universal gas constant)

A particle is moving in a circular path of radius $50cm$ with an uniform speed, completing one revolution in $10s$. The magnitude of centripetal acceleration of the particle is

An air column is constructed by fitting a movable piston in a long cylindrical tube. Longitudinal waves are sent in the tube by a tuning fork of frequency $416\text{Hz}$. How far from the open end should the piston be so that the air column in the tube may vibrate in its first overtone neglecting the end corrections?

$[$ speed of sound in air $=333\text{m/s}]$

A thin copper wire of length $l$ increases its length by 1% when heated from temperature $T_1$ to $T_2$. What is the percentage change in area when a thin copper plate having dimensions $2l\times l$ is heated from $T_1$ to $T_2$?

A block of mass m is placed on an inclined plane. With what acceleration should the system move towards right on a horizontal surface so that m does not slide on the surface of inclined plane? Assume all surfaces are smooth.

A sound wave of frequency $10\text{kHz}$ is travelling in air with speed of $340\text{m/s}$. Find the minimum separation between two points in the medium where the phase difference is $2\pi$.

A non-uniform thin rod of length $10\text{m}$ is placed along $x$- axis such that one of its ends is at the origin. The linear mass density of the rod varies with respect to distance from the end as $\lambda =l_{0}x$, where $l_0$ is a constant. Then the position of the centre of mass of the rod w.r.t the origin will be:

A block released from rest from the top of a smooth inclined plane of inclination ${45}^{\circ }$ takes $t$ seconds to reach the bottom. The same block released from rest from the top of a rough inclined plane of the same inclination of ${45}^{\circ }$ takes $2t$ seconds to reach the bottom. The coefficient of friction is

A ball is projected from a certain point on the ground at an angle with the horizontal surface. The horizontal and vertical displacement x and y(in metres) vary with time (in seconds) as

$x=10\sqrt{3}t$

$y=10t-5t^2$

What is the magnitude and direction of the velocity of projection?

Four particles of masses m, 2m, 3m and 4m are kept in sequence at the corners of a square of side a. The magnitude of gravitational force acting on a particle of mass m placed at the centre of the square will be

A $900kg$ car moving at $10m/s$ takes a turn around a circular path of radius $25m$. The centripetal acceleration and the centripetal force acting on the car is given by (Neglect friction between all surfaces)

uniform circular disc of mass 200 g and radius 4.0 cm is rotated about one of its diameter at an angular speed of 10 rad/s. Find the angular momentum about the axis of rotation

The mean length of an object is $5\text{cm}$. Which of the following measurements is most accurate?

A space-ship is put into a circular orbit close to Earth's surface. What additional velocity must be imparted to the ship so that it is able to escape the gravitational pull of Earth(approximately)? $(R=6400km,g=9.8\frac{m}{s^2})$

[IIT-JEE (Screening) 1999; MP PMT 2002; KCET 2003]

Find I

The Van Der Waal’s equation of state for some gases can be expressed as: $(P+\frac{a}{V^2})(V-b)=RT$ Where $P$ is the pressure, $V$ is the molar volume and $T$ is the absolute temperature of the given sample of gas and $a,b$ and $R$ are constants.



Which of the following does not have the same dimension formula as that of $RT$ ?

A bead constrained to move in a tube of diameter $10\text{cm}$ is initially placed at the bottommost point of the tube. What minimum speed should be given to the bead so that it just reaches the topmost point in the tube?

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

The height of the point vertically above the earth’s surface, at which acceleration due to gravity becomes $1\%$ of its value at the surface is (Radius of the earth = R)

Two particles of equal masses moving with same speed collide perfectly in-elastically. After the collision, the combined mass moves with half of the speed of the individual masses. The angle between the initial momenta of individual particle is .

In a meter bridge, the balancing length from left is found to be $20cm$ when standard resistance of $1\Omega$ is in right gap. The value of unknown resistance is

The velocity of a particle moving along the positive $x$-axis is given by $v\left(t\right)=-t^3+6t^2+2t$. What is the magnitude of maximum acceleration and also find the time when it is attained?

An ideal monoatomic gas is confined in a cylinder by a spring-loaded piston of cross-sectional area $8\times {10}^{-3}{\text{m}}^2$. Initially the gas is at $300\text{K}$ and occupies a volume of $2.4\times {10}^{-3}{\text{m}}^3$. The spring is in a relaxed state. The gas is heated by a small heater coil $H$. The force constant of the spring is $8000\text{N/m}$ and the atmospheric pressure is $1.0\times {10}^5\text{Pa}$. The cylinder and piston are thermally insulated. The piston and the spring are massless and there is no friction between the piston and cylinder. There is no heat loss through the heater coil wire and thermal capacity of the heater coil is negligible. With all the above assumptions, if the gas is heated by the heater until the piston moves out slowly by $0.1\text{m}$, then the final temperature is -

The second overtone of an open organ pipe has the same frequency as the first overtone of a closed pipe $L$ metres long. The length of the open pipe will be

A car travels with the speed of $15\text{m/s}$ for $0<t\le 10\text{sec}$, $20\text{m/s}$ for $10<t\le 20\text{sec}$ and $5\text{m/s}$ for $20<t\le 25\text{sec}$. The total distance travelled by the car is

Steam is passed into $56\text{g}$ of water at ${30}^{\circ }\text{C}$ till the temperature of mixture becomes ${90}^{\circ }\text{C}$. If the latent heat of condensation of steam is $536\text{cal/g}$, the mass of the mixture will be:

(Specific heat capacity of water is $1{\text{cal/g}}^{\circ }\text{C}$)

When a long wire carrying a steady current is bent into a circular coil of one turn, the magnetic induction at its centre is B. When the same wire carrying the same current is bent to form a circular coil of n turns of a smaller radius, the magnetic induction at the centre will be

A vernier caliper has $1\text{mm}$ marks on the main scale. It has $20$ equal divisions on the vernier scale which match with $16$ main scale divisions. For this vernier caliper, the least count is

A body of mass m having an initial velocity v, makes head on elastic collision with a stationary body of mass M. After the collision, the body of mass m comes to rest and only the body having mass M moves. This will happen only when

The coefficient of apparent expansion of mercury in a glass vessel is 153 × ${10}^{-6}{/}^{\circ }C$ and in a steel vessel is 144 × ${10}^{-6}{/}^{\circ }C$. If afor steel is 12 × ${10}^{-6}{/}^{\circ }C$, then that of glass is

A load of 4.0 kg is suspended from a ceiling through a steel wire of radius 2.0 mm. Find the tensile stress developed in the wire when equilibrium is achieved. Take $g=3.1\pi m{s}^{-2}.$

Evaluate: ${\int }_1^4(cosx-\frac{3}{x^5})dx$

Three identical rods, each of mass $m$ and length $l$, form an equilateral triangle. Moment of inertia about one of the sides is

A simple pendulum has time period $T_1$. The point of suspension is now moved upward according to the relation $y=k{t}^2,(k=\frac{1m}{s^2})$ where y is the vertical displacement. The time period now becomes $T_2$. The ratio of$\frac{T_1^2}{T_2^2}$ is (Take g =$10\frac{m}{s^2})$

A uniform chain of length $L$ has one of its ends attached to the wall at point A, while $\frac{3L}{4}$ length of the chain is lying on the table as shown in figure. Then, minimum co-efficient of static friction between table and chain, so that the chain remains in equilibrium is

Nuclear binding energy is equivalent to

A network consisting of three resistors, three batteries, and a capacitor is shown in the figure



$\begin{array}{ccc}& \text{Column I}& \text{Column II}\\ \left(A\right)& \text{Current in branch EB is}& \text{(P)}10\mu C\\ \left(B\right)& \text{Current in branch CB is}& \text{(Q)}0.5\mu C\\ \left(C\right)& \text{Current in branch ED is}& \text{(R)}1.5\mu C\\ \left(D\right)& \text{Charge on capacitor is}& \text{(S)}5\mu C\\ & & \text{(T)}7.5\mu C\end{array}$

Which of the following option has the correct combination considering column - I and Column - II.

Distance of the centre of mass of a solid uniform cone from its vertex is $Z_0$. If the radius of its base is R and its heights is h, then $Z_0$ is equal to ?

A solid cylinder of mass $m$ and radius $r$ starts rolling down on inclined plane of inclination $\theta$. Friction is sufficient enough to prevent slipping. Find the speed of centre of mass of solid cylinder, when it's centre of mass has fallen through a height of $h=9\text{m}$. (Take $g=10{\text{m/s}}^2$)

The motion of a body is described by the equation $\frac{dv\left(t\right)}{dt}=6-3v\left(t\right)$ where $v\left(t\right)$ is the speed (in $m{s}^{-1}$) at time t (in s). If the body was at rest at t = 0, the magnitude of the initial acceleration is

A small ball 'A' of mass 'm' is attached rigidly to the end of a light rod of length 'd'. The structure rotates about an axis perpendicular to the rod and passing through its other end with an angular velocity '$\omega$'. The angular momentum of the system about this axis is.

A particle is thrown from point B as shown in figure. At this moment of time, the angular speed of particle with respect to an axis passing through point A and perpendicular to plane of figure is

Two spheres $A$ and $B$ have diameters in the ratio $1:2$, densities in the ratio $2:1$ and thermal capacities in the ratio $1:12$. Find the ratio of their specific heat capacities.

Consider the following data and choose the correct option.



$1.$ The time period of a simple pendulum, whose bob is a hollow metallic sphere is $T$.

$2.$ The time period of a simple pendulum is $T_1,$ when the bob is filled with sand,

$3.$ The time period of a simple pendulum is $T_2,$ when it is filled with mercury and

$4.$ The time period of a simple pendulum is $T_3,$ when it is half-filled with mercury as shown.

While measuring the acceleration due to gravity by a simple pendulum, a student makes a positive error of 1% in the length of the pendulum and a negative error of 3% in the value of time period. His percentage error in the measurement of g by the relation $g=4{\pi }^2\left(\frac{l}{T^2}\right)$ will be