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Three light strings are connected at the point $P$. A weight $W$ is suspended from one of the strings. End $A$ of string $AP$ and end $B$ of string $PB$ are fixed as shown. In equilibrium, $PB$ is horizontal and $PA$ makes an angle of ${60}^{\circ }$ with the horizontal. If the tension in $PB$ is $30N$, then the tension in $PA$ and weight $W$ are respectively given by

A box contains $N$ molecules of a perfect gas at temperature $T_1$ and pressure $P_1$. The number of molecules in the box is doubled keeping the total kinetic energy of the gas same as before. If the new pressure and temperature are $P_2$ are $T_2$ respectively, then

A small object is placed $10\text{cm}$ in front of a plane mirror. If you stand behind the object $30\text{cm}$ from the mirror and look at its image, the distance between the observer and the image of the object will be

A particle is projected vertically upwards from the surface of the earth (radius $R)$ with a speed equal to one-fourth of escape velocity. The maximum height attained by it from the surface of the earth is

In the figure, the blocks $A,B$ and $C$ of mass $m$ each have acceleration $a_1,a_2$ and $a_3$ respectively. $F_1$ and $F_2$ are external forces of magnitudes $2mg$ each. What is the relation between the acceleration of the masses $A$, $B$ and $C$?

The ratio of the adiabatic to isothermal elasticities of a triatomic gas is

Two coherent sources, emitting waves of wavelength $\lambda$, are placed at a separation of $3\lambda$. A detector is moving around them on a large circle. The number of minima detected by it in one complete rotation is

The angular frequency of a particle executing SHM about the origin is $10rad{s}^{-1}$ and it's amplitude is 0.2 m. If at t=0 the particle was at 0.1 m. What is the equation of it's motion?

Two mutually perpendicular insulated long conducting wires carrying equal currents $I$, intersect at origin. Then the resultant magnetic induction at point $P(2m,3m)$ will be

The change in internal energy of two moles of a monoatomic gas, when the temperature increases by ${50}^{\circ }\text{C}$ is

[Take $R=2{\text{cal mol}}^{-1}{}^{\circ }{\text{C}}^{-1}$]

A person standing on a road has to hold his umbrella at 60${}^{\circ }$ with the vertical to keep away rain. He throws the umbrella and starts running in horizontal direction at a speed of $20\text{m/s}$. He finds that the rain drops are hitting his head vertically. Find the speed of rain drops with respect to ground.

The bob of the pendulum shown in the figure describes an arc of a circle in a vertical plane. If the tension in the cord is $3$ times of the weight of the bob for the position shown, the velocity of the bob at this position is

A cyclist goes round a circular path of circumference 34.3 m in $\sqrt{22}$ sec. the angle made by him, with the vertical, will be

Find slope of the Straight line passing through the points $(-3,8)$ and $(1,4)$.

A particle $A$ is projected with an initial velocity of $60\text{m/s}$ at an angle ${30}^{\circ }$ to the horizontal. At the same time a second particle $B$ is projected in opposite direction with initial speed of $50\text{m/s}$ from a point at a distance of $100\text{m}$ from $A$. If the particles collide in air find the angle of projection $\alpha$ for particle $B$

A wheel of radius $r=1\text{m}$ rolls without slipping with an angular velocity of $\omega =\frac{\pi }{4}\text{rad/s}$ about its centre and velocity of its centre of mass is $v_{COM}=2\text{m/s}$. If the point $P(0,0)$ is in contact with the ground initially, find its position vector after time $t=1\text{s}$.

A simple pendulum is oscillating without damping.When the displacement of the bob is less than maximum, its acceleration vector $\overrightarrow{a}$ is correctly shown in

A block of mass $2\text{kg}$ rests on a rough inclined plane making an angle of ${30}^{\circ }$ with the horizontal. The coefficient of static friction between the block and the plane is $0.7$. The frictional force on the block is

If $|\overrightarrow{A}|=4$, $|\overrightarrow{B}|=3$ and the angle between $\overrightarrow{A}$ and $\overrightarrow{B}$ is ${30}^{\circ }$, then the scalar product of these two vectors is

A stone weighing $3\text{kg}$ falls from the top of a tower $100\text{m}$ high and buries itself $2\text{m}$ deep in sand. Find the time (in seconds) of penetration $(g=10{\text{m/s}}^2)$

Two simple pendulums whose lengths are $100\text{cm}$ and $121\text{cm}$ are suspended side by side. Their bobs are pulled together and then released. After how many minimum oscillations of the longer pendulum, will the two be in phase again.

A uniform chain of mass M and length L lies on a horizontal table such that one third of its length hangs from the edge of the table. The work done is pulling the hanging part on the table will be

If a square shape of side $5\text{cm}$ is cut from a square sheet of side $10\text{cm}$. Find the sum of $x$ and $y$ coordinates of the centre of mass of the remaining part as shown in the figure below. Consider the origin at the corner as shown in figure. Material is considered to be of uniform density.

The velocity $v$ of a particle as a function of its position $\left(x\right)$ is expressed as $v=\sqrt{c_1-c_{2}x}$, where $c_1$ and $c_2$ are positive constants. The acceleration of the particle is

Find the speed of block $B$ if the speed of wedge $A$ is $2\text{m/s}$ as shown in the figure. Assume the surface to be smooth and string inextensible.

A person of mass $60kg$ is inside a lift of mass $940kg$ and he presses the button on the control panel. The lift starts moving upwards with an acceleration of $1{m/s}^2$. If $g=10{m/s}^2$, the tension in the supporting cable is

The value of coefficient of volume expansion of glycerin is $5\times {10}^{-4}{{}^{\circ }\text{C}}^{-1}$. The fractional change in the density of glycerin for a rise of $40{}^{\circ }\text{C}$ in its temperature is

The container $A$ contains ice at $0^{\circ }\text{C}$. A conducting uniform rod $PQ$ of length $4R$ is used to transfer heat to the ice in the container. The end $P$ of the rod is maintained at ${100}^{\circ }\text{C}$ and the other end $Q$ is kept inside the container $A$. The complete ice melts in $23\text{minutes}$. In another experiments, two conductors in the shape of quarter circles of radii $2R$ and $R$ are welded to the conductor $PQ$ at $M$ and $N$ respectively and their other ends are inserted inside the container $A$. All conductors are made of the same material and have the same cross sectional area. Once again, the end $P$ is maintained at ${100}^{\circ }\text{C}$ and this time, the complete ice melts in $t\text{minutes}$. Find $t$.

[Assume no heat loss from the curved surface of the rods and take $\pi \simeq 3.0$]

Statement 1 – A square of side 2m lying on y – z plane can be represented as $4\hat{i}or-4\hat{i}$ vectorially.

Statement – Any planar area can have two directions perpendicular to it.

Select the correct option:

A particle is undergoing SHM along a straight line so that its period is 12 s. The time it takes in traversing a distance equal to half its amplitude from the equilibrium position is

A neutron mass $1.67\times {10}^{-27}kg$ moving with a velocity $1.2\times {10}^{7}m{s}^{-1}$ collides head-on with a deuteron of mass $3.34\times {10}^{-27}kg$ initially at rest. If the collision is perfectly elastic,, what would be speed of deuteron after the collision?

If velocity of a particle is $v\left(t\right)=2t-4\text{m/s}$, the average speed and magnitude of average velocity of the particle in $10\text{sec}$ respectively are

A large plastic plate of area $40{\text{m}}^2$ is floating on the surface of a river. Determine the tangential force that should be applied on the plastic plate to move the plate at $4\text{m/s}$ on the surface of the river. The depth of the river is $2\text{m}$ and coefficient of viscosity of water is ${10}^{-2}$ poise.

Find the fundamental, first overtone and second overtone frequencies of an open organ pipe of length 20 cm. Speed of sound in air is 340$m{s}^{-1}$.

A body is projected from the top of a tower of height $32\text{m}$ with a velocity of $20\text{m/s}$ at an angle of ${37}^{\circ }$ above the horizontal. The time of flight of the projectile is

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

Harry carried out an experiment to find out how different surfaces (A, B, C and D) affect the distance of a car travelling at 50 km/h needed to stop once the brakes applied. The results are shown below in given table. Which type of road will provide the most friction for the car to stop?



हैरी यह ज्ञात करने के लिए एक प्रयोग करता है, कि भिन्न-भिन्न पृष्ठ (A, B, C तथा D) 50 km/h से गतिशील एक कार को ब्रेक लगाकर रोकने के लिए आवश्यक दूरी को किस प्रकार प्रभावित करते हैं। परिणाम नीचे दी गई सारणी में दर्शाए गए हैं तथा किस प्रकार की सड़क कार के रूकने के लिए सर्वाधिक घर्षण प्रदान करेगी?

A block of mass $m$, attached to a spring of spring constant $k$, oscillates on a smooth horizontal table. The other end of the spring is fixed to a wall. The block has a speed $v,$ when the spring is at its natural length. How far will the block move on the table before coming to an instantaneous rest?

The ratio of radii of two wires of the same material is $2:1$. If these wires are stretched by equal forces, then the ratio of stresses produced in them will be

Given that $x=2^p$ and $y=4^q$, then value of ${\log }_2\left(x^{3}y\right)$ is

A block of mass $m$ lying on a smooth horizontal surface is attached to a spring (of negligible mass) of spring constant $k$. The other end of the spring is fixed as shown in the figure. The block is initially at rest in its equilibrium position. If now the block is pulled with a constant force $F$, the maximum speed of the block is

An electron jumps from the $4^{th}$ orbit to the $2^{nd}$ orbit of hydrogen atom. Given the Rydberg's constant $R={10}^{5}c{m}^{-1}$. The frequency in Hz of the emitted radiation will be

When there is no change of state in a process, which of the following is not required to calculate the heat lost or gained in the process?

The penetrating power of X-rays increases with the

A point object $S$ is placed midway between two concave mirrors having equal focal length $f$ as shown in the figure. Find the possible value(s) of $d$ for which only one image is formed of the object.

A bock of mass $0.9\text{kg}$ is attached to a spring of force constant k is lying on a frictionless floor. The spring is compressed to $\sqrt{2}\text{cm}$ and the block is at a distance of $\frac{1}{\sqrt{2}}\text{cm}$ from the wall as shown in the figure. When the block is released, it makes an elastic collision with the wall and its period of motion is $0.2\text{s}$. Find the approximate value of $k$.

de-Broglie wavelength of a body of mass m and kinetic energy E is given by