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If $x=sec\theta -tan\theta ,y=cosec\theta +cot\theta$, then $xy+1=$ _____

Two stones are dropped down simultaneously from different heights with initial speed $u_1=0$ and $u_2=30\text{m/s}$ respectively. If the initial distance between them is $10\text{m}$ and mass of the first stone is twice that of the second stone, find the velocity of their $COM$ after $2\text{s}$. [Take $g=10{\text{m/s}}^2$]

Distance travelled by a particle at any instant $t$ can be represented as $S=A(t+B)+C{t}^2$. The dimensions of $B$ are

Three vectors $\overrightarrow{P},\overrightarrow{Q}$ and $\overrightarrow{R}$ are such that $|\overrightarrow{P}|=|\overrightarrow{Q}|,|\overrightarrow{R}|=\sqrt{2}|\overrightarrow{P}|$ and $\overrightarrow{P}+\overrightarrow{Q}+\overrightarrow{R}=0$. The angle between $\overrightarrow{P}$ and $\overrightarrow{Q},\overrightarrow{Q}$ and $\overrightarrow{R}$, and $\overrightarrow{P}$ and $\overrightarrow{R}$ are

Two spheres $A$ and $B$ having radii $3\text{cm}$ and $5\text{cm}$ respectively are coated with carbon black on their outer surface. The wavelengths of maximum intensity of emission of radiation are $300\text{nm}$ and $500\text{nm}$ respectively. The respective powers radiated by them are in the ratio of :

A block of mass $m$ is attached rigidly to a spring of stiffness $k_1$ & touches another spring of stiffness $k_2$ as shown in the figure. Assuming the surface is smooth, the period of oscillation of the block is

Water flows through a horizontal tube of variable cross section. The area of cross section at A and B are 4 mm² and 2 mm² respectively. If 1 cc of water enters per second through A, find the speed of water at B?

A stone tied to an inextensible string of length $l=1\text{m}$ is kept horizontal. If it is released, find the angular speed (rad/s) of the stone when the string makes an angle $\theta ={30}^{\circ }$ with horizontal.

What is the maximum speed with which a vehicle can negotiate a convex bridge of radius $10\text{m}$ without leaving the surface of the road.

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

A body is rotated in the vertical plane by means of a thread of length $l$ with minimum possible velocity. When the body goes up and reaches at the highest point B of its path, the thread breaks and the body moves on a parabolic path under the influence the gravitational field as shown in the diagram. The horizontal range AC would be

If the speed of the particle is $v=8t^3-3t^2+2\text{m/s}$, the distance of the particle as a function of time $t$ is

A particle is thrown vertically upwards. Its velocity at half of the maximum height is $20\text{m/s}$. Then the maximum height attained by it will be: (Take $g=10{\text{m/s}}^2$)

The position $x$ of a particle having mass $m$ moving along $x$ - axis at time $t$ is given by the equation $t=\sqrt{x}+2$, where $x$ is in $\text{m}$ and $t$ in $\text{s}$. Find the work done by the force in first $4\text{s}$.

A sound wave travels a distance $l$ in helium gas in time $t$ at a particular temperature. If at the same temperature a sound wave is propagated in oxygen gas, it will cover the same distance $l$ in what time (approximately)?

A disc is suspended at a point $\frac{R}{2}$ above its center, Find its period of oscillation.

Given two vectors a vector and b vector where b makes an angle theta with the positive direction of a as shown. Find the magnitude of the resultant according to triangle law in this case.

A manometer connected to a closed tap reads 3.5 $\times$ ${10}^5$ N/$m^2$. When the valve is opened, the reading of manometer falls to 3.0 $\times$ ${10}^5$ N/$m^2$, then velocity of flow of water is

A bus moves from stop $A$ to the next stop $B$. Its acceleration varies as $a=\alpha -\beta x$, where $\alpha$ and $\beta$ are positive constants and $x$ is the distance from the stop $A$ to stop $B$. The distance between $A$ and $B$ and the maximum speed of the bus are

Two parallel rail tracks run north-south. Train A moves north with a speed of 54 km/h and train B moves south with a speed of 90 km/hr. What is the velocity of a monkey running on the roof of the train A against its motion (with its velocity of 18 km/h with respect to the train A) as observed by a man standing on the ground?

During a tennis match, a player serves at $23.6{\text{ms}}^{-1}$, the ball leaving the racquet horizontally $2.37\text{m}$ above the court surface. By how much does the ball clear the net which is $12\text{m}$ away and $0.9\text{m}$ high?

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

Paragraph for below question



नीचे दिए गए प्रश्न के लिए अनुच्छेद



A uniform rod of mass M and length l = 1 m hinged at A in the vertical plane, is released from rest when it is in horizontal position. At angular position θ = 60°, angular velocity of rod is ω. Then [g = 10 m/s²]



जब ऊर्ध्वाधर तल में A पर किलकित द्रव्यमान M व लम्बाई l = 1 m की एक समरूप छड़ क्षैतिज स्थिति में है, तब इसे विराम से छोड़ा जाता है। कोणीय स्थिति θ = 60° पर, छड़ का कोणीय वेग ω है। तब [g = 10 m/s²]





Q. Tangential acceleration of end B at angular position, θ = 60° is



प्रश्न - कोणीय स्थिति θ = 60° पर, सिरे B का स्पर्शरेखीय त्वरण है

Find the direction of the reaction force on the block for the figure shown.

A body of mass 2 kg moving with a velocity of 3 m/sec collides head on with a body of mass 1 kg moving in opposite direction with a velocity of 4 m/sec. After collision, two bodies stick together and move with a common velocity which in m/sec is equal to

One quadrant of a circular disc is removed from the original disc of radius $5\text{cm}$. Take reference axes and origin $O$ as shown in the figure. Find the sum of the magnitudes (in $\text{cm}$) of the $x$ and $y$ coordinates of the $\text{COM}$ of the new shape. Consider the material to be of uniform density.

A cylindrical tank has a hole of area $1{\text{cm}}^2$ at its bottom. If water is allowed to flow into the tank from a tube above it at the rate of $70{\text{cm}}^3/\text{sec}$, then the maximum height up to which water can rise in the tank is -

[Take $g=980{\text{cm/s}}^2$]

A body of mass $100\text{grams}$ is allowed to fall freely under gravity. Find its momentum after $10$ seconds (in $\text{N- s}$)

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 center of the hoop is zero. What will be the velocity of the center of the hoop when it ceases to slip?

Browsing through Mr. Fox's old scientific notes, Bruce Wayne encounters a secret temperature scale, which Mr. Fox called Z, in order to keep the Batmobile's technology a secret. On that scale, the freezing and the boiling points of water were recorded to be -7⁰Z and 78⁰Z, respectively. The records instruct to maintain the coolant temperature at -24⁰Z for top speed. Mr. Wayne, naturally comfortable with the Fahrenheit scale due to his American upbringing, needs to calculate the coolant temperature in ⁰F. Help him choose the correct value.

The maximum acceleration of a body moving in SHM is $a_0$ and maximum velocity is $v_0$. The amplitude of the oscillation is given by,

A bob of mass $500\text{g}$ attached to a light inextensible string revolving vertically in a radius $R$ is given an initial velocity $v=\sqrt{7gR}$ at the bottom most point. The tension in the string at the bottom most point is

One mole of an ideal monoatomic gas has initial temperature $T_0$ is made to go through the cyclic process $abca$ as shown in figure. If $\text{U}$ denotes the internal energy, then choose the correct alternative.

Consider the situation shown in figure. Find the maximum angle $\theta$ for which the light suffers total internal reflection at the vertical surface.

In a SHM, potential energy of a particle at mean position is $E_1$ and kinetic energy is $E_2,$ then

Two thin convex lenses of focal length $f_{1}and{f}_2$ are separated by a horizontal distance d (where $d<f_1,d<f_2$) and their centers are displaced by a vertical separation $\Delta$ as shown in figure.



Taking the origin of coordinates O at the center of the first lens, the x and y coordinates of the focal point of this lens system, for a parallel beam of rays coming from the left, are given by:

Two satellites $S_1$ and $S_2$ revolve round a planet in coplanar circular orbits in the same sense. Their periods of revolution are 1 hour and 8 hours respectively. The radius of the orbit of $S_1={10}^4$km When $S_2$ is closest to $S_1$ find speed of $S_2$ relative to $S_2$ and the angular speed of $S_1$ actually observed by an astronaut at $S_1$.

The temperature of gas is increased from $400\text{K}$ to $600\text{K}$ keeping the volume of the gas constant. If initially the gas exerted a pressure of $3\text{kPa}$ on the walls of the container, then the final pressure of the gas is (in $\text{kPa}$)

In the figure shown the cart of mass '6m' is initially at rest. A particle of mass 'm' is attached to the end of the light rod which can rotate freely about A. If the rod is released from rest in a horizontal position shown, determine the velocity $v_{rel}$of the particle with respect to the cart when the rod is vertical.

Spring fitted doors close by themselves when released. You want to keep the door open for a long time, say for an hour. If you put a half kg stone in front of the open door, it does not help. The stone slides with the door and the door gets closed. However, if you sandwich a 20 gm piece of wood in the small gap between the door and the floor, the door stays open. Explain why a much lighter piece of wood is able to keep the door open while the heavy stone fails. Assume coefficient of friction between stone and ground and wood and ground are the same.

What happens in a ${\beta }^{+}$ (beta positive) decay?

A force of 50 dynes is acted on a body of mass 5 g which is at rest for an interval of 3 seconds, then impulse is

A small bead of mass $m$ moving with velocity $v$ gets threaded on a stationary semicircular ring of mass $m$ and radius $R$ kept on a horizontal table. The ring can freely rotate about its centre. The bead comes to rest relative to the ring. What will be the final angular velocity of the system?

If $\overrightarrow{A}=\hat{i}+2\hat{j}-\hat{k}$ and $\overrightarrow{B}=2\hat{i}+3\hat{j}+4\hat{k}$, Then the vector which is perpendicular to both vectors $\overrightarrow{A}$ and $\overrightarrow{B}$ is

A particle is moving along a circular path with uniform speed. Through what angle does its angular velocity changes when it completes half of the circular path?

A bullet of mass $20\text{g}$ strikes a block of mass $180\text{g}$. The bullet remains embedded in the block. Find the amplitude of the resulting SHM (all surfaces are fricitonless]

Two particles having position vectors ${\overrightarrow{r}}_1=(3\hat{i}+5\hat{j})m$ and ${\overrightarrow{r}}_2=(-5\hat{i}-3\hat{j})m$ are moving with velocities

$\overrightarrow{v_1}=(4\hat{i}+3\hat{j})m/s$ and ${\overrightarrow{v}}_2=(a\hat{i}+7\hat{j})m/s$.If they collide after 2s, find the value of a.

A force $F=-k(yi+xj)$ (where $k$ is a positive constant) acts on a particle moving in the $xy$ - plane. Starting from the origin, the particle is taken along the positive $x$ - axis to the point $(a,0)$ and then parallel to the $y$ - axis to the point $(a,a)$. The total work done by the force $F$ on the particle is