A man stretches a spring attached to the front wall of a railway carriage through a distance in time t opposite to the motion of the carriage , during which the train itself moved through L Then the work done by the man in a system fixed to the train will be :

A body of mass m sliding down a movable inclined plane of mass 2 m , assuming friction is absent everywhere the kinetic energy of 2 m as a function of time is :

In the figure shown the velocity of platform while string is winding on the motor shaft with velocity 5 m/s will be :

Two blocks are arranged as shown in the figure The relation between acceleration a₁ and a₂ is :

A particle P is moving on a circle under the action of only one force acting always towards fixed point O on the circumference . The ratio of and

In the pulleys system as shown in the figure a particle of mass m falls from a height h₀ on the mass m and gets stick to it. Speed with which the system moves just after the collision is

A 10 kg block is pulled along a frictionless surface in the form of an arc of a circle of radius 10 m The applied force F is 200 N as shown in figure. If the block started from rest at P :

A square plate of mass m , length a is free to rotate about the axis XX¢ shown in figure . At a distance from an edge . If is released from horizontal position shown in figure , find the angular speed attained by the plate , when it will pass through vertical position .

A bob of mass m is suspended by a string from a train of mass M, free to move on a horizontal surface The bob is given a horizontal velocity V₀ The maximum height attained by the bob is

A small block is released from a point A of a rough semicircular track as shown in figure. When it moves along the track :

A man standing on flat car under the shed of a small umbrella as shown in figure Rainfall starts at 30 m/s at an angle 30° with the vertical as shown. Find the speed and direction in which flatcar should be moved in order to save man from rain fall .

In the diagram shown, the rod is rigid and massless. A constant force F = mg has been applied vertically downwards on the rod The wall and the horizontal surface are perfectly smooth If the initial value of the angle is equal to 45° then the minimum value of coefficient of friction between the blocks for on relative motion to start between the blocks, is equal to

A particle of mass m is projected at an angle of 60° with a velocity of 20 m/s relative to the ground from a plank of same mass m which is placed on smooth surface Initially plank was at rest The minimum length of the plank for which the ball will fall on the plank itself is

Statement‐I:: The values of a for which the point lies inside the triangle formed by the lines x=0, x+y=2, and 3y=x is (0,1) .
Statement‐II:: The parabola meets the line at (1, 1) .

Tangents are drawn from the point (-1,2) on the parabola The length, these tangents will intercept on the line :

Hence finally the length is 

Let A and B be two points on a parabola with vertex V such that VA is perpendicular to VB and θ is the angle between the chord VA and the axis of the parabola The value of is