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

The equation of the parabola whose vertex and focus lie on the axis of at distances a and from the origin, respectively, is

Consider, where p is a real number, and
Statement‐l:: If line is a chord of circle , then line is not always adiameter ofcircle C.
Statement‐II :: If line is a diameter ofcircle , then line is not a chord ofcircle C.

The equation of the circle passing through the points (1, 0) and (0,1) and having the smallest radius is:

Therefore the correct option is Choice 3

If a circle passes through the point (a, b) and cuts the circle orthogonally, then the equation of the locus of its centre is‐

Hence locus of the given condition is optionD

The square of the length of tangent from (3, -4) on the circle

Hence option C is correct

A hollow sphere of mass M and radius r is immersed in a tank of water (density r_w). The sphere would float if it were set free. The sphere is tied to the bottom of the tank by two wires which makes angle 45° with the horizontal as shown in the figure. The tension T₁ in the wire is :

A small wooden ball of density r is immersed in water of density to depth h and then released. The height H above the surface of water up to which the ball will jump out of water is