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

A dumbbell is placed in water of density r. It is observed that by attaching a mass m to the rod, the dumbbell floats with the rod horizontal on the surface of water and each sphere exactly half submerged as shown in the figure. The volume of the mass m is negligible. The value of length l is

A slender homogeneous rod of length 2L floats partly immersed in water, being supported by a string fastened to one of its ends, as shown. The specific gravity of the rod is 0.75. The length of rod that extends out of water is :

Two cubes of size 1.0 m sides, one of relative density 0.60 and another of relative density = 1.15 are connected by weightless wire and placed in a large tank of water. Under equilibrium the lighter cube will project above the water surface to a height of

An open-ended U-tube of uniform cross-sectional area contains water (density 1.0 gram/centimeter³) standing initially 20 centimeters from the bottom in each arm. An immiscible liquid of density 4.0 grams/ centimeter³ is added to one arm until a layer 5 centimeters high forms, as shown in the figure above. What is the ratio h₂/h₁ of the heights of the liquid in the two arms?

A light semi cylindrical gate of radius R is piovted at its mid point O, of the diameter as shown in the figure holding liquid of density r. The force F required to prevent the rotation of the gate is equal to