A thin rod of mass m and length is hinged to a ceiling and it is free to rotate in a vertical plane. A particle of mass m, moving with speed v strikes it as shown in the figure and gets stick with the rod. The value of v , for which the rod becomes horizontal after collision is

An impulsive force F acts horizontally on a solid sphere of radius R placed on a horizontal surface. The line of action of the impulsive force is at a height h above the centre of the sphere. If the rotational and translational kinetic energies of the sphere just after the impulse are equal, then the value of h will be

The disc of radius r is confined to roll without slipping at A and B. If the plates have the velocities shown, then

The figure shows a uniform rod lying along the x-axis. The locus of all the points lying on the xy-plane, about which the moment of inertia of the rod is same as that about O is :

Figure shows a uniform disk, with mass M = 2.4 kg and radius R = 20 cm, mounted on a fixed horizontal axle. A block of mass m = 1.2 kg hangs from a massless cord that is wrapped around the rim of the disk. The tension in cord is

A small block of mass 'm' is rigidly attached at 'P' to a ring of mass '3m' and radius 'r'. The system is released from rest at and rolls without sliding. The angular acceleration of hoop just after release is

x – 2 = t², y = 2t are the parametric equations of the parabola

parametric form gives us the general coordinates of the curve. we can solve for the two to build the relationship between the x and y coordinates which gives us the locus

Vertex of the parabola y² + 2y + x = 0 lies in the quadrant

vertex of the parabola is the point that divides the curve into two symmetric parts.

The equation of the parabola with focus (3, 0) and the directrix x + 3 = 0 is

the locus of all points which are equidistant from a point called focus and a aline called directrix is known as a parabola. as per this definition, we can solve the  given question.

Length of the shortest normal chord of the parabola y² = 4x is-

If the tangents and normals at the extremities of a focal chord of a parabola intersect at (x₁, y₁) and (x₂, y₂) respectively, then-

The common tangents to the circle x² + y² = a²/2 and the parabola y² = 4ax intersect at the focus of the parabola-

If on a given base triangle b be described such that the sum of the tangents of the base angles is constant (k), then the locus of third vertex is -

Set of values of ‘h’ for which the number of distinct common normals of (x – 2)² = 4(y – 3) and x² + y² – 2x – hy – c = 0 (c > 0) is 3, is -

PA and PB are the tangents drawn to y² = 4x from point P. These tangents meet the y-axis at the points A₁ and B₁ respectively. If the area of triangle PA₁ B₁ is 2 sq. units, then locus of ‘p’ is -

If the chord of contact of tangents from a point P to the parabola y² = 4ax, touches the parabola x² = 4by, then the locus of P is a/an -