The ends of a line segments are P (1, 3) and Q (1, 1). R is a point on the line segment PQ such that PR : QR = 1 : λ. If R is an interior point of the parabola y² = 4x then -

The circle x² + y² + 2λx = 0, λ  R touches the parabola y² = 4x externally, then -

If b and c are the length of the segments of and focal chord of a parabola y² = 4ax. Then the length of the semi-latus rectum is -

A parabola is drawn with its focus (3, 4) and vertex at the focus of the parabola y² – 12 x – 4y + 4 = 0. The equation of the parabola is -

The range of values of λ for which the point (λ, – 1) is exterior to both the parabola y² = |x| is -

The value of P such that the vertex of y = x² + 2px + 13 is 4 unit above the x axis is -

In the figure, the blocks have unequal masses and . has a downward acceleration a. The pulley P has a radius r, and some mass. The string does not slip on the pulley–

A uniform rod AB of length is free to rotate about a horizontal axis passing through A. The rod is released from rest from horizontal position. If the rod gets broken at midpoint C when it becomes vertical, then just after breaking of the rod :

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.