JEE Advanced Mi...

A football of radius R is kept on a hole of radius r (𝑟 < 𝑅) made on a plank kept horizontally. One end of the plank is now lifted so that it gets tilted making an angle 𝜃 from the horizontal as shown in the figure below. The maximum value of 𝜃 so that the football does not start rolling down the plank satisfies (figure is schematic and not drawn to scale) -

A spherical shell has a radius R and mass M . The gravitational force on a particle of mass m located at a distance r from the center of the shell is calculated using integral calculus. Which of the following integrals correctly represents the force acting on the particle inside the shell?

A projectile is launched from the origin at an angle of 45^∘ with respect to the horizontal axis. At the peak of its trajectory (point P), determine the radial and transverse components of its acceleration, expressed in terms of gravitational acceleration  g .

A first-order gas-phase reaction A(g) → B(g) + C(g) has a half-life of 30 minutes. If the initial pressure of A is 800 torr what will be the total pressure of the system after 60 minutes?

Consider the circle x² + y² = 9 and the parabola y² = 8x. They intersect at P and Q in the first and the fourth quadrants, respectively. Tangents to the circle at P and Q intersect the x-axis at R and tangents to the parabola at P and Q intersect the x-axis at S.

The ratio of the areas of ΔPQS and ΔPQR is