The position vector $$\overrightarrow r $$ of a particle of mass m is given by the following equation $$$\overrightarrow r \left( t \right) = \alpha {t^3}\widehat i + \beta {t^2}\widehat j,$$$where $$\alpha = {{10} \over 3}m{s^{ - 3}}$$, $$\beta = 5\,m{s^{ - 2}}$$ and m = 0.1 kg. At t = 1 s, which of the following statement(s) is(are) true about the particle?

Consider two solid spheres P and Q each of density 8 gm cm^–3 and diameters 1 cm and 0.5 cm, respectively. Sphere P is dropped into a liquid of density 0.8 gm cm^–3 and viscosity $$\eta $$ = 3 poiseulles. Sphere Q is dropped into a liquid of density 1.6 gm cm^–3 and viscosity $$\eta $$ = 2 poiseulles. The ratio of the terminal velocities of P and Q is

A metal is heated in a furnace where a sensor is kept above the metal surface to read the power radiated (P) by the metal. The sensor has a scale that displays $${\log _2}\left( {{P \over {{P_0}}}} \right)$$, where P₀ is a constant. When the metal surface is at a temperature of 487^oC, the sensor shows a value 1. Assume that the emissivity of the metallic surface remains constant. What is the value displayed by the sensor when the temperature of the metal surface is raised to 2767^oC?

A parallel beam of light is incident from air at an anglea on the side PQ of a right angled triangular prism of refractive index n = $$\sqrt 2 $$. Light undergoes total internal reflection in the prism at the face PR when a has a minimum value of 45°. The angle q of the prism is :