25. A particle moves in the x-y plane under the action of a force F such that the value of its linear momentum (overrightarrow (P)) at anytime t is P_(x)=2cost,p_(y)=2sint The angle Theta between overrightarrow (F) and overrightarrow (P) at a given time t. will be [MNR 1991;UPSEAT 2000] (a) Theta =0^circ (b) Theta =30^circ (c) Theta =90^circ (d) Theta =180^circ

To find the angle between the force and the linear momentum , we need to use the relationship between the force, momentum, and acceleration. The force acting on the particle is given by Newton's second law: where is the mass of the particle and is the acceleration. The acceleration can be found from the time derivative of the velocity, which in turn can be found from the time derivative of the momentum: Given the momentum components: we find the velocity components by differentiating with respect to time: Next, we find the acceleration components by differentiating the velocity components with respect to time: The force components can be found using Newton's second law: The angle between the force and the momentum is given by: Since \(\tan t = \tan(\pi - t)\), we have: This implies that .Therefore, the correct answer is:(c)