JEE Main 2024 — 5 April, Shift 2 — Question 37

$\quad \overrightarrow{\mathrm{P}}=\cos (\mathrm{kt}) \hat{\mathrm{i}}-\sin (\mathrm{kt}) \hat{\mathrm{j}} ;|\overrightarrow{\mathrm{P}}|=1$

$\because \overrightarrow{\mathrm{P}}=\mathrm{m} \overrightarrow{\mathrm{v}}$

$\therefore \hat{\mathrm{P}}=\hat{\mathrm{V}}$

$\Rightarrow \hat{\mathrm{v}}=\cos (\mathrm{kt}) \hat{\mathrm{i}}-\sin (\mathrm{kt}) \hat{\mathrm{j}}$

$\hat{a}=\frac{-k \sin (k t) \hat{i}-k \cos (k t) \hat{j}}{k}$

$\Rightarrow \hat{a}=-\sin k t \hat{i}-\cos k t \hat{j}$

$\because \hat{F}=\hat{a}=-\sin k t \hat{i}-\cos k t \hat{j}$

$\cos \theta=\frac{\hat{\mathrm{F}} \cdot \hat{\mathrm{P}}}{|\hat{\mathrm{F}}||\hat{\mathrm{P}}|}=-\frac{\sin \mathrm{kt} \cos \mathrm{t}+\sin \mathrm{kt} \cos t}{1 \times 1}=0$

$\Rightarrow \theta=\frac{\pi}{2}$