Assertion : For projection angle ( ^ -1 (4)), the horizontal and...

- Assertion: The assertion states that for a projection angle of (\tan ^{-1}(4)), the horizontal range ($R$) and the maximum height ($H$) are equal. This statement is true. The formulas for range and maximum height are: (R=\frac{u^2 \sin (2 \theta)}{g} \text{ and } H=\frac{u^2 \sin ^2 \theta}{2 g}) For $R=H$, we set the equations equal to each other: ($$\begin{aligned} & \frac{u^2 \sin (2 \theta)}{g}=\frac{u^2 \sin ^2 \theta}{2 g} \\ & 2 \sin \theta \cos \theta=\frac{\sin ^2 \theta}{2} \\ & 4 \cos \theta=\sin \theta \\ & \frac{\sin \theta}{\cos \theta}=4 \\ & \tan \theta=4 \\ & \theta=\tan ^{-1}(4) \end{aligned}$$) This confirms the assertion is correct. - Reason: The reason states that the maximum range of a projectile is directly proportional to the square of the velocity and inversely proportional to the acceleration due to gravity. The formula for the maximum range is (R_{\text{max }}=\frac{u^2}{g}). This is obtained when the projection angle (\theta=45^{\circ}). This formula shows that (R_{\text{max }}) is directly proportional to the square of the initial velocity ($u^2$) and inversely proportional to the acceleration due to gravity ($g$). Thus, the reason is true. Conclusion Both the Assertion and the Reason are correct. However, the reason discusses the maximum range and its dependencies, while the assertion is about the condition for any range to be equal to the height. The reason, therefore, is a correct statement about projectile motion but does not explain the specific condition in the assertion. This corresponds to option (2).