(A): The maximum horizontal range of projectile is proportional to square of velocity.
(R): The maximum horizontal range of projectile is equal to maximum height at tained by projectile

Options

A. Both (A) and (R) are true and (R) is the correct explanation of (A)
B. Both (A) and (R) are true and (R) is not the correct explanation of (A)
C. (A) is true but (R) is false
D. Both (A) and (R) are false

Answer

To analyze the given statements and determine their validity, we will break down the assertions step by step.

### Step 1: Understanding Assertion A
**Assertion A:** The maximum horizontal range of a projectile is proportional to the square of its velocity.

- The formula for the range \( R \) of a projectile is given by:
\[ R = \frac{u^2 \sin 2\theta}{g} \]
where \( u \) is the initial velocity, \( \theta \) is the angle of projection, and \( g \) is the acceleration due to gravity.

- To find the maximum range, we set \( \theta = 45^\circ \) (since \( \sin 90^\circ = 1 \)):
\[ R_{max} = \frac{u^2}{g} \]

- From this equation, we can see that the maximum range \( R_{max} \) is indeed proportional to the square of the initial velocity \( u \). Thus, **Assertion A is true**.

### Step 2: Understanding Reason R
**Reason R:** The maximum horizontal range of a projectile is equal to the maximum height attained by the projectile.

- The formula for the maximum height \( h \) of a projectile is given by:
\[ h = \frac{u^2 \sin^2 \theta}{2g} \]

- For maximum height, we again set \( \theta = 45^\circ \):
\[ h_{max} = \frac{u^2 \sin^2 45^\circ}{2g} = \frac{u^2 \cdot \left(\frac{1}{2}\right)}{2g} = \frac{u^2}{4g} \]

- Now, comparing the maximum range and maximum height:
\[ R_{max} = \frac{u^2}{g} \quad \text{and} \quad h_{max} = \frac{u^2}{4g} \]

- From this, we can see that:
\[ R_{max} = 4 \cdot h_{max} \]

- Therefore, the maximum horizontal range is **not equal** to the maximum height attained by the projectile; instead, it is four times the maximum height. Thus, **Reason R is false**.

### Conclusion
- **Assertion A is true.**
- **Reason R is false.**

### Final Answer
The correct option is **C**: A is true but R is false.