In the following questions a statement of assertion (A) is followed by a statement of reason ( R).
A: Horizontal range of a projectile is always same for angle of projection `theta` with horizontal or `theta` with vertical .
R : Horizontal range depends only on angle of projection .

Options

A. If both Assertion & Reason are true and the reason is the correct explanation of the assertion then mark (1) .
B. If both Assertion & Reason are true but the reason is not the correct explanation of the assertion then mark (2)
C. If Assertion is true statement but Reason is false then mark (3).
D. If both Assertion and Reason are false statements then mark (4) .

Answer

To solve the question, we need to analyze both the assertion (A) and the reason (R) provided.

### Step-by-Step Solution:

**Step 1: Understand the Assertion (A)**
- The assertion states that the horizontal range of a projectile is the same when projected at an angle θ with the horizontal and at an angle θ with the vertical.

**Step 2: Recall the Formula for Horizontal Range**
- The formula for the horizontal range \( R \) of a projectile launched at an angle \( \theta \) with the horizontal is given by:
\[ R = \frac{u^2 \sin(2\theta)}{g} \]
where \( u \) is the initial velocity and \( g \) is the acceleration due to gravity.

**Step 3: Analyze the Case of Projection at Angle θ with Horizontal**
- If the angle of projection is \( \theta \) with respect to the horizontal, the range can be calculated as:
\[ R = \frac{u^2 \sin(2\theta)}{g} \]

**Step 4: Analyze the Case of Projection at Angle θ with Vertical**
- If the angle of projection is \( \theta \) with respect to the vertical, then the angle with respect to the horizontal will be \( 90^\circ - \theta \).
- The range in this case will be:
\[ R' = \frac{u^2 \sin(2(90^\circ - \theta))}{g} = \frac{u^2 \sin(180^\circ - 2\theta)}{g} = \frac{u^2 \sin(2\theta)}{g} \]
- Thus, \( R' = R \) when the initial velocity \( u \) is the same.

**Step 5: Conclusion about the Assertion**
- The assertion is true only if the initial velocity \( u \) is the same for both cases. However, it does not hold universally since the initial velocity can vary. Therefore, the assertion is **false**.

**Step 6: Understand the Reason (R)**
- The reason states that the horizontal range depends only on the angle of projection.
- From our analysis, we can see that the horizontal range actually depends on both the angle of projection and the initial velocity \( u \).

**Step 7: Conclusion about the Reason**
- Since the range depends on both the angle and the initial velocity, the reason is also **false**.

### Final Conclusion
Both the assertion (A) and the reason (R) are false.

### Answer:
The correct option is that both the assertion and reason are false statements.

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