The length of the ladder placed against a vertical wall is twice the distance between the foot of the ladder and the wall.

Assertion(A): The angle that the ladder makes with the wall is 60°.

Reason(R): If ladder makes angle θ with the wall then sin θ = $\dfrac{\text{base}}{\text{hypotenuse}} = \dfrac{x}{2x} = \dfrac{1}{2}$.

1. A is true, R is false.

2. A is false, R is true.

3. Both A and R are true and R is correct reason for A.

4. Both A and R are true and R is incorrect reason for A.

Let AB be the wall and AC be the ladder. Let distance between foot of ladder and wall be x.

So, length of ladder (AC) = 2x.

According to the Pythagoras theorem :

⇒ Hypotenuse² = Base² + Height²

⇒ AC² = BC² + AB²

⇒ (2x)² = x² + AB²

⇒ 4x² = x² + AB²

⇒ AB² = 4x² - x²

⇒ AB² = 3x²

⇒ AB = $\sqrt{3x^2}$

⇒ AB = $\sqrt{3}x$.

Let angle between ladder and wall be θ.

We know that,

$\Rightarrow \text{sin θ} = \dfrac{\text{Perpendicular side}}{\text{Hypotenuse}} \\[1em] \Rightarrow \text{sin θ} = \dfrac{BC}{AC} \\[1em] \Rightarrow \text{sin θ} = \dfrac{x}{2x} \\[1em] \Rightarrow \text{sin θ} = \dfrac{1}{2} \\[1em]$

⇒ sin θ = sin 30°

⇒ θ = 30°.

Assertion (A) is false.

If ladder makes angle θ with the wall then sin θ = $\dfrac{\text{base}}{\text{hypotenuse}} = \dfrac{x}{2x} = \dfrac{1}{2}$.

Here base refers side opposite to angle θ i.e. BC.

Reason (R) is true.

Hence, option 2 is the correct option.