Assertion (A): In the adjoining figure, tan A = $\dfrac{1}{\sqrt{3}}$. Then AC = 2AB.

Reason (R): In right angled ΔABC, AC² = AB² + BC²

1. Assertion (A) is true, Reason (R) is false.

2. Assertion (A) is false, Reason (R) is true.

3. Both Assertion (A) and Reason (R) are true, and Reason (R) is the correct reason for Assertion (A).

4. Both Assertion (A) and Reason (R) are true, but Reason (R) is not the correct reason (or explanation) for Assertion (A).

Given, tan A = $\dfrac{1}{\sqrt{3}}$

As we know,

$\text{tan A }= \dfrac{\text{Perpendicular}}{\text{Base}}\\[1em] \Rightarrow \dfrac{BC}{AB} = \dfrac{1}{\sqrt{3}}$

Let BC = k and AB = $\sqrt{3}$ k

Since, ΔABC is a right angled triangle, using pythagoras theorem,

⇒ AC² = AB² + BC²

∴ Reason (R) is true.

Substituting values we get :

$\Rightarrow AC^2 = (\sqrt{3}k)^2 + k^2\\[1em] \Rightarrow AC^2 = 3k^2 + k^2\\[1em] \Rightarrow AC^2 = 4k^2\\[1em] \Rightarrow AC = \sqrt{4k^2}\\[1em] \Rightarrow AC = 2k \\[1em] \Rightarrow AC = 2k\dfrac{\sqrt{3}}{\sqrt{3}} \\[1em] \Rightarrow AC = 2k \times \dfrac{\sqrt{3}}{\sqrt{3}} \\[1em] \Rightarrow AC = \dfrac{2(\sqrt{3}k)}{\sqrt{3}} \\[1em] \Rightarrow AC = \dfrac{2AB}{\sqrt{3}}.$

∴ Assertion (A) is false.

∴ Assertion (A) is false, Reason (R) is true.

Hence, option 2 is the correct option.