The equation $\sqrt{15 - 2x} = x$.

Assertion (A): x = 3.

Reason (R): $\sqrt{15 - 2x} = x$

$\Rightarrow 15 - 2x = x^2 \\[1em] \Rightarrow x^2 - 2x - 15 = 0 \\[1em] \Rightarrow x = -5 \text{ or } x = 3$

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.

A is true, R is false.

Reason

Given,

$\Rightarrow\sqrt{15 - 2x} = x\\[1em] \Rightarrow 15 - 2x = x^2\\[1em] \Rightarrow x^2 + 2x - 15 = 0\\[1em] \Rightarrow x^2 + 5x - 3x - 15 = 0\\[1em] \Rightarrow x(x + 5) - 3(x + 5) = 0\\[1em] \Rightarrow (x + 5)(x - 3) = 0\\[1em] \Rightarrow (x + 5) = 0 \text{ or } (x - 3) = 0\\[1em] \Rightarrow x = -5 \text{ or } x = 3$

The quadratic equation mentioned in the reason is $x^2 - 2x - 15 = 0$ whereas we see that the correct quadratic equation is $x^2 + 2x - 15 = 0$
∴ Reason (R) is false.

Our solution shows that one of the roots is 3.
∴ Assertion (A) is true.

Hence, option 1 is correct.