GATE CS 2014 Set 2 QGA-5 — General Aptitude

Let the given infinite nested radical expression be represented by $x$:

$$x = \sqrt{12 + \sqrt{12 + \sqrt{12 + \dots}}}$$

Since the expression is infinite, the nested part under the first square root is also equal to $x$. Therefore, we can rewrite the equation as:

$$x = \sqrt{12 + x}$$

Squaring both sides to eliminate the square root:

$$x^2 = 12 + x$$

Rearranging the terms into a standard quadratic equation form:

$$x^2 - x - 12 = 0$$

Factoring the quadratic equation:

$$x^2 - 4x + 3x - 12 = 0$$

$$x(x - 4) + 3(x - 4) = 0$$

$$(x - 4)(x + 3) = 0$$

This gives two possible values for $x$:

$$x = 4 \quad \text{or} \quad x = -3$$

Since $x$ represents the principal square root of a positive real number, it must be positive ($x > 0$). Thus, we discard $x = -3$.

Therefore, the value is $x = 4.000$.