Telangana TET Paper 1 Exam Model Paper 4

Concept:An infinite continued fraction repeats the same pattern. So the whole expression equals the same as the first step.Explanation:Let the given expression equal $x$.Since the pattern repeats infinitely, we write $x = \frac{1}{2 + x}$.Multiply both sides: $x(2 + x) = 1$ → $2x + x^2 = 1$ → $x^2 + 2x - 1 = 0$.Use the quadratic formula: $x = \frac{-2 \pm \sqrt{4 + 4}}{2} = \frac{-2 \pm 2\sqrt{2}}{2} = -1 \pm \sqrt{2}$.The two possible values are $-1 + \sqrt{2}$ (positive) and $-1 - \sqrt{2}$ (negative).Since the original fraction $\frac{1}{2+\cdots}$ is clearly positive, we take the positive root.Answer:$C. \sqrt{2} - 1$