Project ForgeEngineering Mathematics
Calculus
GATE 2007
Medium
If y = x + √(x + √(x + √(x + ⋯∞))), then y(2) =
- A4 or 1
- 4 only
- C1 only
- Dundefined
Explanation
Since the expression under the radicals repeats, y−x=√(x+(y−x))=√y, so (y−x)²=y. At x=2: (y−2)²=y ⟹ y²−5y+4=0 ⟹ y=4 or y=1. Since the nested radical is non-negative, y−x=√y≥0 requires y≥x=2, which rejects y=1 (extraneous, introduced by squaring) and leaves y=4 only.
Source: GATE ME 2007 (2007)
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