Given, √x + 1⁄√x = 4, then what is x + 1⁄x?

Correct — B. The two terms are reciprocals, so squaring makes their product collapse to 1.

Given: √x + 1⁄√x = 4
Square both sides: (√x)² + 2 × √x × 1⁄√x + (1⁄√x)² = 16
Simplify each term: x + 2 + 1⁄x = 16

Subtract 2: x + 1⁄x = 16 − 2 = 14 → option (b)

Why the others are wrong

• (a)
  15 — 15 would make the square 17: x + 1⁄x + 2 = 17 gives √x + 1⁄√x = √17, not the given 4.
• (c)
  16 — 16 is 4² with the middle term dropped. (p + q)² = p² + 2pq + q², and here 2pq = 2 × √x × 1⁄√x = 2, which must come off.
• (d)
  17 — 17 would make the square 19: x + 1⁄x + 2 = 19 gives √x + 1⁄√x = √19, not the given 4.

When a quantity is added to its own reciprocal, squaring is the move, because the cross term is fixed: 2 × √x × 1⁄√x = 2.

So (√x + 1⁄√x)² = x + 1⁄x + 2, and x + 1⁄x is always the square minus 2. The same identity one level up gives (x + 1⁄x)² = x² + 1⁄x² + 2.

Key facts

• (a + 1⁄a)² = a² + 1⁄a² + 2, because a × 1⁄a = 1.
• If √x + 1⁄√x = k, then x + 1⁄x = k² − 2.
• (a − 1⁄a)² = a² + 1⁄a² − 2.
• Squaring once more here: x² + 1⁄x² = 14² − 2 = 194.

Common traps

• Squaring 4 to 16 and stopping, forgetting the cross term 2 × √x × 1⁄√x.
• Adding the 2 instead of subtracting it: x + 1⁄x + 2 = 16 means x + 1⁄x = 14, not 18.

No directly related past PYQ was found.