Identify the incorrect relationship(s) from the list below: (i) √6 + √2 = √5 + √3 (ii) √6 + √2 < √5 + √3 (iii) √6 + √2 > √5 + √3

Correct — C. Both sides are positive, so square them and compare.

Square the left: (√6 + √2)² = 6 + 2 + 2√12 = 8 + 2√12
Square the right: (√5 + √3)² = 5 + 3 + 2√15 = 8 + 2√15
Compare: the 8s match and √12 < √15

So √6 + √2 < √5 + √3, and relation (ii) is the true one.
The stem asks for the incorrect ones: (i) and (iii) → option (c).

Why the others are wrong

• (a)
  (i) — Only half the answer. (i) is incorrect, but so is (iii): it claims √6 + √2 is the larger side, and squaring shows it is the smaller.
• (b)
  (ii) — (ii) is the true relation, since 8 + 2√12 is less than 8 + 2√15. The stem asks for the incorrect relationships, so (ii) is the one to leave out.
• (d)
  (ii) and (iii) — It includes (ii), which is true, and leaves out (i). The squares are 8 + 2√12 and 8 + 2√15, so the two sides are not equal and (i) is incorrect.

To compare sums of square roots, square both sides. For positive numbers squaring keeps the order, so the larger square belongs to the larger sum.

Here both pairs add to 8 under the roots (6 + 2 = 5 + 3), so the comparison comes down to the cross terms, 2√12 against 2√15.

With a fixed sum, the pair of numbers closer together has the larger product: 5 × 3 = 15 beats 6 × 2 = 12.

Key facts

• (√a + √b)² = a + b + 2√(ab).
• For positive numbers, x < y exactly when x² < y².
• √6 + √2 ≈ 3.864 and √5 + √3 ≈ 3.968.
• When a + b = c + d, √a + √b is larger for the pair with the larger product.

Common traps

• Marking the one true relation, (ii), when the stem asks for the incorrect ones.
• Squaring √6 + √2 as 6 + 2 and dropping the cross term 2√12, which makes both sides look equal at 8.

No directly related past PYQ was found.