Choose the correct relation: (i) √10 < 3.2 (ii) √11 < √12 (iii) √13 < √14

Correct — D.
For positive numbers, compare squares instead of roots.

(i) Square 3.2: 3.2² = 10.24
10 < 10.24, so √10 < 3.2 → true

(ii) 11 < 12, so √11 < √12 → true
(iii) 13 < 14, so √13 < √14 → true

All three hold → (i), (ii) and (iii) → option (d)

Why the others are wrong

• (a)
  (i) only — (i) only drops (ii) and (iii), yet both follow at once: the square root keeps order, so 11 < 12 gives √11 < √12 and 13 < 14 gives √13 < √14.
• (b)
  (i) and (ii) — (i) and (ii) leaves out (iii), which is true for the same reason as (ii): 13 < 14, so √13 < √14.
• (c)
  (ii) and (iii) — (ii) and (iii) leaves out (i), but 3.2² = 10.24 is more than 10, so √10 ≈ 3.162 really is below 3.2.

Order survives squaring and square roots for non-negative numbers: if 0 ≤ a < b, then a² < b² and √a < √b.

So (ii) and (iii) need no arithmetic, because the numbers under the roots are already in order.

Relation (i) compares a root with a decimal. Square the decimal instead: 3.2² = 10.24, just above 10, so √10 sits a little under 3.2.

Key facts

• For 0 ≤ a < b, √a < √b
• 3.2² = 10.24
• √10 ≈ 3.162

Common traps

• Rounding √10 to 3.2 and rejecting (i) as 'not less than'
• Picking (ii) and (iii) because they need no arithmetic, without testing (i)

No directly related past PYQ was found.