In the following number-pairs, the second number is obtained by applying certain mathematical operations to the first number. Select the set in which the numbers are related in the same way as are the numbers of the following sets. (NOTE : Operations should be performed on the whole numbers, without breaking down the numbers into its constituent digits. E.g. 13 – Operations on 13 such as adding /deleting /multiplying etc., to 13 can be performed. Breaking down 13 into 1 and 3 and then performing mathematical operations on 1 and 3 is not allowed.) (12, 145) (9, 82)

Correct — C. Rule: second number = (first number)² + 1.

Check both given pairs.
12² = 144, and 144 + 1 = 145.
9² = 81, and 81 + 1 = 82.

Now apply it to the options.
14² = 196, and 196 + 1 = 197.

That is the pair (14, 197), option (c).

Why the others are wrong

• (a)
  (22, 483) — 22² = 484, so the rule wants 485. 483 is 484 minus one, the same square with the +1 pointing the wrong way.
• (b)
  (16, 256) — 256 is 16² exactly. The rule adds one after squaring, so this pair would have to read (16, 257).
• (d)
  (8, 69) — 8² + 1 = 65, not 69. A gap of 5 fits no operation that both given pairs support.

The note printed with the question is the instruction that matters: operate on the whole number, never on its digits. That rules out reading 12 as a 1 and a 2, and rules in squaring, cubing and products.

With two sample pairs you hold two equations, so test the cheapest families first — squares, then cubes, then n(n+1).

Here 145 and 82 sit one above 144 and 81, both perfect squares, so n² + 1 is fixed by the second pair every bit as much as by the first.

Key facts

• 12² + 1 = 145 and 9² + 1 = 82, so the rule is n² + 1.
• 14² = 196, which makes the keyed pair (14, 197).
• The question forbids breaking a number into its constituent digits.

Common traps

• Settling on plain n² after noticing only that 145 and 82 are close to squares.
• Testing the rule on one given pair and never on the second.

No directly related past PYQ was found.