Select the set in which the numbers are related in the same way as are the numbers of the given sets. (6, 31) (10, 91) (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)

Correct — B. Rule: second number = n² − n + 1, which is the same as n(n − 1) + 1.

6² − 6 + 1 = 36 − 6 + 1 = 31
10² − 10 + 1 = 100 − 10 + 1 = 91

Both given sets obey it, so apply it to the options:
8² − 8 + 1 = 64 − 8 + 1 = 57

That is exactly the printed set (8, 57) — option (b).

Why the others are wrong

• (a)
  (7, 68) — For 7 the rule gives 7² − 7 + 1 = 43, not 68, so this set does not share the relation.
• (c)
  (12, 146) — For 12 the rule gives 144 − 12 + 1 = 133, not 146. 146 is 12² + 2, and the same 'square plus 2' would make 6 give 38, not 31.
• (d)
  (9, 81) — 81 is simply 9², and squaring alone fails the samples — 6² is 36 where the set prints 31. The rule gives 9² − 9 + 1 = 73.

A set analogy hands you two solved pairs and asks for a third built the same way.

Form the rule from the first pair, then verify it on the second before touching the options. Two pairs kill most candidate rules, and plain squaring fails here: 6² is 36, not 31.

The gap from the square is the giveaway here: 36 − 31 = 5 and 100 − 91 = 9. In each case that gap is n − 1, so the relation reads n² − (n − 1), the same thing as n² − n + 1.

The printed note bars digit-splitting, so 146 is read as one number.

Key facts

• n² − n + 1 is the same as n(n − 1) + 1.
• For n = 6, 8 and 10 the rule gives 31, 57 and 91.
• The gap between n² and the second number is n − 1 in both given sets.
• A candidate rule must survive the second sample set before it is used on the options.

Common traps

• Settling on 'square it' because 9 and 81 look convincing, without testing 6 and 10
• Getting n² − 5 from the first set and never checking that the 5 is itself n − 1
• Doing the arithmetic on the digits of 146 instead of the whole number

No directly related past PYQ was found.