If a $ b = 2, where a $ b = a² − b, what is the value of a and b?

Correct — A. Substitute each option into a² − b and keep the pair that gives 2.

Rule: a $ b = a² − b.

(2, 2): 2² − 2 = 4 − 2 = 2
(8, 1): 8² − 1 = 63
(4, 5): 4² − 5 = 11
(9, 6): 9² − 6 = 75

Only 2, 2 gives 2: option (a).

Why the others are wrong

• (b)
  8, 1 — 8² − 1 = 64 − 1 = 63, not 2. Squaring 8 far outruns the 1 taken away; with a = 8 the rule would need b = 62.
• (c)
  4, 5 — 4² − 5 = 16 − 5 = 11, not 2. Subtracting 5 is not enough; with a = 4 the rule would need b = 14.
• (d)
  9, 6 — 9² − 6 = 81 − 6 = 75, not 2. With a = 9 the rule would need b = 79, so 6 falls far short.

A defined operator states its formula outright, so the work is substitution. With four candidate pairs, test each one in the formula rather than solving.

Solving would not settle it anyway: a² − b = 2 has endless solutions, such as (3, 7) and (4, 14). The options narrow them to one.

Key facts

• a $ b = a² − b: square the first number, then subtract the second.
• (2, 2) is the option pair that gives 4 − 2 = 2.
• a² − b = 2 alone has many solutions, such as (3, 7).

Common traps

• Trying to solve a² − b = 2 for one pair instead of testing the options.
• Squaring b instead of a: a − b² gives −2 for (2, 2) and hides the answer.

No directly related past PYQ was found.