If 5 * 4 = 41 and 6 * 3 = 45, what is 7 * 2?

Correct — C.
Rule: a * b = a² + b².

5 * 4 = 25 + 16 = 41 ✓
6 * 3 = 36 + 9 = 45 ✓

7 * 2 = 7² + 2²
= 49 + 4 = 53 → option (c).

Why the others are wrong

• (a)
  48 — 48 is less than 7² = 49 on its own. The rule adds 2² = 4 on top of 49, so the result must be 53.
• (b)
  51 — 51 is 49 + 2, adding the 2 without squaring it. In 5 * 4 = 41 the 4 enters as 16, so here the 2 enters as 4.
• (d)
  55 — 55 is 49 + 6, but the second number adds its square, and 2² = 4. The examples show it: 4 adds 16 and 3 adds 9.

When a sign is defined only by examples, test the simplest rules first: sum, product, difference, then squares.

The product fails at once: 5 × 4 = 20, far from 41. Squares fit: 25 + 16 = 41 and 36 + 9 = 45.

Check a rule on both examples before using it. One example can be matched by many rules; each further example rules more of them out.

Key facts

• Both examples fit a² + b²: 25 + 16 = 41 and 36 + 9 = 45.
• The product rule fails at once: 5 × 4 = 20, not 41.
• a² + b² differs from (a + b)²: 7² + 2² = 53, but (7 + 2)² = 81.

Common traps

• Adding the second number without squaring it, which gives 51
• Squaring the sum, (7 + 2)² = 81, instead of summing the squares

No directly related past PYQ was found.