The value of the expression (1.55² + 0.55² + 1.55 × 0.55) is equal to:

Correct — D. The expression is the second factor of the difference of cubes.

a³ − b³ = (a − b)(a² + ab + b²)

With a = 1.55 and b = 0.55, a − b = 1.
So a² + ab + b² = (a³ − b³) ÷ 1 = a³ − b³, which is option (d).

Check: 2.4025 + 0.3025 + 0.8525 = 3.5575, and 1.55³ − 0.55³ = 3.723875 − 0.166375 = 3.5575.

Why the others are wrong

• (a)
  Option (a) is the sum of cubes, and its factorisation (a + b)(a² − ab + b²) needs a minus on the middle term. Its value is 3.890250, not 3.5575.
• (b)
  Option (b) drops the ab term. Since 1.55 × 0.55 = 0.8525, option (b) comes to 2.7050 while the expression in the question is 3.5575.
• (c)
  Option (c) is the difference of squares, (a + b)(a − b) = 2.10 × 1 = 2.10. The expression has a plus between the squares and an extra ab term on top.

Two standard identities differ only by the sign in the middle:

a³ − b³ = (a − b)(a² + ab + b²)
a³ + b³ = (a + b)(a² − ab + b²)

A plus in front of ab points to the difference of cubes. SSC then picks numbers with a − b = 1, so the bracket you are handed is the whole of a³ − b³ and no multiplication is left to do.

Spotting that 1.55 − 0.55 = 1 is the entire question.

Key facts

• a³ − b³ = (a − b)(a² + ab + b²).
• 1.55 − 0.55 = 1, which is why the bracket equals a³ − b³ exactly here.
• 1.55² + 0.55² + 1.55 × 0.55 = 3.5575.

Common traps

• Choosing the sum of cubes because every sign in the expression is a plus.
• Multiplying the decimals out and then finding no option in that form.
• Forgetting the shortcut works only because a − b happens to be 1.

No directly related past PYQ was found.