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S6 Maths: The Binomial Theorem: Paper 2 (Intermediate)
S6 Maths: The Binomial Theorem: Paper 2 (Intermediate)
Ten items with minus signs, fractions and an unknown n, all needing the general term before substituting.
Direct answers Items 1-3: 40, -8, 6. Items 4-6: n = 10, 792, 270. Items 7-9: -20, 36, the general term. Item 10: 54.
01 Paper details (check answers afterwards)
| Item | Detail |
|---|---|
| Questions | 10 |
| Suggested time | 20 |
| Difficulty mix | Foundation 3 · Intermediate 3 · Challenge 3 · Error redo 1 |
| Chapter | Chapter: S6 Maths: The Binomial Theorem |
| Level | Intermediate |
02 Questions (10 items)
- FoundationIn the expansion of (1 + 2x)^5, what number multiplies the x² term?
- FoundationIn the expansion of (2 – x)^4, what number multiplies the x³ term?
- FoundationWhat is the constant term in the expansion of (x + 1/x)^4?
- IntermediateGiven that the x² term of (1 + x)^n carries the number 45, find n.
- IntermediateEvaluate C(12, 5).
- IntermediateIn the expansion of (3 + x)^5, what number multiplies the x² term?
- ChallengeIn the expansion of (1 – x)^6, what number multiplies the x³ term?
- ChallengeIn the expansion of (2x + 3)^3, what number multiplies the x² term?
- ChallengeWrite the general term of (a + b)^n and explain the symbols.
- Error redoA student answers 6 for the number in front of x² in (1 + 3x)^4. What went wrong?
03 Answers and one-line notes
| # | Answer | One-line note |
|---|---|---|
| 1 | 40 | The general term is C(5, r) (2x)^r; r = 2 gives 10 × 4 = 40. |
| 2 | −8 | With r = 3, C(4, 3) × 2 × (-1)³ = -8; the minus sign comes from -x. |
| 3 | 6 | The powers add to x^(4-2r), so r = 2 and the term is C(4, 2) = 6. |
| 4 | n = 10 | C(n, 2) = n(n – 1)/2 = 45, so n(n – 1) = 90 and n = 10. |
| 5 | 792 | C(12, 5) = (12 × 11 × 10 × 9 × 8) ÷ 120 = 792. |
| 6 | 270 | With r = 2, C(5, 2) × 3³ = 10 × 27 = 270; the 3 must be cubed. |
| 7 | −20 | With r = 3, C(6, 3) × (-1)³ = 20 × (-1) = -20. |
| 8 | 36 | With r = 1, C(3, 1) × (2x)² × 3 = 3 × 4 × 3 = 36; note (2x)² = 4x². |
| 9 | C(n, r) a^(n-r) b^r | With r running from 0 to n, term r + 1 is C(n, r) a^(n-r) b^r, where C(n, r) counts combinations. |
| 10 | The 3² factor was dropped; the answer is 54 | The general term is C(4, r) (3x)^r, so r = 2 needs 3² = 9 and 6 × 9 = 54. |
04 What to do after marking
- After marking, redo only the wrong items and write down which step went wrong (formula, unit, or misread number).
- Repeat the same item two days later; move to the next paper only when both attempts are correct.
- Parents only need the answer and note columns; there is no need to explain every item.
Write the general term C(n, r) a^(n-r) b^r first, then put in r.Tutor Registration →
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