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S6 Maths: The Binomial Theorem: Paper 1 (Basics)
S6 Maths: The Binomial Theorem: Paper 1 (Basics)
Ten items starting from the number of terms and C(n, r), then reading off the number in front of a chosen power.
Direct answers Items 1-3: 7 terms, 35, 6. Items 4-6: 1, the full cubic expansion, 9. Items 7-9: 2, 12, 45. Item 10: a³ + 3a²b + 3ab² + b³.
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 | Basics |
02 Questions (10 items)
- FoundationHow many terms are there in the expansion of (a + b)^6?
- FoundationEvaluate C(7, 3).
- FoundationIn the expansion of (1 + x)^4, what number multiplies the x² term?
- IntermediateWhat is the constant term in the expansion of (x + 1)^5?
- IntermediateWrite out the full expansion of (1 + x)^3.
- IntermediateEvaluate C(9, 1) and state how it relates to 9.
- ChallengeEvaluate C(8, 0) + C(8, 8).
- ChallengeIn the expansion of (2 + x)^3, what number multiplies the x term?
- ChallengeEvaluate C(10, 2).
- Error redoA student writes (a + b)^3 = a³ + b³. What is wrong, and what is correct?
03 Answers and one-line notes
| # | Answer | One-line note |
|---|---|---|
| 1 | 7 terms | A binomial to the power n gives n + 1 terms, so 6 + 1 = 7 terms. |
| 2 | 35 | C(7, 3) = (7 × 6 × 5) ÷ (3 × 2 × 1) = 35, the same as C(7, 4). |
| 3 | 6 | The general term is C(4, r) x^r; putting r = 2 gives C(4, 2) = 6. |
| 4 | 1 | The constant term takes x^0, so r = 5 and C(5, 5) × 1^5 = 1. |
| 5 | 1 + 3x + 3x² + x³ | The numbers are C(3, 0), C(3, 1), C(3, 2), C(3, 3), that is 1, 3, 3, 1. |
| 6 | 9 | Choosing 1 object from 9 gives 9 ways, so C(n, 1) always equals n. |
| 7 | 2 | Both C(8, 0) and C(8, 8) equal 1, so the total is 2. |
| 8 | 12 | The general term is C(3, r) 2^(3-r) x^r; r = 1 gives C(3, 1) × 2² = 12. |
| 9 | 45 | C(10, 2) = (10 × 9) ÷ 2 = 45, the number of ways to choose 2 from 10. |
| 10 | The two middle terms are missing; it is a³ + 3a²b + 3ab² + b³ | The expansion has n + 1 = 4 terms, and the middle numbers are C(3, 1) and C(3, 2), both 3. |
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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