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S6 Maths: The Binomial Theorem: Paper 3 (Challenge)
S6 Maths: The Binomial Theorem: Paper 3 (Challenge)
Ten items with fractional bases, an unknown parameter, a substitution trick and comparing term sizes.
Direct answers Items 1-3: 120, 60, 240. Items 4-6: a = 2, 720, 84. Items 7-9: 256, 30, the x^4 term 240. Item 10: 15.
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 | Challenge |
02 Questions (10 items)
- FoundationIn the expansion of (1 + x)^10, what number multiplies the x^7 term?
- FoundationIn the expansion of (2 + x)^6, what number multiplies the x^4 term?
- FoundationWhat is the constant term in the expansion of (x² + 2/x)^6?
- IntermediateGiven that the x² term of (1 + ax)^5 carries the number 40 with a > 0, find a.
- IntermediateIn the expansion of (3 + 2x)^5, what number multiplies the x³ term?
- IntermediateIn the expansion of (1 – 2x)^7, what number multiplies the x² term?
- ChallengeWhat is the sum of all the numbers in front of the terms of (x + 3)^4?
- ChallengeIn (1 + x)^6 + (1 – x)^6, what number multiplies the x² term?
- ChallengeIn (1 + 2x)^6, which power of x carries the largest number, and what is it?
- Error redoA student computes C(6, 2) as 6 × 5 = 30. What went wrong?
03 Answers and one-line notes
| # | Answer | One-line note |
|---|---|---|
| 1 | 120 | With r = 7, C(10, 7) = C(10, 3) = 120. |
| 2 | 60 | With r = 4, C(6, 4) × 2² = 15 × 4 = 60. |
| 3 | 240 | Powers of x total 12 – 3r, so r = 4 and the constant term is C(6, 4) × 2^4 = 240. |
| 4 | a = 2 | C(5, 2) a² = 10a² = 40, so a² = 4 and the positive root is a = 2. |
| 5 | 720 | With r = 3, C(5, 3) × 3² × 2³ = 10 × 9 × 8 = 720. |
| 6 | 84 | With r = 2, C(7, 2) × (-2)² = 21 × 4 = 84; squaring removes the minus sign. |
| 7 | 256 | Putting x = 1 gives 4^4 = 256, which is far quicker than adding term by term. |
| 8 | 30 | Both expansions give C(6, 2) = 15 for the x² term, so the total is 30. |
| 9 | The x^4 term, 240 | The numbers C(6, r) 2^r run 1, 12, 60, 160, 240, 192, 64, so r = 4 gives the largest. |
| 10 | Combinations divide by 2!; the answer is 15 | C(6, 2) ignores order, so divide by 2 × 1: 30 ÷ 2 = 15. The value 30 is P(6, 2). |
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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