| Exam Board | OCR |
|---|---|
| Module | C2 (Core Mathematics 2) |
| Year | 2012 |
| Session | June |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Binomial Theorem (positive integer n) |
| Type | Sum/difference of two binomials simplification |
| Difficulty | Moderate -0.8 This is a straightforward application of the binomial theorem requiring routine expansion and simplification. Part (i) is direct formula application, and part (ii) simply observes that odd-power terms cancel. No problem-solving insight needed, just careful arithmetic—easier than average A-level questions. |
| Spec | 1.04a Binomial expansion: (a+b)^n for positive integer n |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Attempt expansion – products of powers of 3 and \(2x\) | M1* | Must attempt at least 5 terms. Each term must be an attempt at a product including binomial coeffs if used. Allow M1 for no, or incorrect, binomial coeffs. Powers of 3 and \(2x\) must be intended to sum to 5 within each term. |
| Attempt to use correct binomial coefficients | M1d* | At least 5 correct from 1, 5, 10, 10, 5, 1 - allow missing or incorrect (but not if raised to a power). Must be numerical. They must be part of a product within each term. |
| Obtain at least four correct simplified terms | A1 | Either linked by '+' or as part of a list. |
| \((3+2x)^5 = 243 + 810x + 1080x^2 + 720x^3 + 240x^4 + 32x^5\) | A1 | With all coefficients simplified. Terms must be linked by '+' and not just commas. |
| [4] |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Attempt to change signs of relevant terms | M1 | Must change the sign on all relevant terms from their expansion, and no others. Expansion in part (i) must have at least 5 terms. |
| \(= 486 + 2160x^2 + 480x^4\), from their (i) | A1 FT | Must have been a 6 term quintic in (i) to get FT mark. A0 if subsequent division by a common factor. |
| [2] |
## Question 1:
### Part (i): Expand $(3+2x)^5$
| Answer | Marks | Guidance |
|--------|-------|----------|
| Attempt expansion – products of powers of 3 and $2x$ | M1* | Must attempt at least 5 terms. Each term must be an attempt at a product including binomial coeffs if used. Allow M1 for no, or incorrect, binomial coeffs. Powers of 3 and $2x$ must be intended to sum to 5 within each term. |
| Attempt to use correct binomial coefficients | M1d* | At least 5 correct from 1, 5, 10, 10, 5, 1 - allow missing or incorrect (but not if raised to a power). Must be numerical. They must be part of a product within each term. |
| Obtain at least four correct simplified terms | A1 | Either linked by '+' or as part of a list. |
| $(3+2x)^5 = 243 + 810x + 1080x^2 + 720x^3 + 240x^4 + 32x^5$ | A1 | With all coefficients simplified. Terms must be linked by '+' and not just commas. |
| | **[4]** | |
### Part (ii): $(3+2x)^5 + (3-2x)^5$
| Answer | Marks | Guidance |
|--------|-------|----------|
| Attempt to change signs of relevant terms | M1 | Must change the sign on all relevant terms from their expansion, and no others. Expansion in part (i) must have at least 5 terms. |
| $= 486 + 2160x^2 + 480x^4$, from their (i) | A1 FT | Must have been a 6 term quintic in (i) to get FT mark. A0 if subsequent division by a common factor. |
| | **[2]** | |
---1 (i) Find the binomial expansion of $( 3 + 2 x ) ^ { 5 }$, simplifying the terms.\\
(ii) Hence find the binomial expansion of $( 3 + 2 x ) ^ { 5 } + ( 3 - 2 x ) ^ { 5 }$.
\hfill \mbox{\textit{OCR C2 2012 Q1 [6]}}