| Exam Board | OCR |
|---|---|
| Module | C2 (Core Mathematics 2) |
| Year | 2006 |
| Session | January |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Binomial Theorem (positive integer n) |
| Type | Standard product of two binomials |
| Difficulty | Moderate -0.8 This is a straightforward binomial expansion question requiring routine application of the binomial theorem for part (i), followed by a simple multiplication in part (ii). The 'Hence' signpost makes the method clear, and the calculations are mechanical with no problem-solving insight needed. Easier than average for A-level. |
| Spec | 1.04a Binomial expansion: (a+b)^n for positive integer n |
| Answer | Marks | Guidance |
|---|---|---|
| \((1 - 2x)^{12} = 1 - 24x + 264x^2\) | B1 | Obtain 1 and \(-24x\)... |
| M1 | Attempt \(x^2\) term, including attempt at binomial coeff. | |
| A1 | Obtain \(...264x^2\) |
| Answer | Marks | Guidance |
|---|---|---|
| \((1 \times 264) + (3x - 24) = 192\) | M1 | Attempt coefficient of \(x^2\) from two pairs of terms |
| A1 | Obtain correct unsimplified expression | |
| M1 | ||
| A1 | Obtain 192 |
**(i)**
$(1 - 2x)^{12} = 1 - 24x + 264x^2$ | B1 | Obtain 1 and $-24x$...
| M1 | Attempt $x^2$ term, including attempt at binomial coeff.
| A1 | Obtain $...264x^2$
**(ii)**
$(1 \times 264) + (3x - 24) = 192$ | M1 | Attempt coefficient of $x^2$ from two pairs of terms
| A1 | Obtain correct unsimplified expression
| M1 |
| A1 | Obtain 192
---3 (i) Find the first three terms of the expansion, in ascending powers of $x$, of $( 1 - 2 x ) ^ { 12 }$.\\
(ii) Hence find the coefficient of $x ^ { 2 }$ in the expansion of
$$( 1 + 3 x ) ( 1 - 2 x ) ^ { 12 } .$$
\hfill \mbox{\textit{OCR C2 2006 Q3 [6]}}