| Exam Board | OCR |
|---|---|
| Module | C4 (Core Mathematics 4) |
| Year | 2006 |
| Session | June |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Generalised Binomial Theorem |
| Type | Multiply by polynomial |
| Difficulty | Moderate -0.8 Part (i) is a direct application of the binomial expansion formula for negative powers, requiring only substitution and simplification. Part (ii) requires multiplying two expansions and collecting terms, which is slightly more involved but still a standard textbook exercise with no problem-solving insight needed. The question tests routine algebraic manipulation of binomial series. |
| Spec | 1.04c Extend binomial expansion: rational n, |x|<1 |
| Answer | Marks | Guidance |
|---|---|---|
| \(1 + (-2)(-3x) + \frac{(-2)(-3)}{1.2}(-3x)^2 + \ldots \text{ ignore}\) | M1 | State or imply; accept \(-3x^2\) & \(-9x^2\) |
| \(= 1 + 6x \ldots + 27x^2\) | B1 | Correct first 2 terms |
| A1 | 3 Correct third term |
| Answer | Marks | Guidance |
|---|---|---|
| \((1 + 2x)^2(1-3x)^{-2}\) | M1 | For changing into suitable form, seen/implied |
| Attempt to expand \((1+2x)^2\) & select (at least) 2 relevant products and add | M1 | Selection may be after multiplying out |
| 55 (Accept \(55x^2\)) | A2√ | 4 If (i) is \(a + bx + cx^2\), f.t. \(4(a+b)+c\) |
| SR 1 For expansion of \((1+2x)^2\) with 1 error, A1√ | ||
| SR 2 For expansion of \((1+2x)^2\) & > 1 error, A0 |
| Answer | Marks |
|---|---|
| For correct method idea of long division | M1 |
| \(1 \ldots +10x \ldots +55x^2\) | A1, A1, A1(4) |
### (i)
$1 + (-2)(-3x) + \frac{(-2)(-3)}{1.2}(-3x)^2 + \ldots \text{ ignore}$ | M1 | State or imply; accept $-3x^2$ & $-9x^2$
$= 1 + 6x \ldots + 27x^2$ | B1 | Correct first 2 terms
| A1 | 3 Correct third term
---
### (ii)
$(1 + 2x)^2(1-3x)^{-2}$ | M1 | For changing into suitable form, seen/implied
Attempt to expand $(1+2x)^2$ & select (at least) 2 relevant products and add | M1 | Selection may be after multiplying out
55 (Accept $55x^2$) | A2√ | 4 If (i) is $a + bx + cx^2$, f.t. $4(a+b)+c$
**SR 1** For expansion of $(1+2x)^2$ with 1 error, A1√ | |
**SR 2** For expansion of $(1+2x)^2$ & > 1 error, A0 | |
**Alternative Method**
For correct method idea of long division | M1 |
$1 \ldots +10x \ldots +55x^2$ | A1, A1, A1(4) |
---\begin{enumerate}[label=(\roman*)]
\item Expand $(1 - 3x)^{-2}$ in ascending powers of $x$, up to and including the term in $x^2$. [3]
\item Find the coefficient of $x^2$ in the expansion of $\frac{(1 + 2x)^2}{(1 - 3x)^2}$ in ascending powers of $x$. [4]
\end{enumerate}
\hfill \mbox{\textit{OCR C4 2006 Q2 [7]}}