| Exam Board | OCR |
|---|---|
| Module | C4 (Core Mathematics 4) |
| Year | 2009 |
| Session | June |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Generalised Binomial Theorem |
| Type | Single unknown from one coefficient condition |
| Difficulty | Standard +0.3 This is a straightforward application of the binomial expansion for negative powers followed by algebraic manipulation. Part (i) is routine recall of the formula, and part (ii) requires multiplying by (1-x) and equating coefficients—a standard technique with minimal problem-solving demand. Slightly above average due to the two-step process and negative exponent, but still a typical C4 question. |
| Spec | 1.04c Extend binomial expansion: rational n, |x|<11.04d Binomial expansion validity: convergence conditions |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(\left(1+\frac{x}{a}\right)^{-2} = 1+(-2)\frac{x}{a}+\frac{-2-3}{2}\left(\frac{x}{a}\right)^2+\ldots\) | M1 | Check 3rd term; accept \(\frac{x^2}{a}\) |
| \(= 1-\frac{2x}{a}+\ldots\) or \(1+\left(-\frac{2x}{a}\right)\) | B1 | or \(1-2xa^{-1}\) (Ind of M1) |
| \(\ldots+\frac{3x^2}{a^2}+\ldots\) (or \(3\left(\frac{x}{a}\right)^2\) or \(3x^2a^{-2}\)) | A1 | Accept \(\frac{6}{2}\) for 3 |
| \((a+x)^{-2}=\frac{1}{a^2}\) {their expansion of \(\left(1+\frac{x}{a}\right)^{-2}\)} mult out | \(\sqrt{}\)A1 4 | \(\frac{1}{a^2}-\frac{2x}{a^3}+\frac{3x^2}{a^4}\); accept eg \(a^{-2}\) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Mult out \((1-x)\)(their exp) to produce all terms/cfs(\(x^2\)) | M1 | Ignore other terms |
| Produce \(\frac{3}{a^2}+\frac{2}{a}(=0)\) or \(\frac{3}{a^2}+\frac{2}{a^2}(=0)\) or AEF | A1 | Accept \(x^2\) if in both terms |
| \(a=-\frac{3}{2}\) www seen anywhere in (i) or (ii) | A1 3 | Disregard any ref to \(a=0\) |
# Question 3:
## Part (i)
| Answer/Working | Mark | Guidance |
|---|---|---|
| $\left(1+\frac{x}{a}\right)^{-2} = 1+(-2)\frac{x}{a}+\frac{-2-3}{2}\left(\frac{x}{a}\right)^2+\ldots$ | M1 | Check 3rd term; accept $\frac{x^2}{a}$ |
| $= 1-\frac{2x}{a}+\ldots$ or $1+\left(-\frac{2x}{a}\right)$ | B1 | or $1-2xa^{-1}$ (Ind of M1) |
| $\ldots+\frac{3x^2}{a^2}+\ldots$ (or $3\left(\frac{x}{a}\right)^2$ or $3x^2a^{-2}$) | A1 | Accept $\frac{6}{2}$ for 3 |
| $(a+x)^{-2}=\frac{1}{a^2}$ {their expansion of $\left(1+\frac{x}{a}\right)^{-2}$} mult out | $\sqrt{}$A1 **4** | $\frac{1}{a^2}-\frac{2x}{a^3}+\frac{3x^2}{a^4}$; accept eg $a^{-2}$ |
## Part (ii)
| Answer/Working | Mark | Guidance |
|---|---|---|
| Mult out $(1-x)$(their exp) to produce all terms/cfs($x^2$) | M1 | Ignore other terms |
| Produce $\frac{3}{a^2}+\frac{2}{a}(=0)$ or $\frac{3}{a^2}+\frac{2}{a^2}(=0)$ or AEF | A1 | Accept $x^2$ if in both terms |
| $a=-\frac{3}{2}$ www seen anywhere in (i) or (ii) | A1 **3** | Disregard any ref to $a=0$ |
---3 (i) Expand $( a + x ) ^ { - 2 }$ in ascending powers of $x$ up to and including the term in $x ^ { 2 }$.\\
(ii) When $( 1 - x ) ( a + x ) ^ { - 2 }$ is expanded, the coefficient of $x ^ { 2 }$ is 0 . Find the value of $a$.
\hfill \mbox{\textit{OCR C4 2009 Q3 [7]}}