| Exam Board | OCR |
|---|---|
| Module | C4 (Core Mathematics 4) |
| Year | 2007 |
| Session | January |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Generalised Binomial Theorem |
| Type | Substitute expression for variable |
| Difficulty | Moderate -0.3 Part (i) is a standard binomial expansion with negative/fractional index requiring routine application of the formula. Part (ii) requires substitution and collecting terms, which adds a modest problem-solving element, but the overall question remains a straightforward textbook exercise testing binomial series manipulation with no novel insight required. |
| Spec | 1.04c Extend binomial expansion: rational n, |x|<1 |
| Answer | Marks | Guidance |
|---|---|---|
| (i) Show clear knowledge of binomial expansion | M1 | \(-3x\) should appear but brackets can be missing; \(-\frac{1}{4}, -\frac{3}{4}\) should appear, not \(-\frac{1}{3}, -\frac{3}{3}\) |
| \(= 1 + x\) | B1 | Correct first 2 terms; not dep on M1 |
| \(+ 2x^2\) | A1 | |
| \(+ \frac{14}{3}x^3\) | A1 | 4 marks |
| (ii) Attempt to substitute \(x + x^3\) for \(x\) in (i) | M1 | Not just in the \(\frac{14}{3}x^3\) term |
| Clear indication that \((x + x^3)^3\) has no term in \(x^3\) | A1 | |
| \(\frac{17}{3}\) | √A1 | 3 marks |
(i) Show clear knowledge of binomial expansion | M1 | $-3x$ should appear but brackets can be missing; $-\frac{1}{4}, -\frac{3}{4}$ should appear, not $-\frac{1}{3}, -\frac{3}{3}$
$= 1 + x$ | B1 | Correct first 2 terms; not dep on M1
$+ 2x^2$ | A1 |
$+ \frac{14}{3}x^3$ | A1 | 4 marks
(ii) Attempt to substitute $x + x^3$ for $x$ in (i) | M1 | Not just in the $\frac{14}{3}x^3$ term
Clear indication that $(x + x^3)^3$ has no term in $x^3$ | A1 |
$\frac{17}{3}$ | √A1 | 3 marks | f.t. cf$(x) + $cf$(x^3)$ in part (i) |
---\begin{enumerate}[label=(\roman*)]
\item Expand $(1 - 3x)^{-\frac{1}{2}}$ in ascending powers of $x$, up to and including the term in $x^3$. [4]
\item Hence find the coefficient of $x^3$ in the expansion of $\left(1 - 3(x + x^3)\right)^{-\frac{1}{2}}$. [3]
\end{enumerate}
\hfill \mbox{\textit{OCR C4 2007 Q5 [7]}}