| Exam Board | OCR MEI |
|---|---|
| Module | Paper 2 (Paper 2) |
| Year | 2024 |
| Session | June |
| Marks | 5 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Generalised Binomial Theorem |
| Type | Expand and state validity |
| Difficulty | Moderate -0.8 This is a straightforward application of the binomial expansion formula for fractional powers. Part (a) requires factoring out 8^(1/3) and expanding (1 + 3x/8)^(1/3) using the standard formula—routine calculation with no conceptual challenges. Part (b) tests basic knowledge that the expansion is valid when |3x/8| < 1, requiring simple algebraic manipulation. Both parts are standard textbook exercises with no problem-solving or novel insight required. |
| Spec | 1.04c Extend binomial expansion: rational n, |x|<11.04d Binomial expansion validity: convergence conditions |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(8^{\frac{1}{3}}\) or \(2\) seen | B1 | |
| \(1 + \left(\frac{1}{3}\right)\left(\frac{3x}{8}\right) + \left(\frac{1}{3}\right)\left(\frac{1}{3}-1\right)\frac{\left(\frac{3x}{8}\right)^2}{2!} + \cdots\) | M1 | Two of first three terms correct; ignore terms in \(x^3\) and above; may be embedded; must see at least substitution for third term |
| \(\left(1 + \frac{x}{8} - \frac{x^2}{64} + \cdots\right)\) | A1 | May be unsimplified; may be embedded |
| \(2 + \frac{x}{4} - \frac{x^2}{32}\) or \(2\left(1 + \frac{x}{8} - \frac{x^2}{64} + \cdots\right)\) isw | A1 | All three terms correct; ignore extra terms |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \( | x | < \frac{8}{3}\) or \(-\frac{8}{3} < x < \frac{8}{3}\) |
## Question 10:
### Part (a):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $8^{\frac{1}{3}}$ or $2$ seen | B1 | |
| $1 + \left(\frac{1}{3}\right)\left(\frac{3x}{8}\right) + \left(\frac{1}{3}\right)\left(\frac{1}{3}-1\right)\frac{\left(\frac{3x}{8}\right)^2}{2!} + \cdots$ | M1 | Two of first three terms correct; ignore terms in $x^3$ and above; may be embedded; must see at least substitution for third term |
| $\left(1 + \frac{x}{8} - \frac{x^2}{64} + \cdots\right)$ | A1 | May be unsimplified; may be embedded |
| $2 + \frac{x}{4} - \frac{x^2}{32}$ or $2\left(1 + \frac{x}{8} - \frac{x^2}{64} + \cdots\right)$ isw | A1 | All three terms correct; ignore extra terms |
### Part (b):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $|x| < \frac{8}{3}$ **or** $-\frac{8}{3} < x < \frac{8}{3}$ | B1FT | Allow $|x| \leq \frac{8}{3}$ or $-\frac{8}{3} \leq x \leq \frac{8}{3}$; FT their $\left(1 + \frac{a}{b}x\right)$ |10
\begin{enumerate}[label=(\alph*)]
\item Determine the first three terms in ascending powers of $x$ of the binomial expansion of $( 8 + 3 x ) ^ { \frac { 1 } { 3 } }$.
\item State the range of values of $x$ for which this expansion is valid.
\end{enumerate}
\hfill \mbox{\textit{OCR MEI Paper 2 2024 Q10 [5]}}