| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2020 |
| Session | November |
| 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 with fractional index (1/3), requiring routine substitution into the formula and simplification of coefficients. Part (b) asks for the standard validity condition |6x| < 1, which is direct recall. Both parts are mechanical with no problem-solving required. |
| Spec | 1.04c Extend binomial expansion: rational n, |x|<11.04d Binomial expansion validity: convergence conditions |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| State a correct unsimplified version of the \(x\) or \(x^2\) or \(x^3\) term | M1 | For the given expression |
| State correct first two terms \(1 + 2x\) | A1 | |
| Obtain the next two terms \(-4x^2 + \dfrac{40}{3}x^3\) | A1 + A1 | One mark for each correct term. ISW Accept \(13\frac{1}{3}\). The question asks for simplified coefficients, so candidates should cancel fractions |
| Total | 4 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| State answer \(\ | x\ | < \dfrac{1}{6}\) |
| Total | 1 |
## Question 2(a):
| Answer | Marks | Guidance |
|--------|-------|----------|
| State a correct unsimplified version of the $x$ or $x^2$ or $x^3$ term | **M1** | For the given expression |
| State correct first two terms $1 + 2x$ | **A1** | |
| Obtain the next two terms $-4x^2 + \dfrac{40}{3}x^3$ | **A1 + A1** | One mark for each correct term. ISW Accept $13\frac{1}{3}$. The question asks for simplified coefficients, so candidates should cancel fractions |
| **Total** | **4** | |
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## Question 2(b):
| Answer | Marks | Guidance |
|--------|-------|----------|
| State answer $\|x\| < \dfrac{1}{6}$ | **B1** | OE. Strict inequality |
| **Total** | **1** | |2
\begin{enumerate}[label=(\alph*)]
\item Expand $\sqrt [ 3 ] { 1 + 6 x }$ in ascending powers of $x$, up to and including the term in $x ^ { 3 }$, simplifying the coefficients.
\item State the set of values of $x$ for which the expansion is valid.
\end{enumerate}
\hfill \mbox{\textit{CAIE P3 2020 Q2 [5]}}