| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2020 |
| 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 for negative powers with standard coefficient simplification and validity condition |3x| < 2. Requires only direct recall of the formula and routine algebraic manipulation, making it easier than average but not trivial due to the negative power and coefficient handling. |
| Spec | 1.04c Extend binomial expansion: rational n, |x|<11.04d Binomial expansion validity: convergence conditions |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| State a correct unsimplified version of the \(x\) or \(x^2\) term of the expansion of \((2-3x)^{-2}\) or \(\left(1-\dfrac{3}{2}x\right)^{-2}\) | M1 | |
| State correct first term \(\dfrac{1}{4}\) | B1 | |
| Obtain the next two terms \(\dfrac{3}{4}x + \dfrac{27}{16}x^2\) | A1+A1 | |
| Total | 4 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| State answer \(\ | x\ | < \dfrac{2}{3}\), or equivalent |
| Total | 1 |
## Question 2(a):
| Answer | Mark | Guidance |
|--------|------|----------|
| State a correct unsimplified version of the $x$ or $x^2$ term of the expansion of $(2-3x)^{-2}$ or $\left(1-\dfrac{3}{2}x\right)^{-2}$ | M1 | |
| State correct first term $\dfrac{1}{4}$ | B1 | |
| Obtain the next two terms $\dfrac{3}{4}x + \dfrac{27}{16}x^2$ | A1+A1 | |
| **Total** | **4** | |
---
## Question 2(b):
| Answer | Mark | Guidance |
|--------|------|----------|
| State answer $\|x\| < \dfrac{2}{3}$, or equivalent | B1 | |
| **Total** | **1** | |2
\begin{enumerate}[label=(\alph*)]
\item Expand $( 2 - 3 x ) ^ { - 2 }$ in ascending powers of $x$, up to and including the term in $x ^ { 2 }$, 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]}}