| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2020 |
| Session | Specimen |
| Marks | 4 |
| 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 negative/fractional powers with standard validity condition |3x| < 1. Requires only direct substitution into the formula and simplification of coefficients—routine bookwork with no problem-solving element. |
| 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 \(x^2\) term | M1 | Symbolic coefficients, e.g. \(\binom{-1}{2}\), are not sufficient for the M mark |
| State correct first two terms \(1 - x\) | A1 | |
| State the next term \(+ 2x^2\) | A1 | |
| Total | 3 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \( | x | < \frac{1}{3}\) |
| Total | 1 |
## Question 2(a):
| Answer | Marks | Guidance |
|--------|-------|----------|
| State a correct unsimplified version of the $x$ or $x^2$ term | M1 | Symbolic coefficients, e.g. $\binom{-1}{2}$, are not sufficient for the M mark |
| State correct first two terms $1 - x$ | A1 | |
| State the next term $+ 2x^2$ | A1 | |
| **Total** | **3** | |
## Question 2(b):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $|x| < \frac{1}{3}$ | B1 | OE |
| **Total** | **1** | |
---2 (a) Expand $( 1 + 3 x ) ^ { - \frac { 1 } { 3 } }$ in ascending powers of $x$, up to and including the term in $x ^ { 2 }$, simplifying the coefficients.\\
(b) State the set of values of $x$ for which the expansion is valid.\\
\hfill \mbox{\textit{CAIE P3 2020 Q2 [4]}}