| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2024 |
| Session | November |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Generalised Binomial Theorem and Partial Fractions |
| Type | Partial fractions with validity range |
| Difficulty | Standard +0.3 This is a standard partial fractions question with binomial expansion and validity range. Part (a) is routine decomposition, part (b) requires expanding two binomial terms to x², and part (c) asks for the standard validity condition (smaller of the two convergence radii). Slightly easier than average as it's a well-practiced technique with no novel insight required. |
| Spec | 1.02y Partial fractions: decompose rational functions1.04c Extend binomial expansion: rational n, |x|<1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| State or imply the form \(\frac{A}{a-2x} + \frac{B}{3a+x}\) and use a correct method to find a constant | M1 | |
| Obtain \(A = 2a\) or \(B = a\) | A1 | |
| Obtain \(A = 2a\) and \(B = a\) | A1 | |
| Total | 3 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Use a correct method to obtain the first two terms of the expansion of \((a-2x)^{-1}\) or \(\left(1 - \frac{2x}{a}\right)^{-1}\) or \((3a+x)^{-1}\) or \(\left(1 + \frac{x}{3a}\right)^{-1}\) | M1 | |
| Obtain \(+2\left(1 + \frac{2x}{a} + \frac{4x^2}{a^2} + \ldots\right)\) | A1ft | OE. May be unsimplified. Follow *their* \(A\), \(B\) for an expansion involving \(a\). |
| Obtain \(+\frac{1}{3}\left(1 - \frac{x}{3a} + \frac{x^2}{9a^2} + \ldots\right)\) | A1ft | OE. May be unsimplified. Follow *their* \(A\), \(B\) for an expansion involving \(a\). |
| Obtain \(+\frac{7}{3} + \frac{35x}{9a} + \frac{217x^2}{27a^2}\) | A1 | Or simplified equivalent. Final answer. Ignore any terms in higher powers of \(x\). Do not ISW, e.g. multiplying by \(27a^2\). Condone different order of terms. |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Expanding \(7a^2(a-2x)^{-1}(3a+x)^{-1}\) from the original question. Use a correct method to obtain the first two terms of the expansion of \((a-2x)^{-1}\) or \(\left(1-\frac{2x}{a}\right)^{-1}\) or \((3a+x)^{-1}\) or \(\left(1+\frac{x}{3a}\right)^{-1}\) | M1 | |
| Obtain \(+7a\left(1 + \frac{2x}{a} + \frac{4x^2}{a^2} + \ldots\right)\) or \(+\frac{7a}{3}\left(1 - \frac{x}{3a} + \frac{x^2}{9a^2} + \ldots\right)\) | A1 | OE. May be unsimplified. May be implied by the expression shown for the next A1. |
| Obtain \(+\frac{7}{3}\left(1 + \frac{2x}{a} + \frac{4x^2}{a^2} + \ldots\right)\left(1 - \frac{x}{3a} + \frac{x^2}{9a^2} + \ldots\right)\) | A1 | OE. May be unsimplified. |
| Obtain \(+\frac{7}{3} + \frac{35x}{9a} + \frac{217x^2}{27a^2}\) | A1 | Or simplified equivalent. Final answer. Ignore any terms in higher powers of \(x\). Do not ISW, e.g. multiplying by \(27a^2\). Condone different order of terms. |
| Total | 4 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(\ | x\ | < \frac{a}{2}\) |
| Total | 1 |
## Question 8(a):
| Answer | Marks | Guidance |
|--------|-------|----------|
| State or imply the form $\frac{A}{a-2x} + \frac{B}{3a+x}$ and use a correct method to find a constant | M1 | |
| Obtain $A = 2a$ or $B = a$ | A1 | |
| Obtain $A = 2a$ and $B = a$ | A1 | |
| **Total** | **3** | |
## Question 8(b):
| Answer | Marks | Guidance |
|--------|-------|----------|
| Use a correct method to obtain the first two terms of the expansion of $(a-2x)^{-1}$ or $\left(1 - \frac{2x}{a}\right)^{-1}$ or $(3a+x)^{-1}$ or $\left(1 + \frac{x}{3a}\right)^{-1}$ | M1 | |
| Obtain $+2\left(1 + \frac{2x}{a} + \frac{4x^2}{a^2} + \ldots\right)$ | A1ft | OE. May be unsimplified. Follow *their* $A$, $B$ for an expansion involving $a$. |
| Obtain $+\frac{1}{3}\left(1 - \frac{x}{3a} + \frac{x^2}{9a^2} + \ldots\right)$ | A1ft | OE. May be unsimplified. Follow *their* $A$, $B$ for an expansion involving $a$. |
| Obtain $+\frac{7}{3} + \frac{35x}{9a} + \frac{217x^2}{27a^2}$ | A1 | Or simplified equivalent. Final answer. Ignore any terms in higher powers of $x$. Do not ISW, e.g. multiplying by $27a^2$. Condone different order of terms. |
**Alternative Method for Question 8(b):**
| Answer | Marks | Guidance |
|--------|-------|----------|
| Expanding $7a^2(a-2x)^{-1}(3a+x)^{-1}$ from the original question. Use a correct method to obtain the first two terms of the expansion of $(a-2x)^{-1}$ or $\left(1-\frac{2x}{a}\right)^{-1}$ or $(3a+x)^{-1}$ or $\left(1+\frac{x}{3a}\right)^{-1}$ | M1 | |
| Obtain $+7a\left(1 + \frac{2x}{a} + \frac{4x^2}{a^2} + \ldots\right)$ or $+\frac{7a}{3}\left(1 - \frac{x}{3a} + \frac{x^2}{9a^2} + \ldots\right)$ | A1 | OE. May be unsimplified. May be implied by the expression shown for the next A1. |
| Obtain $+\frac{7}{3}\left(1 + \frac{2x}{a} + \frac{4x^2}{a^2} + \ldots\right)\left(1 - \frac{x}{3a} + \frac{x^2}{9a^2} + \ldots\right)$ | A1 | OE. May be unsimplified. |
| Obtain $+\frac{7}{3} + \frac{35x}{9a} + \frac{217x^2}{27a^2}$ | A1 | Or simplified equivalent. Final answer. Ignore any terms in higher powers of $x$. Do not ISW, e.g. multiplying by $27a^2$. Condone different order of terms. |
| **Total** | **4** | |
## Question 8(c):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $\|x\| < \frac{a}{2}$ | B1 | Or $-\frac{a}{2} < x < \frac{a}{2}$. Mark final answer. Must make a clear statement. |
| **Total** | **1** | |
---8 Let $\mathrm { f } ( x ) = \frac { 7 a ^ { 2 } } { ( a - 2 x ) ( 3 a + x ) }$, where $a$ is a positive constant.
\begin{enumerate}[label=(\alph*)]
\item Express $\mathrm { f } ( x )$ in partial fractions.\\
\includegraphics[max width=\textwidth, alt={}, center]{6280ab81-0bdb-47b4-8651-bff1261a0adf-12_2718_40_107_2009}\\
\includegraphics[max width=\textwidth, alt={}, center]{6280ab81-0bdb-47b4-8651-bff1261a0adf-13_2726_33_97_22}
\item Hence obtain the expansion of $\mathrm { f } ( x )$ in ascending powers of $x$, up to and including the term in $x ^ { 2 }$. [4]
\item State the set of values of $x$ for which the expansion in part (b) is valid.
\end{enumerate}
\hfill \mbox{\textit{CAIE P3 2024 Q8 [8]}}