| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2024 |
| Session | November |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Generalised Binomial Theorem and Partial Fractions |
| Type | Partial fractions with specific coefficient |
| Difficulty | Standard +0.8 This question requires partial fractions with a quadratic factor (leading to Bx+C form), then binomial expansion of three separate terms with careful coefficient tracking. The combination of algebraic manipulation and series expansion with specific coefficient extraction is more demanding than standard A-level questions, though still within P3 scope. |
| Spec | 1.02y Partial fractions: decompose rational functions1.04c Extend binomial expansion: rational n, |x|<1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| State or imply the form \(\frac{A}{1+2x}+\frac{Bx+C}{2+x^2}\) | B1 | |
| Use a correct method to find a constant | M1 | |
| Obtain one of \(A=1\), \(B=2\) and \(C=3\) | A1 | |
| Obtain a second value | A1 | |
| Obtain the third value | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| State \(\frac{-1\cdot-2\cdot-3}{3!}(2x)^3\) or \(-8\) | B1 FT | Correct term in \(x^3\) or coefficient of \(x^3\) in the expansion of \(A(1+2x)^{-1}\). Any equivalent form. |
| Use a correct method to obtain the coefficient of \(x^2\) in the expansion of \((2+x^2)^{-1}\) or the coefficient of \(x^2\) in the expansion of \(\left(1+\frac{x^2}{2}\right)^{-1}\) | M1 | Do not need to deal with \(2^{-1}\) at this stage. |
| Obtain \((Bx+C)\times\frac{1}{2}\times-\frac{1}{2}x^2\) or \(-\frac{B}{4}x^3\) or \(-\frac{B}{4}\) | A1 FT | Follow *their* \(B\) (and \(C\)). |
| Obtain final answer \(-8\frac{1}{2}\) or \(-8\frac{1}{2}x^3\) | A1 | Or simplified equivalent. Ignore additional terms for other powers of \(x\). |
## Question 7(a):
| Answer | Mark | Guidance |
|--------|------|----------|
| State or imply the form $\frac{A}{1+2x}+\frac{Bx+C}{2+x^2}$ | B1 | |
| Use a correct method to find a constant | M1 | |
| Obtain one of $A=1$, $B=2$ and $C=3$ | A1 | |
| Obtain a second value | A1 | |
| Obtain the third value | A1 | |
**Total: 5 marks**
## Question 7(b):
| Answer | Mark | Guidance |
|--------|------|----------|
| State $\frac{-1\cdot-2\cdot-3}{3!}(2x)^3$ or $-8$ | B1 FT | Correct term in $x^3$ or coefficient of $x^3$ in the expansion of $A(1+2x)^{-1}$. Any equivalent form. |
| Use a correct method to obtain the coefficient of $x^2$ in the expansion of $(2+x^2)^{-1}$ or the coefficient of $x^2$ in the expansion of $\left(1+\frac{x^2}{2}\right)^{-1}$ | M1 | Do not need to deal with $2^{-1}$ at this stage. |
| Obtain $(Bx+C)\times\frac{1}{2}\times-\frac{1}{2}x^2$ or $-\frac{B}{4}x^3$ or $-\frac{B}{4}$ | A1 FT | Follow *their* $B$ (and $C$). |
| Obtain final answer $-8\frac{1}{2}$ or $-8\frac{1}{2}x^3$ | A1 | Or simplified equivalent. Ignore additional terms for other powers of $x$. |
**Total: 4 marks**
---7\\
Let $f ( x ) = \frac { 5 x ^ { 2 } + 8 x + 5 } { ( 1 + 2 x ) \left( 2 + x ^ { 2 } \right) }$.
\begin{enumerate}[label=(\alph*)]
\item Express $\mathrm { f } ( x )$ in partial fractions.\\
\includegraphics[max width=\textwidth, alt={}, center]{656df2a8-fc4d-49f3-a649-746103b4576e-13_2726_34_97_21}
\item Hence find the coefficient of $x ^ { 3 }$ in the expansion of $\mathrm { f } ( x )$.
\end{enumerate}
\hfill \mbox{\textit{CAIE P3 2024 Q7 [9]}}