| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2021 |
| Session | June |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Generalised Binomial Theorem and Partial Fractions |
| Type | Partial fractions then binomial expansion |
| Difficulty | Standard +0.3 This is a straightforward two-part question combining routine partial fractions decomposition with standard binomial expansions. Part (a) requires setting up and solving for constants in the form A/(2+x) + (Bx+C)/(3+x²), which is textbook technique. Part (b) involves expanding (2+x)^(-1) and (3+x²)^(-1) using the binomial theorem and collecting terms—mechanical but requiring care with coefficients. Both parts are standard exam exercises with no novel insight required, making this slightly easier than average. |
| 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 \(\frac{A}{2+x}+\frac{B+Cx}{3+x^2}\) | B1 | |
| Use a correct method for finding a constant | M1 | SOI |
| Obtain one of \(A=4\), \(B=1\) and \(C=-2\) | A1 | |
| Obtain a second value | A1 | |
| Obtain the third value | A1 | ISW |
| Total | 5 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Use correct method to find the first two terms of the expansion of \((2+x)^{-1}\), \(\left(1+\frac{1}{2}x\right)^{-1}\), \((3+x^2)^{-1}\) or \(\left(1+\frac{1}{3}x^2\right)^{-1}\) | M1 | Allow unsimplified but not if still including \(^nC_r\) |
| Obtain correct unsimplified expansions up to the term in \(x^2\) of each partial fraction | A1 FT, A1 FT | \(2\left(1-\frac{1}{2}x+\left(\frac{1}{2}x\right)^2\ldots\right)\) \(+\frac{1}{3}(1-2x)\left(1-\frac{1}{3}x^2\ldots\right)\) The FT is on *their* \(A\), \(B\) and \(C\) |
| Multiply out, up to the terms in \(x^2\), by \(B+Cx\), where \(BC\neq 0\) | M1 | Allow with \(B\) and \(C\) as implied in part (b) |
| Obtain final answer \(\frac{7}{3}-\frac{5}{3}x+\frac{7}{18}x^2\) | A1 | Or equivalent in form \(p+qx+rx^2\). A0 if they multiply through by 18. |
| Total | 5 |
## Question 9(a):
| Answer | Mark | Guidance |
|--------|------|----------|
| State or imply the form $\frac{A}{2+x}+\frac{B+Cx}{3+x^2}$ | B1 | |
| Use a correct method for finding a constant | M1 | SOI |
| Obtain one of $A=4$, $B=1$ and $C=-2$ | A1 | |
| Obtain a second value | A1 | |
| Obtain the third value | A1 | ISW |
| **Total** | **5** | |
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## Question 9(b):
| Answer | Mark | Guidance |
|--------|------|----------|
| Use correct method to find the first two terms of the expansion of $(2+x)^{-1}$, $\left(1+\frac{1}{2}x\right)^{-1}$, $(3+x^2)^{-1}$ or $\left(1+\frac{1}{3}x^2\right)^{-1}$ | M1 | Allow unsimplified but not if still including $^nC_r$ |
| Obtain correct unsimplified expansions up to the term in $x^2$ of each partial fraction | A1 FT, A1 FT | $2\left(1-\frac{1}{2}x+\left(\frac{1}{2}x\right)^2\ldots\right)$ $+\frac{1}{3}(1-2x)\left(1-\frac{1}{3}x^2\ldots\right)$ The FT is on *their* $A$, $B$ and $C$ |
| Multiply out, up to the terms in $x^2$, by $B+Cx$, where $BC\neq 0$ | M1 | Allow with $B$ and $C$ as implied in part (b) |
| Obtain final answer $\frac{7}{3}-\frac{5}{3}x+\frac{7}{18}x^2$ | A1 | Or equivalent in form $p+qx+rx^2$. A0 if they multiply through by 18. |
| **Total** | **5** | |
---9 Let $\mathrm { f } ( x ) = \frac { 14 - 3 x + 2 x ^ { 2 } } { ( 2 + x ) \left( 3 + x ^ { 2 } \right) }$.
\begin{enumerate}[label=(\alph*)]
\item Express $\mathrm { f } ( x )$ in partial fractions.
\item Hence obtain the expansion of $\mathrm { f } ( x )$ in ascending powers of $x$, up to and including the term in $x ^ { 2 }$.\\
\includegraphics[max width=\textwidth, alt={}, center]{459b8403-a481-4ece-88c0-e7600a47c8e4-14_292_732_264_705}
The diagram shows a trapezium $A B C D$ in which $A D = B C = r$ and $A B = 2 r$. The acute angles $B A D$ and $A B C$ are both equal to $x$ radians. Circular arcs of radius $r$ with centres $A$ and $B$ meet at $M$, the midpoint of $A B$.
\end{enumerate}
\hfill \mbox{\textit{CAIE P3 2021 Q9 [10]}}