| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2022 |
| Session | November |
| 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 a repeated linear factor) and standard binomial expansion. Part (a) requires setting up and solving for constants A, B, C, which is a textbook exercise. Part (b) involves expanding (1+x)^{-1} and (2+x)^{-2} using the binomial theorem and collecting terms—mechanical but requiring care. Both techniques are core P3 syllabus 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|<11.10c Magnitude and direction: of vectors1.10d Vector operations: addition and scalar multiplication1.10f Distance between points: using position vectors1.10g Problem solving with vectors: in geometry |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| State or imply the form \(\frac{A}{1+x} + \frac{B}{2+x} + \frac{C}{(2+x)^2}\) | B1 | |
| Use a correct method to find a constant | M1 | |
| Obtain one of \(A=3\), \(B=-1\) and \(C=-2\) | A1 | SR after B0 can score M1A1 for one correct value |
| Obtain a second value | A1 | |
| Obtain the third value | A1 | \(\frac{A}{1+x} + \frac{Dx+E}{(2+x)^2}\), where \(A=3\), \(D=-1\) and \(E=-4\), is awarded B1 M1 A1 A1 A1 as above |
| Total | 5 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Use a correct method to find the first two terms of the expansion of \((1+x)^{-1}\), \((2+x)^{-1}\), \(\left(1+\frac{1}{2}x\right)^{-1}\), \((2+x)^{-2}\) or \(\left(1+\frac{1}{2}x\right)^{-2}\) | M1 | For the \(A, D, E\) form of fractions, award M1 A1FT A1FT for the expanded partial fractions, then if \(D\neq 0\), M1 for multiplying out fully, and A1 for the final answer |
| Obtain correct unsimplified expansions up to the term in \(x^2\) of each partial fraction | A3 FT | \(3\!\left(1-x+x^2\ldots\right)\); \(-\frac{1}{2}\!\left(1-\frac{x}{2}+\frac{x^2}{4}\ldots\right)\); \(-\frac{2}{4}\!\left(1-x+\frac{3}{4}x^2\right)\) |
| Obtain final answer \(2 - \frac{9}{4}x + \frac{5}{2}x^2\) | A1 | |
| Total | 5 |
## Question 10(a):
| Answer | Mark | Guidance |
|--------|------|----------|
| State or imply the form $\frac{A}{1+x} + \frac{B}{2+x} + \frac{C}{(2+x)^2}$ | B1 | |
| Use a correct method to find a constant | M1 | |
| Obtain one of $A=3$, $B=-1$ and $C=-2$ | A1 | SR after B0 can score M1A1 for one correct value |
| Obtain a second value | A1 | |
| Obtain the third value | A1 | $\frac{A}{1+x} + \frac{Dx+E}{(2+x)^2}$, where $A=3$, $D=-1$ and $E=-4$, is awarded B1 M1 A1 A1 A1 as above |
| **Total** | **5** | |
## Question 10(b):
| Answer | Mark | Guidance |
|--------|------|----------|
| Use a correct method to find the first two terms of the expansion of $(1+x)^{-1}$, $(2+x)^{-1}$, $\left(1+\frac{1}{2}x\right)^{-1}$, $(2+x)^{-2}$ or $\left(1+\frac{1}{2}x\right)^{-2}$ | M1 | For the $A, D, E$ form of fractions, award M1 A1FT A1FT for the expanded partial fractions, then if $D\neq 0$, M1 for multiplying out fully, and A1 for the final answer |
| Obtain correct unsimplified expansions up to the term in $x^2$ of each partial fraction | A3 FT | $3\!\left(1-x+x^2\ldots\right)$; $-\frac{1}{2}\!\left(1-\frac{x}{2}+\frac{x^2}{4}\ldots\right)$; $-\frac{2}{4}\!\left(1-x+\frac{3}{4}x^2\right)$ |
| Obtain final answer $2 - \frac{9}{4}x + \frac{5}{2}x^2$ | A1 | |
| **Total** | **5** | |10 Let $\mathrm { f } ( x ) = \frac { 2 x ^ { 2 } + 7 x + 8 } { ( 1 + x ) ( 2 + x ) ^ { 2 } }$.
\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]{98001cfe-46a1-4c8f-9230-c140ebff6176-18_737_1034_262_552}
In the diagram, $O A B C D$ is a solid figure in which $O A = O B = 4$ units and $O D = 3$ units. The edge $O D$ is vertical, $D C$ is parallel to $O B$ and $D C = 1$ unit. The base, $O A B$, is horizontal and angle $A O B = 90 ^ { \circ }$. Unit vectors $\mathbf { i } , \mathbf { j }$ and $\mathbf { k }$ are parallel to $O A , O B$ and $O D$ respectively. The midpoint of $A B$ is $M$ and the point $N$ on $B C$ is such that $C N = 2 N B$.\\
(a) Express vectors $\overrightarrow { M D }$ and $\overrightarrow { O N }$ in terms of $\mathbf { i } , \mathbf { j }$ and $\mathbf { k }$.\\
(b) Calculate the angle in degrees between the directions of $\overrightarrow { M D }$ and $\overrightarrow { O N }$.
\item Show that the length of the perpendicular from $M$ to $O N$ is $\sqrt { \frac { 22 } { 5 } }$.\\
If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.
\end{enumerate}
\hfill \mbox{\textit{CAIE P3 2022 Q10 [10]}}