| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2010 |
| Session | June |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Generalised Binomial Theorem and Partial Fractions |
| Type | Partial fractions with repeated linear factor |
| Difficulty | Standard +0.3 This is a standard two-part question combining partial fractions with binomial expansion. Part (i) requires decomposing into the form A/(1-2x) + B/(2+x) + C/(2+x)², which is routine but involves a squared factor. Part (ii) applies standard binomial expansions to each term. While it requires careful algebraic manipulation and multiple steps (10 marks total), it follows a well-practiced procedure with no novel insight required, making it slightly easier than average. |
| Spec | 1.02y Partial fractions: decompose rational functions1.04c Extend binomial expansion: rational n, |x|<1 |
| Answer | Marks | Guidance |
|---|---|---|
| (i) State or imply partial fractions of the form \(\frac{A}{1-2x} + \frac{B}{2+x} + \frac{C}{(2+x)^2}\) | B1 | — |
| Use any relevant method to determine a constant | M1 | — |
| Obtain one of the values \(A = 1\), \(B = 1\), \(C = -2\) | A1 | — |
| Obtain a second value | A1 | — |
| Obtain the third value | A1 | [5] |
| [The form \(\frac{A}{1-2x} + \frac{Dx+E}{(2+x)^2}\), where \(A = 1\), \(D = 1\), \(E = 0\), is acceptable scoring B1M1A1A1A1 as above.] | — | — |
| [For the \(A, D, E\) form of partial fractions, give M1A1√A1√ for the expansions then, if \(D \neq 0\), M1 for multiplying out fully and A1 for the final answer.] | — | — |
| [In the case of an attempt to expand \((4 + 5x - x^2)(1-2x)^{-1}(2+x)^{-2}\), give M1A1A1 for the expansions, M1 for multiplying out fully, and A1 for the final answer.] | — | — |
| [SR: If \(B\) or \(C\) omitted from the form of fractions, give B0M1A0A0A0 in (i); M1A1√A1√ in (ii).] | — | — |
| [SR: If \(D\) or \(E\) omitted from the form of fractions, give B0M1A0A0A0 in (i); M1A1√A1√ in (ii).] | — | — |
| (ii) Use correct method to obtain the first two terms of the expansion of \((1-2x)^{-1}\), \((2+x)^{-1}\), \((2+x)^{-2}\), \((1+\frac{1}{2}x)^{-1}\), or \((1+\frac{1}{4}x)^{-2}\) | M1 | — |
| Obtain correct unsimplified expansions up to the term in \(x^2\) of each partial fraction \(A1√\) | A1√ | — |
| Obtain correct unsimplified expansions up to the term in \(x^2\) of each partial fraction \(A1√\) | A1√ | — |
| Obtain answer \(1 + \frac{9}{4}x + \frac{15}{4}x^2\), or equivalent | A1 | [5] |
| [Symbolic binomial coefficients, e.g. \(\binom{-1}{1}\), are not sufficient for the M1. The f.t. is on \(A, B, C\).] | — | — |
**(i)** State or imply partial fractions of the form $\frac{A}{1-2x} + \frac{B}{2+x} + \frac{C}{(2+x)^2}$ | B1 | —
Use any relevant method to determine a constant | M1 | —
Obtain one of the values $A = 1$, $B = 1$, $C = -2$ | A1 | —
Obtain a second value | A1 | —
Obtain the third value | A1 | [5]
| [The form $\frac{A}{1-2x} + \frac{Dx+E}{(2+x)^2}$, where $A = 1$, $D = 1$, $E = 0$, is acceptable scoring B1M1A1A1A1 as above.] | — | —
| [For the $A, D, E$ form of partial fractions, give M1A1√A1√ for the expansions then, if $D \neq 0$, M1 for multiplying out fully and A1 for the final answer.] | — | —
| [In the case of an attempt to expand $(4 + 5x - x^2)(1-2x)^{-1}(2+x)^{-2}$, give M1A1A1 for the expansions, M1 for multiplying out fully, and A1 for the final answer.] | — | —
| [SR: If $B$ or $C$ omitted from the form of fractions, give B0M1A0A0A0 in (i); M1A1√A1√ in (ii).] | — | —
| [SR: If $D$ or $E$ omitted from the form of fractions, give B0M1A0A0A0 in (i); M1A1√A1√ in (ii).] | — | —
**(ii)** Use correct method to obtain the first two terms of the expansion of $(1-2x)^{-1}$, $(2+x)^{-1}$, $(2+x)^{-2}$, $(1+\frac{1}{2}x)^{-1}$, or $(1+\frac{1}{4}x)^{-2}$ | M1 | —
Obtain correct unsimplified expansions up to the term in $x^2$ of each partial fraction $A1√$ | A1√ | —
Obtain correct unsimplified expansions up to the term in $x^2$ of each partial fraction $A1√$ | A1√ | —
Obtain answer $1 + \frac{9}{4}x + \frac{15}{4}x^2$, or equivalent | A1 | [5]
| [Symbolic binomial coefficients, e.g. $\binom{-1}{1}$, are not sufficient for the M1. The f.t. is on $A, B, C$.] | — | —
---\begin{enumerate}[label=(\roman*)]
\item Express $\frac{4 + 5x - x^2}{(1 - 2x)(2 + x)^2}$ in partial fractions. [5]
\item Hence obtain the expansion of $\frac{4 + 5x - x^2}{(1 - 2x)(2 + x)^2}$ in ascending powers of $x$, up to and including the term in $x^2$. [5]
\end{enumerate}
\hfill \mbox{\textit{CAIE P3 2010 Q9 [10]}}