| Exam Board | Edexcel |
|---|---|
| Module | Paper 1 (Paper 1) |
| Year | 2021 |
| Session | October |
| Marks | 11 |
| 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 A-level question combining partial fractions decomposition with binomial expansion. Part (a) uses routine cover-up/substitution methods, and part (b) applies standard binomial expansion to each term. The validity range requires comparing |5x+2| > 1 and |2x| < 1, which is straightforward. While it has multiple parts, each step follows textbook procedures without requiring novel insight. |
| Spec | 1.05l Double angle formulae: and compound angle formulae1.05o Trigonometric equations: solve in given intervals1.05p Proof involving trig: functions and identities |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(50x^2 + 38x + 9 \equiv A(5x+2)(1-2x) + B(1-2x) + C(5x+2)^2\); obtains \(B=...\) or \(C=...\) | M1 | Uses correct identity with appropriate strategy e.g. sub \(x = \frac{1}{2}\); may be implied by one correct value |
| \(B = 1\) and \(C = 2\) | A1 | Both values correct |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| E.g. \(x=0 \Rightarrow 9 = 2A + B + 4C \Rightarrow 9 = 2A + 1 + 8 \Rightarrow A = ...\) | M1 | Uses appropriate method to establish equation connecting \(A\) with \(B\) and/or \(C\); uses their \(B\)/\(C\) values |
| \(A = 0\) | A1* | Fully correct proof with no errors |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(\frac{1}{(5x+2)^2} = (5x+2)^{-2} = 2^{-2}\left(1+\frac{5}{2}x\right)^{-2}\) or \((5x+2)^{-2} = 2^{-2} + ...\) | M1 | Applies key steps: writes as \((5x+2)^{-2}\), takes out factor of \(2^{-2}\); or uses direct expansion to obtain \(2^{-2}+...\) |
| \(\left(1+\frac{5}{2}x\right)^{-2} = 1 - 2\left(\frac{5}{2}x\right) + \frac{(-2)(-3)}{2!}\left(\frac{5}{2}x\right)^2 + ...\) | M1 | Correct attempt at binomial expansion of \((1+*x)^{-2}\); look for \(1+(-2)*x + \frac{(-2)(-3)}{2}*x^2\) where \(*\) is not 5 or 1 |
| \(2^{-2}\left(1+\frac{5}{2}x\right)^{-2} = \frac{1}{4} - \frac{5}{4}x + \frac{75}{16}x^2 + ...\) | A1 | Fully correct expansion of \((2+5x)^{-2}\) |
| \(\frac{1}{(1-2x)} = (1-2x)^{-1} = 1 + 2x + \frac{(-1)(-2)}{2!}(2x)^2 + ...\) | M1 | Correct attempt at binomial expansion of \((1-2x)^{-1}\) |
| \(\frac{1}{(5x+2)^2} + \frac{2}{1-2x} = \frac{1}{4} - \frac{5}{4}x + \frac{75}{16}x^2 + ... + 2 + 4x + 8x^2 + ...\) | dM1 | Fully correct strategy dependent on previous TWO method marks; must attempt to use values of \(B\) and \(C\) |
| \(= \frac{9}{4} + \frac{11}{4}x + \frac{203}{16}x^2 + ...\) | A1 | Correct expression or correct values for \(p\), \(q\) and \(r\) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \( | x | < \frac{2}{5}\) |
## Question 9:
### Part (a)(i):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $50x^2 + 38x + 9 \equiv A(5x+2)(1-2x) + B(1-2x) + C(5x+2)^2$; obtains $B=...$ or $C=...$ | M1 | Uses correct identity with appropriate strategy e.g. sub $x = \frac{1}{2}$; may be implied by one correct value |
| $B = 1$ and $C = 2$ | A1 | Both values correct |
**(shown in part a(i) section)**
### Part (a)(ii):
| Answer/Working | Mark | Guidance |
|---|---|---|
| E.g. $x=0 \Rightarrow 9 = 2A + B + 4C \Rightarrow 9 = 2A + 1 + 8 \Rightarrow A = ...$ | M1 | Uses appropriate method to establish equation connecting $A$ with $B$ and/or $C$; uses their $B$/$C$ values |
| $A = 0$ | A1* | Fully correct proof with no errors |
**(4 marks total for (a))**
### Part (b)(i):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $\frac{1}{(5x+2)^2} = (5x+2)^{-2} = 2^{-2}\left(1+\frac{5}{2}x\right)^{-2}$ or $(5x+2)^{-2} = 2^{-2} + ...$ | M1 | Applies key steps: writes as $(5x+2)^{-2}$, takes out factor of $2^{-2}$; or uses direct expansion to obtain $2^{-2}+...$ |
| $\left(1+\frac{5}{2}x\right)^{-2} = 1 - 2\left(\frac{5}{2}x\right) + \frac{(-2)(-3)}{2!}\left(\frac{5}{2}x\right)^2 + ...$ | M1 | Correct attempt at binomial expansion of $(1+*x)^{-2}$; look for $1+(-2)*x + \frac{(-2)(-3)}{2}*x^2$ where $*$ is not 5 or 1 |
| $2^{-2}\left(1+\frac{5}{2}x\right)^{-2} = \frac{1}{4} - \frac{5}{4}x + \frac{75}{16}x^2 + ...$ | A1 | Fully correct expansion of $(2+5x)^{-2}$ |
| $\frac{1}{(1-2x)} = (1-2x)^{-1} = 1 + 2x + \frac{(-1)(-2)}{2!}(2x)^2 + ...$ | M1 | Correct attempt at binomial expansion of $(1-2x)^{-1}$ |
| $\frac{1}{(5x+2)^2} + \frac{2}{1-2x} = \frac{1}{4} - \frac{5}{4}x + \frac{75}{16}x^2 + ... + 2 + 4x + 8x^2 + ...$ | dM1 | Fully correct strategy dependent on previous TWO method marks; must attempt to use values of $B$ and $C$ |
| $= \frac{9}{4} + \frac{11}{4}x + \frac{203}{16}x^2 + ...$ | A1 | Correct expression or correct values for $p$, $q$ and $r$ |
### Part (b)(ii):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $|x| < \frac{2}{5}$ | B1 | Correct range; also allow $-\frac{2}{5} < x < \frac{2}{5}$ or $x \in \left(-\frac{2}{5}, \frac{2}{5}\right)$; do not allow multiple answers |
**(7 marks total for (b))**
---9.
$$f ( x ) = \frac { 50 x ^ { 2 } + 38 x + 9 } { ( 5 x + 2 ) ^ { 2 } ( 1 - 2 x ) } \quad x \neq - \frac { 2 } { 5 } \quad x \neq \frac { 1 } { 2 }$$
Given that $\mathrm { f } ( x )$ can be expressed in the form
$$\frac { A } { 5 x + 2 } + \frac { B } { ( 5 x + 2 ) ^ { 2 } } + \frac { C } { 1 - 2 x }$$
where $A$, $B$ and $C$ are constants
\begin{enumerate}[label=(\alph*)]
\item \begin{enumerate}[label=(\roman*)]
\item find the value of $B$ and the value of $C$
\item show that $A = 0$
\end{enumerate}\item \begin{enumerate}[label=(\roman*)]
\item Use binomial expansions to show that, in ascending powers of $x$
$$f ( x ) = p + q x + r x ^ { 2 } + \ldots$$
where $p , q$ and $r$ are simplified fractions to be found.
\item Find the range of values of $x$ for which this expansion is valid.
\end{enumerate}\end{enumerate}
\hfill \mbox{\textit{Edexcel Paper 1 2021 Q9 [11]}}