| Exam Board | Edexcel |
|---|---|
| Module | P4 (Pure Mathematics 4) |
| Year | 2020 |
| Session | October |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Generalised Binomial Theorem |
| Type | Finding unknown constant from coefficient |
| Difficulty | Standard +0.3 This is a standard two-part binomial expansion question requiring routine application of the generalized binomial theorem followed by algebraic manipulation to find unknown constants by comparing coefficients. While it involves multiple steps, each step follows a predictable pattern with no novel insight required, making it slightly easier than average. |
| Spec | 1.04c Extend binomial expansion: rational n, |x|<1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(4^{-\frac{1}{2}}\) or \(\frac{1}{4^{\frac{1}{2}}}\) or \(\frac{1}{2}\) | B1 | For taking out factor of \(4^{-\frac{1}{2}}\) or \(\frac{1}{2}\). For direct expansion look for \(4^{-\frac{1}{2}}+\ldots\) |
| \((4-5x)^{-\frac{1}{2}} = \frac{1}{2}\left(1-\frac{5x}{4}\right)^{-\frac{1}{2}} = \ldots\left(1+\left(-\frac{1}{2}\right)\times\left(-\frac{5x}{4}\right)+\frac{\left(-\frac{1}{2}\right)\times\left(-\frac{3}{2}\right)}{2!}\times\left(-\frac{5x}{4}\right)^2+\ldots\right)\) | M1A1 | M1: Form \((1+ax)^{-\frac{1}{2}}\) where \(a\neq 1\) or \(-5\). Sufficient to see either term two or term three. Allow sign slip. A1: Any correct unsimplified binomial expansion of \(\left(1-\frac{5x}{4}\right)^{-\frac{1}{2}}\) |
| \(= \frac{1}{2}+\frac{5}{16}x+\frac{75}{256}x^2+\ldots\) | A1 | Must be simplified |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(\frac{2+kx}{(2-3x)^3} = (2+kx)\left(\frac{1}{2}+\frac{5}{16}x+\frac{75}{256}x^2+\ldots\right)\); compares \(x\) terms: \(\frac{10}{16}+\frac{k}{2}=\frac{3}{10} \Rightarrow k=-\frac{13}{20}\) | M1 A1 | M1: Correct equation in \(k\) from comparing \(x\) terms, must lead to value for \(k\). Look for \(Pk+2Q=\frac{3}{10}\). Condone slips (copying errors) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Compares \(x^2\) terms: \(m = \frac{75}{128}+\frac{5}{16}\times\left(-\frac{13}{20}\right) \Rightarrow m=\frac{49}{128}\) | M1 A1 | M1: Correctly compares \(x^2\) terms following through on expansion and value of \(k\). Look for \(Qk+2R=m\). Condone slips. A1: \(m=\frac{49}{128}\), condone sight of \(\frac{49}{128}x^2\) |
# Question 2(a):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $4^{-\frac{1}{2}}$ or $\frac{1}{4^{\frac{1}{2}}}$ or $\frac{1}{2}$ | B1 | For taking out factor of $4^{-\frac{1}{2}}$ or $\frac{1}{2}$. For direct expansion look for $4^{-\frac{1}{2}}+\ldots$ |
| $(4-5x)^{-\frac{1}{2}} = \frac{1}{2}\left(1-\frac{5x}{4}\right)^{-\frac{1}{2}} = \ldots\left(1+\left(-\frac{1}{2}\right)\times\left(-\frac{5x}{4}\right)+\frac{\left(-\frac{1}{2}\right)\times\left(-\frac{3}{2}\right)}{2!}\times\left(-\frac{5x}{4}\right)^2+\ldots\right)$ | M1A1 | M1: Form $(1+ax)^{-\frac{1}{2}}$ where $a\neq 1$ or $-5$. Sufficient to see either term two or term three. Allow sign slip. A1: Any correct unsimplified binomial expansion of $\left(1-\frac{5x}{4}\right)^{-\frac{1}{2}}$ |
| $= \frac{1}{2}+\frac{5}{16}x+\frac{75}{256}x^2+\ldots$ | A1 | Must be simplified |
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# Question 2(b):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $\frac{2+kx}{(2-3x)^3} = (2+kx)\left(\frac{1}{2}+\frac{5}{16}x+\frac{75}{256}x^2+\ldots\right)$; compares $x$ terms: $\frac{10}{16}+\frac{k}{2}=\frac{3}{10} \Rightarrow k=-\frac{13}{20}$ | M1 A1 | M1: Correct equation in $k$ from comparing $x$ terms, must lead to value for $k$. Look for $Pk+2Q=\frac{3}{10}$. Condone slips (copying errors) |
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# Question 2(c):
| Answer/Working | Mark | Guidance |
|---|---|---|
| Compares $x^2$ terms: $m = \frac{75}{128}+\frac{5}{16}\times\left(-\frac{13}{20}\right) \Rightarrow m=\frac{49}{128}$ | M1 A1 | M1: Correctly compares $x^2$ terms following through on expansion and value of $k$. Look for $Qk+2R=m$. Condone slips. A1: $m=\frac{49}{128}$, condone sight of $\frac{49}{128}x^2$ |
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\item (a) Use the binomial expansion to expand
\end{enumerate}
$$( 4 - 5 x ) ^ { - \frac { 1 } { 2 } } \quad | x | < \frac { 4 } { 5 }$$
in ascending powers of $x$, up to and including the term in $x ^ { 2 }$ giving each coefficient as a fully simplified fraction.
$$f ( x ) = \frac { 2 + k x } { \sqrt { 4 - 5 x } } \quad \text { where } k \text { is a constant and } | x | < \frac { 4 } { 5 }$$
Given that the series expansion of $\mathrm { f } ( x )$, in ascending powers of $x$, is
$$1 + \frac { 3 } { 10 } x + m x ^ { 2 } + \ldots \quad \text { where } m \text { is a constant }$$
(b) find the value of $k$,\\
(c) find the value of $m$.\\
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\hfill \mbox{\textit{Edexcel P4 2020 Q2 [8]}}