| Exam Board | Edexcel |
|---|---|
| Module | P4 (Pure Mathematics 4) |
| Year | 2021 |
| Session | October |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Generalised Binomial Theorem |
| Type | Non-zero terms only |
| Difficulty | Standard +0.3 This is a straightforward application of the binomial expansion for (1+x)^n with n=1/2 and a substitution of -4x². Part (a) requires routine application of the formula with coefficient simplification, while part (b) is a direct numerical substitution. The question is slightly easier than average as it follows a standard template with no problem-solving or novel insight required. |
| Spec | 1.04c Extend binomial expansion: rational n, |x|<11.04d Binomial expansion validity: convergence conditions |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(\sqrt{1-4x^2} = 1 - \frac{1}{2}\times 4x^2\) | B1 | Correct first two terms; \(1+\frac{1}{2}\times(-4x^2)\) is fine |
| \(\dfrac{\frac{1}{2}\times-\frac{1}{2}\times(-4x^2)^2}{2}\) or \(\dfrac{\frac{1}{2}\times-\frac{1}{2}\times-\frac{3}{2}\times(-4x^2)^3}{3!}\) | M1 | Correct attempt at term 3 or term 4; condone failure to square or cube the 4 |
| \(= 1-2x^2-2x^4-4x^6+\ldots\) | A1, A1 | A1: any two correct simplified terms from \(-2x^2,\ -2x^4,\ -4x^6\). A1: fully correct; ignore additional terms |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Substitutes \(x=\frac{1}{4}\) into both sides of (a): \(\sqrt{\frac{3}{4}} \approx 1-2\left(\frac{1}{4}\right)^2-2\left(\frac{1}{4}\right)^4-4\left(\frac{1}{4}\right)^6\) | M1 | Achieves LHS of \(\sqrt{\frac{3}{4}}\) or \(\frac{\sqrt{3}}{2}\) |
| \(\sqrt{3} \approx 1.7324\) | A1 | Correct answer only; cao (calculator answer is 1.7321) |
# Question 4(a):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $\sqrt{1-4x^2} = 1 - \frac{1}{2}\times 4x^2$ | B1 | Correct first two terms; $1+\frac{1}{2}\times(-4x^2)$ is fine |
| $\dfrac{\frac{1}{2}\times-\frac{1}{2}\times(-4x^2)^2}{2}$ or $\dfrac{\frac{1}{2}\times-\frac{1}{2}\times-\frac{3}{2}\times(-4x^2)^3}{3!}$ | M1 | Correct attempt at term 3 or term 4; condone failure to square or cube the 4 |
| $= 1-2x^2-2x^4-4x^6+\ldots$ | A1, A1 | A1: any two correct simplified terms from $-2x^2,\ -2x^4,\ -4x^6$. A1: fully correct; ignore additional terms |
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# Question 4(b):
| Answer/Working | Mark | Guidance |
|---|---|---|
| Substitutes $x=\frac{1}{4}$ into both sides of (a): $\sqrt{\frac{3}{4}} \approx 1-2\left(\frac{1}{4}\right)^2-2\left(\frac{1}{4}\right)^4-4\left(\frac{1}{4}\right)^6$ | M1 | Achieves LHS of $\sqrt{\frac{3}{4}}$ or $\frac{\sqrt{3}}{2}$ |
| $\sqrt{3} \approx 1.7324$ | A1 | Correct answer only; cao (calculator answer is 1.7321) |4.
$$\mathrm { f } ( x ) = \sqrt { 1 - 4 x ^ { 2 } } \quad | x | < \frac { 1 } { 2 }$$
\begin{enumerate}[label=(\alph*)]
\item Find, in ascending powers of $x$, the first four non-zero terms of the binomial expansion of $\mathrm { f } ( x )$. Give each coefficient in simplest form.
\item By substituting $x = \frac { 1 } { 4 }$ into the binomial expansion of $\mathrm { f } ( x )$, obtain an approximation for $\sqrt { 3 }$
Give your answer to 4 decimal places.\\
\includegraphics[max width=\textwidth, alt={}, center]{08756c4b-6619-42da-ac8a-2bf065c01de8-13_42_63_2606_1852}
\end{enumerate}
\hfill \mbox{\textit{Edexcel P4 2021 Q4 [6]}}