| Exam Board | Edexcel |
|---|---|
| Module | C2 (Core Mathematics 2) |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Binomial Theorem (positive integer n) |
| Type | Express result in specific form |
| Difficulty | Moderate -0.8 Part (a) is a straightforward binomial expansion with small integer power requiring only Pascal's triangle or the binomial theorem. Part (b) involves substitution and simplification with surds, but follows directly from part (a) with routine algebraic manipulation. The question tests standard C2 techniques without requiring problem-solving insight or complex multi-step reasoning. |
| Spec | 1.02b Surds: manipulation and rationalising denominators1.04a Binomial expansion: (a+b)^n for positive integer n |
| Answer | Marks | Guidance |
|---|---|---|
| (a) \(= 1 + 4x + 6x^2 + 4x^3 + x^4\) | M1 A1 | |
| (b)(i) \(= 1 + 4(\sqrt{2}) + 6(\sqrt{2})^2 + 4(\sqrt{2})^3 + (\sqrt{2})^4\) | M1 | |
| \(= 1 + 4\sqrt{2} + 6(2) + 4(2\sqrt{2}) + 4\) | M1 | |
| \(= 17 + 12\sqrt{2}\) | A1 | |
| (b)(ii) \((1-\sqrt{2})^4 = 17 - 12\sqrt{2}\) | B1 | |
| \((1-\sqrt{2})^8 = [(1-\sqrt{2})^2]^2 = (17 - 12\sqrt{2})^2\) | M1 | |
| \(= 289 - 408\sqrt{2} + 288\) | M1 | |
| \(= 577 - 408\sqrt{2}\) | A1 | (9 marks) |
**(a)** $= 1 + 4x + 6x^2 + 4x^3 + x^4$ | M1 A1 |
**(b)(i)** $= 1 + 4(\sqrt{2}) + 6(\sqrt{2})^2 + 4(\sqrt{2})^3 + (\sqrt{2})^4$ | M1 |
$= 1 + 4\sqrt{2} + 6(2) + 4(2\sqrt{2}) + 4$ | M1 |
$= 17 + 12\sqrt{2}$ | A1 |
**(b)(ii)** $(1-\sqrt{2})^4 = 17 - 12\sqrt{2}$ | B1 |
$(1-\sqrt{2})^8 = [(1-\sqrt{2})^2]^2 = (17 - 12\sqrt{2})^2$ | M1 |
$= 289 - 408\sqrt{2} + 288$ | M1 |
$= 577 - 408\sqrt{2}$ | A1 | (9 marks)\begin{enumerate}[label=(\alph*)]
\item Expand $(1 + x)^4$ in ascending powers of $x$. [2]
\item Using your expansion, express each of the following in the form $a + b\sqrt{2}$, where $a$ and $b$ are integers.
\begin{enumerate}[label=(\roman*)]
\item $(1 + \sqrt{2})^4$
\item $(1 - \sqrt{2})^8$ [7]
\end{enumerate}
\end{enumerate}
\hfill \mbox{\textit{Edexcel C2 Q4 [9]}}