| Exam Board | Edexcel |
|---|---|
| Module | C1 (Core Mathematics 1) |
| Year | 2012 |
| Session | January |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Indices and Surds |
| Type | Rationalize denominator simple |
| Difficulty | Easy -1.3 This is a routine C1 surds question requiring standard techniques: simplifying surds by extracting square factors, then rationalizing a denominator. Part (a) is pure recall of surd simplification, and part (b) applies the standard multiply-by-conjugate method with no problem-solving insight needed. Easier than average A-level content. |
| Spec | 1.02b Surds: manipulation and rationalising denominators |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(\sqrt{32} = 4\sqrt{2}\) or \(\sqrt{18} = 3\sqrt{2}\) | B1 | Award for either surd simplified |
| \((\sqrt{32} + \sqrt{18} =)\ 7\sqrt{2}\) | B1 | Accept \(a = 7\). Answer only scores B1B1. Common error \(\sqrt{50} = 5\sqrt{2}\) scores B0B0 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(\times \dfrac{3-\sqrt{2}}{3-\sqrt{2}}\) or \(\times \dfrac{-3+\sqrt{2}}{-3+\sqrt{2}}\) seen | M1 | Attempt to multiply by \(\dfrac{3-\sqrt{2}}{3-\sqrt{2}}\); allow poor use of brackets |
| \(\dfrac{a\sqrt{2}(3-\sqrt{2})}{[9-2]} \rightarrow \dfrac{3a\sqrt{2}-2a}{[9-2]}\) | dM1 | Using \(a\sqrt{2}\) to correctly obtain numerator of form \(p + q\sqrt{2}\), \(p,q\) non-zero integers. Allow slips e.g. \(21\sqrt{2}-28\) or \(3\sqrt{2}\times\sqrt{2}=3\). Dependent on 1st M1 |
| \(= 3\sqrt{2}, -2\) | A1, A1 | 1st A1 for \(3\sqrt{2}\) or \(b=3\); 2nd A1 for \(-2\) or \(c=-2\), from correct working |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \((b\sqrt{2}+c)(3+\sqrt{2}) = 7\sqrt{2}\) leading to \(3b+c=7\), \(3c+2b=0\) | M1 | Forming 2 simultaneous equations, \(a=7\) |
| e.g. \(3(7-3b)+2b=0\) | dM1 | Solving simultaneous equations: reducing to linear equation in one variable |
## Question 2:
### Part (a)
| Answer/Working | Mark | Guidance |
|---|---|---|
| $\sqrt{32} = 4\sqrt{2}$ or $\sqrt{18} = 3\sqrt{2}$ | B1 | Award for either surd simplified |
| $(\sqrt{32} + \sqrt{18} =)\ 7\sqrt{2}$ | B1 | Accept $a = 7$. Answer only scores B1B1. Common error $\sqrt{50} = 5\sqrt{2}$ scores B0B0 |
### Part (b)
| Answer/Working | Mark | Guidance |
|---|---|---|
| $\times \dfrac{3-\sqrt{2}}{3-\sqrt{2}}$ or $\times \dfrac{-3+\sqrt{2}}{-3+\sqrt{2}}$ seen | M1 | Attempt to multiply by $\dfrac{3-\sqrt{2}}{3-\sqrt{2}}$; allow poor use of brackets |
| $\dfrac{a\sqrt{2}(3-\sqrt{2})}{[9-2]} \rightarrow \dfrac{3a\sqrt{2}-2a}{[9-2]}$ | dM1 | Using $a\sqrt{2}$ to correctly obtain numerator of form $p + q\sqrt{2}$, $p,q$ non-zero integers. Allow slips e.g. $21\sqrt{2}-28$ or $3\sqrt{2}\times\sqrt{2}=3$. Dependent on 1st M1 |
| $= 3\sqrt{2}, -2$ | A1, A1 | 1st A1 for $3\sqrt{2}$ or $b=3$; 2nd A1 for $-2$ or $c=-2$, from correct working |
**ALT:**
| Answer/Working | Mark | Guidance |
|---|---|---|
| $(b\sqrt{2}+c)(3+\sqrt{2}) = 7\sqrt{2}$ leading to $3b+c=7$, $3c+2b=0$ | M1 | Forming 2 simultaneous equations, $a=7$ |
| e.g. $3(7-3b)+2b=0$ | dM1 | Solving simultaneous equations: reducing to linear equation in one variable |
---\begin{enumerate}[label=(\alph*)]
\item Simplify
$$\sqrt { } 32 + \sqrt { } 18$$
giving your answer in the form $a \sqrt { } 2$, where $a$ is an integer.
\item Simplify
$$\frac { \sqrt { } 32 + \sqrt { } 18 } { 3 + \sqrt { } 2 }$$
giving your answer in the form $b \sqrt { } 2 + c$, where $b$ and $c$ are integers.
\end{enumerate}
\hfill \mbox{\textit{Edexcel C1 2012 Q2 [6]}}