| Exam Board | Edexcel |
|---|---|
| Module | C1 (Core Mathematics 1) |
| Year | 2010 |
| Session | January |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Indices and Surds |
| Type | Rationalize denominator simple |
| Difficulty | Easy -1.2 This is a straightforward C1 question testing basic surd manipulation: part (a) is routine expansion using FOIL, and part (b) is standard rationalizing the denominator by multiplying by the conjugate. Both are textbook exercises requiring only procedural recall with minimal problem-solving. |
| Spec | 1.02b Surds: manipulation and rationalising denominators |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \((7+\sqrt{5})(3-\sqrt{5}) = 21-5+3\sqrt{5}-7\sqrt{5}\) | M1 | Expand to get 3 or 4 terms, at least 2 correct (even if unsimplified) |
| \(= 16, -4\sqrt{5}\) | A1, A1 | 1st A1 for 16, 2nd A1 for \(-4\sqrt{5}\); i.s.w. if necessary e.g. \(16-4\sqrt{5} \to 4-\sqrt{5}\) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(\frac{7+\sqrt{5}}{3+\sqrt{5}} \times \frac{3-\sqrt{5}}{3-\sqrt{5}}\) | M1 | This is sufficient for the M mark |
| Correct denominator without surds: \(9-5\) or \(4\) | A1 | |
| \(4-\sqrt{5}\) or \(4-1\sqrt{5}\) | A1 | Answer only: \(4-\sqrt{5}\) scores full marks; ignore subsequent working e.g. \(4-\sqrt{5}\) so \(a=4\), \(b=1\) |
| Answer | Marks | Guidance |
|---|---|---|
| Alternative for (b): \((a+b\sqrt{5})(3+\sqrt{5})=7+\sqrt{5}\), form simultaneous equations | M1 | Correct equations: \(3a+5b=7\) and \(3b+a=1\) |
## Question 2:
### Part (a):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $(7+\sqrt{5})(3-\sqrt{5}) = 21-5+3\sqrt{5}-7\sqrt{5}$ | M1 | Expand to get 3 or 4 terms, at least 2 correct (even if unsimplified) |
| $= 16, -4\sqrt{5}$ | A1, A1 | 1st A1 for 16, 2nd A1 for $-4\sqrt{5}$; i.s.w. if necessary e.g. $16-4\sqrt{5} \to 4-\sqrt{5}$ |
**Subtotal: (3)**
### Part (b):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $\frac{7+\sqrt{5}}{3+\sqrt{5}} \times \frac{3-\sqrt{5}}{3-\sqrt{5}}$ | M1 | This is sufficient for the M mark |
| Correct denominator without surds: $9-5$ or $4$ | A1 | |
| $4-\sqrt{5}$ or $4-1\sqrt{5}$ | A1 | Answer only: $4-\sqrt{5}$ scores full marks; ignore subsequent working e.g. $4-\sqrt{5}$ so $a=4$, $b=1$ |
**Subtotal: (3)**
**Total: [6]**
**Alternative for (b):** $(a+b\sqrt{5})(3+\sqrt{5})=7+\sqrt{5}$, form simultaneous equations | M1 | Correct equations: $3a+5b=7$ and $3b+a=1$ | A1 | $a=4$ and $b=-1$ | A1
---\begin{enumerate}[label=(\alph*)]
\item Expand and simplify $( 7 + \sqrt { 5 } ) ( 3 - \sqrt { 5 } )$.
\item Express $\frac { 7 + \sqrt { 5 } } { 3 + \sqrt { 5 } }$ in the form $a + b \sqrt { 5 }$, where $a$ and $b$ are integers.
\end{enumerate}
\hfill \mbox{\textit{Edexcel C1 2010 Q2 [6]}}