| Exam Board | Pre-U |
|---|---|
| Module | Pre-U 9794/2 (Pre-U Mathematics Paper 2) |
| Year | 2012 |
| Session | June |
| Marks | 5 |
| Topic | Completing the square and sketching |
| Type | Complete square then solve equation |
| Difficulty | Easy -1.3 This is a straightforward two-part question testing basic algebraic skills: (i) completing the square or using the quadratic formula with simple manipulation to express in surd form, and (ii) routine expansion of brackets with surds. Both parts are standard textbook exercises requiring only direct application of techniques with no problem-solving or insight needed. The surd manipulation is elementary compared to typical A-level questions. |
| Spec | 1.02b Surds: manipulation and rationalising denominators1.02f Solve quadratic equations: including in a function of unknown |
| Answer | Marks | Guidance |
|---|---|---|
| (i) \(x = \frac{8 \pm \sqrt{64-16}}{2} = 4 \pm 2\sqrt{3}\) | M1, A1 [2] | up to 1 error |
| (ii) \((6 + 2\sqrt{3})(2 - \sqrt{3}) = 12 - 6\sqrt{3} + 4\sqrt{3} - 2\sqrt{3}\sqrt{3}\) | M1 | Multiply out |
| \(= 12 - 6\sqrt{3} + 4\sqrt{3} - 6\) | B1 | \(\sqrt{3}\sqrt{3} = 3\) |
| \(= 6 - 2\sqrt{3}\) | A1 [3] | [5] |
**(i)** $x = \frac{8 \pm \sqrt{64-16}}{2} = 4 \pm 2\sqrt{3}$ | M1, A1 [2] | up to 1 error
**(ii)** $(6 + 2\sqrt{3})(2 - \sqrt{3}) = 12 - 6\sqrt{3} + 4\sqrt{3} - 2\sqrt{3}\sqrt{3}$ | M1 | Multiply out
$= 12 - 6\sqrt{3} + 4\sqrt{3} - 6$ | B1 | $\sqrt{3}\sqrt{3} = 3$
$= 6 - 2\sqrt{3}$ | A1 [3] | [5]\begin{enumerate}[label=(\roman*)]
\item Solve the equation $x^2 - 8x + 4 = 0$, giving your answer in the form $p \pm q\sqrt{3}$, where $p$ and $q$ are integers. [2]
\item Expand and simplify $(6 + 2\sqrt{3})(2 - \sqrt{3})$. [3]
\end{enumerate}
\hfill \mbox{\textit{Pre-U Pre-U 9794/2 2012 Q1 [5]}}