| Exam Board | CAIE |
|---|---|
| Module | P1 (Pure Mathematics 1) |
| Year | 2022 |
| Session | June |
| Marks | 4 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Completing the square and sketching |
| Type | Complete the square |
| Difficulty | Easy -1.2 This is a straightforward completing-the-square exercise followed by a routine equation solve. Part (a) requires only the standard algorithm (halve the coefficient of x, square it, adjust constant), and part (b) is direct substitution and taking square roots. Both are textbook procedures with no problem-solving or insight required, making this easier than average. |
| Spec | 1.02e Complete the square: quadratic polynomials and turning points1.02f Solve quadratic equations: including in a function of unknown |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(x^2 - 8x + 11 = (x-4)^2 \ldots\) or \(p = -4\) | B1 | If \(p\) and \(q\)-values given after *their* completed square expression, mark the expression and ISW |
| \(\ldots -5\) or \(q = -5\) | B1 | |
| 2 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \((x-4)^2 - 5 = 1\) so \((x-4)^2 = 6\) so \(x - 4 = [\pm]\sqrt{6}\) | M1 | Using *their* \(p\) and \(q\) values or by quadratic formula |
| \(x = 4 \pm \sqrt{6}\) or \(\dfrac{8 \pm \sqrt{24}}{2}\) | A1 | Or exact equivalent. No FT; must have \(\pm\) for this mark. ISW decimals 1.55, 6.45 if exact answers seen. If M0, SC B1 possible for correct answers |
| 2 |
## Question 1:
### Part (a):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $x^2 - 8x + 11 = (x-4)^2 \ldots$ **or** $p = -4$ | B1 | If $p$ and $q$-values given after *their* completed square expression, mark the expression and ISW |
| $\ldots -5$ **or** $q = -5$ | B1 | |
| | **2** | |
### Part (b):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $(x-4)^2 - 5 = 1$ so $(x-4)^2 = 6$ so $x - 4 = [\pm]\sqrt{6}$ | M1 | Using *their* $p$ and $q$ values or by quadratic formula |
| $x = 4 \pm \sqrt{6}$ **or** $\dfrac{8 \pm \sqrt{24}}{2}$ | A1 | Or exact equivalent. No FT; must have $\pm$ for this mark. ISW decimals 1.55, 6.45 if exact answers seen. If M0, **SC B1** possible for correct answers |
| | **2** | |
---1
\begin{enumerate}[label=(\alph*)]
\item Express $x ^ { 2 } - 8 x + 11$ in the form $( x + p ) ^ { 2 } + q$ where $p$ and $q$ are constants.
\item Hence find the exact solutions of the equation $x ^ { 2 } - 8 x + 11 = 1$.
\end{enumerate}
\hfill \mbox{\textit{CAIE P1 2022 Q1 [4]}}