| Exam Board | OCR |
|---|---|
| Module | PURE |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Topic | Completing the square and sketching |
| Type | Complete square then find vertex/turning point |
| Difficulty | Moderate -0.8 This is a straightforward completing the square question with standard follow-up parts. Part (a) is routine algebraic manipulation, part (b) requires reading the vertex from completed square form, and part (c) uses the fact that the normal at a turning point is vertical. All parts are direct application of standard techniques with no problem-solving required, making it easier than average. |
| Spec | 1.02e Complete the square: quadratic polynomials and turning points |
| Answer | Marks | Guidance |
|---|---|---|
| \(5[x^2-4x]+3\) | No marks until attempt to complete the square | |
| \(= 5[(x-2)^2-4]+3\), \(p=5\) | B1 (AO 1.1) | Must be of the form \(5(x\pm\alpha)^2\pm\cdots\) |
| \((x-2)^2\) | B1 (AO 1.1) | |
| \(= 5(x-2)^2-17\), \(r=-17\) | B1 (AO 1.1) |
| Answer | Marks | Guidance |
|---|---|---|
| Minimum point \((2,-17)\) | B1ft (AO 1.1) | Follow through their \(-q\); or by differentiation |
| B1ft (AO 1.1) | Follow through their \(r\) |
| Answer | Marks | Guidance |
|---|---|---|
| \(x=2\) | B1ft (AO 1.1) | Follow through their \(x\) coordinate in part (b) |
**Question 2(a):**
$5[x^2-4x]+3$ | | No marks until attempt to complete the square
$= 5[(x-2)^2-4]+3$, $p=5$ | B1 (AO 1.1) | Must be of the form $5(x\pm\alpha)^2\pm\cdots$
$(x-2)^2$ | B1 (AO 1.1) |
$= 5(x-2)^2-17$, $r=-17$ | B1 (AO 1.1) |
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**Question 2(b):**
Minimum point $(2,-17)$ | B1ft (AO 1.1) | Follow through their $-q$; or by differentiation
| B1ft (AO 1.1) | Follow through their $r$
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**Question 2(c):**
$x=2$ | B1ft (AO 1.1) | Follow through their $x$ coordinate in part (b)2
\begin{enumerate}[label=(\alph*)]
\item Express $5 x ^ { 2 } - 20 x + 3$ in the form $p ( x + q ) ^ { 2 } + r$, where $p , q$ and $r$ are integers.
\item State the coordinates of the minimum point of the curve $y = 5 x ^ { 2 } - 20 x + 3$.
\item State the equation of the normal to the curve $y = 5 x ^ { 2 } - 20 x + 3$ at its minimum point.
\end{enumerate}
\hfill \mbox{\textit{OCR PURE Q2 [6]}}