| Exam Board | Edexcel |
|---|---|
| Module | Paper 1 (Paper 1) |
| Year | 2019 |
| Session | June |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Completing the square and sketching |
| Type | Complete the square |
| Difficulty | Moderate -0.8 This is a routine multi-part question on completing the square and basic transformations. Part (a) is standard algebraic manipulation, part (b) is straightforward sketching once completed square form is found, and parts (c)(i)-(ii) require recognizing transformations and using the minimum value to find range—all textbook exercises with no novel problem-solving required. |
| Spec | 1.02e Complete the square: quadratic polynomials and turning points1.02n Sketch curves: simple equations including polynomials1.02u Functions: definition and vocabulary (domain, range, mapping)1.02w Graph transformations: simple transformations of f(x) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(2x^2 + 4x + 9 = 2(x \pm k)^2 \pm \ldots\), states \(a = 2\) | B1 | Factor of 2 outside bracket identified |
| Full method: \(2x^2 + 4x + 9 = 2(x+1)^2 \pm \ldots\), with \(a=2\) and \(b=1\) | M1 | Deals correctly with first two terms of \(2x^2 + 4x + 9\) |
| \(2x^2 + 4x + 9 = 2(x+1)^2 + 7\) | A1 | Correct completed square form |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| U-shaped curve in any position, not through \((0,0)\) | B1 | Be tolerant of pen slips; curve must not bend back on itself |
| \(y\)-intercept at \((0,9)\) | B1 | Allow intercept marked as 9 or \((0,9)\) but not \((9,0)\) |
| Minimum at \((-1,7)\) | B1ft | Follow through on \((-b,c)\) from their \(a(x+b)^2+c\); must be in quadrant 2 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Deduces translation with one correct aspect | M1 | Allow 'shift' or 'move'; allow \(\begin{pmatrix}2\\-4\end{pmatrix}\) stated with no reference to 'translate' |
| Translate \(\begin{pmatrix}2\\-4\end{pmatrix}\) | A1 | Requires both 'translate' and correct vector; SC: \(\begin{pmatrix}-2\\4\end{pmatrix}\) scores M1 A0 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(h(x) = \dfrac{21}{2(x+1)^2+7}\), maximum value \(= \dfrac{21}{7} = 3\) | M1 | Correct attempt at finding maximum value using part (a) |
| \(0 < h(x) \leqslant 3\) | A1ft | Follow through on \(\dfrac{21}{c}\) from their \(a(x+b)^2+c\); allow \((0,3]\) but not \(0 |
# Question 5:
## Part (a):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $2x^2 + 4x + 9 = 2(x \pm k)^2 \pm \ldots$, states $a = 2$ | B1 | Factor of 2 outside bracket identified |
| Full method: $2x^2 + 4x + 9 = 2(x+1)^2 \pm \ldots$, with $a=2$ and $b=1$ | M1 | Deals correctly with first two terms of $2x^2 + 4x + 9$ |
| $2x^2 + 4x + 9 = 2(x+1)^2 + 7$ | A1 | Correct completed square form |
## Part (b):
| Answer/Working | Mark | Guidance |
|---|---|---|
| U-shaped curve in any position, not through $(0,0)$ | B1 | Be tolerant of pen slips; curve must not bend back on itself |
| $y$-intercept at $(0,9)$ | B1 | Allow intercept marked as 9 or $(0,9)$ but not $(9,0)$ |
| Minimum at $(-1,7)$ | B1ft | Follow through on $(-b,c)$ from their $a(x+b)^2+c$; must be in quadrant 2 |
## Part (c)(i):
| Answer/Working | Mark | Guidance |
|---|---|---|
| Deduces translation with one correct aspect | M1 | Allow 'shift' or 'move'; allow $\begin{pmatrix}2\\-4\end{pmatrix}$ stated with no reference to 'translate' |
| Translate $\begin{pmatrix}2\\-4\end{pmatrix}$ | A1 | Requires both 'translate' and correct vector; SC: $\begin{pmatrix}-2\\4\end{pmatrix}$ scores M1 A0 |
## Part (c)(ii):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $h(x) = \dfrac{21}{2(x+1)^2+7}$, maximum value $= \dfrac{21}{7} = 3$ | M1 | Correct attempt at finding maximum value using part (a) |
| $0 < h(x) \leqslant 3$ | A1ft | Follow through on $\dfrac{21}{c}$ from their $a(x+b)^2+c$; allow $(0,3]$ but not $0<x\leqslant 3$ |
---5.
$$\mathrm { f } ( x ) = 2 x ^ { 2 } + 4 x + 9 \quad x \in \mathbb { R }$$
\begin{enumerate}[label=(\alph*)]
\item Write $\mathrm { f } ( x )$ in the form $a ( x + b ) ^ { 2 } + c$, where $a$, $b$ and $c$ are integers to be found.
\item Sketch the curve with equation $y = \mathrm { f } ( x )$ showing any points of intersection with the coordinate axes and the coordinates of any turning point.
\item \begin{enumerate}[label=(\roman*)]
\item Describe fully the transformation that maps the curve with equation $y = \mathrm { f } ( x )$ onto the curve with equation $y = \mathrm { g } ( x )$ where
$$\mathrm { g } ( x ) = 2 ( x - 2 ) ^ { 2 } + 4 x - 3 \quad x \in \mathbb { R }$$
\item Find the range of the function
$$\mathrm { h } ( x ) = \frac { 21 } { 2 x ^ { 2 } + 4 x + 9 } \quad x \in \mathbb { R }$$
\end{enumerate}\end{enumerate}
\hfill \mbox{\textit{Edexcel Paper 1 2019 Q5 [10]}}