| Exam Board | Edexcel |
|---|---|
| Module | C1 (Core Mathematics 1) |
| Year | 2017 |
| Session | June |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Completing the square and sketching |
| Type | Complete the square |
| Difficulty | Easy -1.2 This is a routine C1 completing-the-square question with standard follow-up parts. Part (a) is pure algebraic manipulation, part (b) requires plotting two points on a parabola, and part (c) involves basic distance formula with surds. All techniques are textbook exercises with no problem-solving or novel insight required. |
| Spec | 1.02e Complete the square: quadratic polynomials and turning points1.02n Sketch curves: simple equations including polynomials1.10f Distance between points: using position vectors |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(f(x) = (x-4)^2 + 3\) | M1A1 | M1: \(f(x) = (x \pm 4)^2 \pm \alpha\), \(\alpha \neq 0\) (where \(\alpha\) is a single number or numerical expression \(\neq 0\)). A1: Allow \((x + {^-}4)^2 + 3\) and ignore any spurious "\(= 0\)". Allow \(a = -4\), \(b = 3\) to score both marks |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| U-shaped curve | B1 | U shape anywhere even with no axes. Do not allow a "V" shape i.e. with an obvious vertex |
| \(P(0, 19)\) | B1 | Allow \((0, 19)\) or just \(19\) marked in correct place as long as curve (or straight line) passes through or touches here. Allow \((19, 0)\) as long as marked in correct place. Correct coordinates may be seen in body of script. There must be a sketch to score this mark |
| \(Q(4, 3)\) | B1 | Correct coordinates that can be scored without a sketch but if sketch is drawn it must have a minimum in first quadrant and no other turning points. Allow if \(4\) is clearly marked on \(x\)-axis below minimum and \(3\) is clearly marked on \(y\)-axis corresponding to the minimum |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(PQ^2 = (0-4)^2 + (19-3)^2\) | M1 | Correct use of Pythagoras' Theorem on 2 points of the form \((0, p)\) and \((q, r)\) where \(q \neq 0\) and \(p \neq r\) with \(p\), \(q\), \(r\) numeric |
| \(PQ = \sqrt{4^2 + 16^2}\) | A1 | Correct un-simplified numerical expression for \(PQ\) including square root. Must come from a correct \(P\) and \(Q\). Allow \(PQ = \sqrt{(0-4)^2 + (19-3)^2}\) or \(\pm\sqrt{(0-4)^2+(19-3)^2}\) |
| \(PQ = 4\sqrt{17}\) | A1 | cao and cso. Must come from a correct \(P\) and \(Q\) |
## Question 5:
### Part (a):
| Answer | Mark | Guidance |
|--------|------|----------|
| $f(x) = (x-4)^2 + 3$ | M1A1 | M1: $f(x) = (x \pm 4)^2 \pm \alpha$, $\alpha \neq 0$ (where $\alpha$ is a single number or numerical expression $\neq 0$). A1: Allow $(x + {^-}4)^2 + 3$ and ignore any spurious "$= 0$". **Allow $a = -4$, $b = 3$ to score both marks** |
### Part (b):
| Answer | Mark | Guidance |
|--------|------|----------|
| U-shaped curve | B1 | U shape anywhere even with no axes. Do not allow a "V" shape i.e. with an obvious vertex |
| $P(0, 19)$ | B1 | Allow $(0, 19)$ or just $19$ marked in correct place as long as curve (or straight line) passes through or touches here. Allow $(19, 0)$ as long as marked in correct place. Correct coordinates may be seen in body of script. **There must be a sketch to score this mark** |
| $Q(4, 3)$ | B1 | Correct coordinates that can be scored without a sketch but if sketch is drawn it must have a minimum in first quadrant and no other turning points. Allow if $4$ is clearly marked on $x$-axis below minimum and $3$ is clearly marked on $y$-axis corresponding to the minimum |
### Part (c):
| Answer | Mark | Guidance |
|--------|------|----------|
| $PQ^2 = (0-4)^2 + (19-3)^2$ | M1 | Correct use of Pythagoras' Theorem on 2 points of the form $(0, p)$ and $(q, r)$ where $q \neq 0$ and $p \neq r$ with $p$, $q$, $r$ numeric |
| $PQ = \sqrt{4^2 + 16^2}$ | A1 | Correct un-simplified numerical expression for $PQ$ including square root. **Must come from a correct $P$ and $Q$**. Allow $PQ = \sqrt{(0-4)^2 + (19-3)^2}$ or $\pm\sqrt{(0-4)^2+(19-3)^2}$ |
| $PQ = 4\sqrt{17}$ | A1 | cao and cso. **Must come from a correct $P$ and $Q$** |
---5.
$$f ( x ) = x ^ { 2 } - 8 x + 19$$
\begin{enumerate}[label=(\alph*)]
\item Express $\mathrm { f } ( x )$ in the form $( x + a ) ^ { 2 } + b$, where $a$ and $b$ are constants.
The curve $C$ with equation $y = \mathrm { f } ( x )$ crosses the $y$-axis at the point $P$ and has a minimum point at the point $Q$.
\item Sketch the graph of $C$ showing the coordinates of point $P$ and the coordinates of point $Q$.
\item Find the distance $P Q$, writing your answer as a simplified surd.
\end{enumerate}
\hfill \mbox{\textit{Edexcel C1 2017 Q5 [8]}}