| Exam Board | Edexcel |
|---|---|
| Module | C1 (Core Mathematics 1) |
| Year | 2018 |
| Session | June |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Completing the square and sketching |
| Type | Solve quadratic by substitution |
| Difficulty | Moderate -0.8 This is a straightforward C1 completing the square question with standard follow-through parts. Part (a) is routine manipulation, part (b) applies the quadratic formula to the completed square form, and part (c) requires the simple substitution y^0.5 = x. All techniques are standard textbook exercises with no problem-solving insight required, making it 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/Working | Mark | Guidance |
| \(x^2 - 10x + 23 = (x \pm 5)^2 \pm A\) | M1 | For an attempt to complete the square. Note that if their \(A = 23\) then this is M0 by the General Principles. |
| \((x-5)^2 - 2\) | A1 | Correct expression. Ignore "\(= 0\)". |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \((x \pm 5)^2 - A \Rightarrow x = \ldots\) or \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \Rightarrow x = \ldots\) | M1 | Uses their completion of the square for positive \(A\) or uses the correct quadratic formula to obtain at least one value for \(x\) |
| \(x = 5 \pm \sqrt{2}\) | A1 | Correct exact values. If using the quadratic formula must reach as far as \(\frac{10 \pm \sqrt{8}}{2}\) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \((5 \pm \sqrt{2})^2 = 27 + 10\sqrt{2}\) | M1 | Attempts to square any solution from part (b). Allow poor squaring e.g. \((5 + \sqrt{2})^2 = 25 + 2 = 27\). Do not allow for substituting e.g. \(5 + \sqrt{2}\) into \(x^2 - 10x + 23\). |
| \(= 27 + 10\sqrt{2}\) | A1 | Accept equivalent forms such as \(27 + \sqrt{200}\). If any extra answers are given, this mark should be withheld. |
# Question 3:
## Part (a)
| Answer/Working | Mark | Guidance |
|---|---|---|
| $x^2 - 10x + 23 = (x \pm 5)^2 \pm A$ | M1 | For an attempt to complete the square. Note that if their $A = 23$ then this is M0 by the General Principles. |
| $(x-5)^2 - 2$ | A1 | Correct expression. Ignore "$= 0$". |
## Part (b)
| Answer/Working | Mark | Guidance |
|---|---|---|
| $(x \pm 5)^2 - A \Rightarrow x = \ldots$ or $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \Rightarrow x = \ldots$ | M1 | Uses their completion of the square for **positive** $A$ or uses the correct quadratic formula to obtain at least one value for $x$ |
| $x = 5 \pm \sqrt{2}$ | A1 | Correct exact values. If using the quadratic formula must reach as far as $\frac{10 \pm \sqrt{8}}{2}$ |
## Part (c)
| Answer/Working | Mark | Guidance |
|---|---|---|
| $(5 \pm \sqrt{2})^2 = 27 + 10\sqrt{2}$ | M1 | Attempts to square **any** solution from part (b). Allow poor squaring e.g. $(5 + \sqrt{2})^2 = 25 + 2 = 27$. **Do not allow** for substituting e.g. $5 + \sqrt{2}$ into $x^2 - 10x + 23$. |
| $= 27 + 10\sqrt{2}$ | A1 | Accept equivalent forms such as $27 + \sqrt{200}$. If any extra answers are given, this mark should be withheld. |3.
$$f ( x ) = x ^ { 2 } - 10 x + 23$$
\begin{enumerate}[label=(\alph*)]
\item Express $\mathrm { f } ( x )$ in the form $( x + a ) ^ { 2 } + b$, where $a$ and $b$ are constants to be found.
\item Hence, or otherwise, find the exact solutions to the equation
$$x ^ { 2 } - 10 x + 23 = 0$$
\item Use your answer to part (b) to find the larger solution to the equation
$$y - 10 y ^ { 0.5 } + 23 = 0$$
Write your solution in the form $p + q \sqrt { r }$, where $p , q$ and $r$ are integers.
\end{enumerate}
\hfill \mbox{\textit{Edexcel C1 2018 Q3 [6]}}