| Exam Board | Edexcel |
|---|---|
| Module | C12 (Core Mathematics 1 & 2) |
| Year | 2019 |
| Session | June |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Discriminant and conditions for roots |
| Type | Show discriminant inequality, then solve |
| Difficulty | Moderate -0.3 This is a standard discriminant question requiring students to set the equations equal, form a quadratic, and apply b²-4ac < 0 for no intersection. Part (b) involves routine factorization and solving a quadratic inequality. While it requires multiple steps, the technique is a core textbook exercise with no novel insight needed, making it slightly easier than average. |
| Spec | 1.02d Quadratic functions: graphs and discriminant conditions1.02g Inequalities: linear and quadratic in single variable |
| Answer | Marks | Guidance |
|---|---|---|
| Working | Mark | Guidance |
| \(2x^2 + x + 6 = mx - 2 \Rightarrow 2x^2 + x - mx + 8 = 0\) | M1 | Rearrange to \(=0\); condone sign slips |
| \(b^2 - 4ac < 0 \Rightarrow (1-m)^2 - 4(2)(8) < 0\) | M1 | Apply discriminant with \(a=\pm2, b=\pm1\pm m, c=\pm8\) or \(\pm4\) |
| \(m^2 - 2m - 63 < 0\) | A1* | Correct inequality; must appear before final line |
| Answer | Marks | Guidance |
|---|---|---|
| Working | Mark | Guidance |
| \(m^2 - 2m - 63 = 0 \Rightarrow (m-9)(m+7) = 0 \Rightarrow m = ...\) | M1 | Solve quadratic to find critical values |
| \(m = -7, 9\) | A1 | Both values only |
| Attempt at inside region | M1 | Must proceed to \(... < m < ...\); allow \(\leq\) for this mark |
| \(-7 < m < 9\) | A1 | Must be in terms of \(m\); do NOT accept \(m > -7\) OR \(m < 9\) |
## Question 11:
### Part (a):
| Working | Mark | Guidance |
|---------|------|----------|
| $2x^2 + x + 6 = mx - 2 \Rightarrow 2x^2 + x - mx + 8 = 0$ | M1 | Rearrange to $=0$; condone sign slips |
| $b^2 - 4ac < 0 \Rightarrow (1-m)^2 - 4(2)(8) < 0$ | M1 | Apply discriminant with $a=\pm2, b=\pm1\pm m, c=\pm8$ or $\pm4$ |
| $m^2 - 2m - 63 < 0$ | A1* | Correct inequality; must appear before final line |
### Part (b):
| Working | Mark | Guidance |
|---------|------|----------|
| $m^2 - 2m - 63 = 0 \Rightarrow (m-9)(m+7) = 0 \Rightarrow m = ...$ | M1 | Solve quadratic to find critical values |
| $m = -7, 9$ | A1 | Both values only |
| Attempt at inside region | M1 | Must proceed to $... < m < ...$; allow $\leq$ for this mark |
| $-7 < m < 9$ | A1 | Must be in terms of $m$; do NOT accept $m > -7$ OR $m < 9$ |
---11. The straight line $l$ has equation $y = m x - 2$, where $m$ is a constant.
The curve $C$ has equation $y = 2 x ^ { 2 } + x + 6$
The line $l$ does not cross or touch the curve $C$.
\begin{enumerate}[label=(\alph*)]
\item Show that $m$ satisfies the inequality
$$m ^ { 2 } - 2 m - 63 < 0$$
\item Hence find the range of possible values of $m$.
\end{enumerate}
\hfill \mbox{\textit{Edexcel C12 2019 Q11 [7]}}