| Exam Board | Edexcel |
|---|---|
| Module | C12 (Core Mathematics 1 & 2) |
| Year | 2018 |
| Session | October |
| Marks | 8 |
| 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 problem requiring rearrangement to standard form, applying b²-4ac > 0, and solving a quadratic inequality. While it involves multiple steps and algebraic manipulation, it follows a well-practiced procedure with no novel insight required, 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 |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(7x^2 + 2kx + k^2 - k - 7(=0)\) or \(a=7, b=2k, c=k^2-k-7\) | M1 | Attempts to collect terms to one side; look for \(7x^2 + 2kx + k^2 \pm k \pm 7(=0)\). May write values of \(a\), \(b\), \(c\) where \(a=7\), \(b=2k\), \(c=k^2 \pm k \pm 7\) |
| \((2k)^2 - 4 \times 7 \times (k^2-k-7) > 0\) | A1 | Correct quadratic inequality (not the printed answer). Allow missing brackets around \(2k\) or \(k^2-k-7\). Do not allow if incorrect rearrangement earlier |
| \(6k^2 - 7k - 49 < 0\) * | A1* | Fully correct proof with no errors, including bracketing, sign errors. Starting with \(7x^2+2kx+k^2-k-7>0\) or \(<0\) would be an error |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(6k^2 - 7k - 49 = 0 \Rightarrow k = \ldots\) | M1 | Attempt to solve the 3TQ from part (a) to obtain 2 values for \(k\). If no working shown and roots are incorrect, score M0 |
| \(k = -\dfrac{7}{3},\ \dfrac{7}{2}\) | A1 | Correct values. Allow \(k = \dfrac{7 \pm 35}{12}\). May be seen as part of inequalities |
| \(-\dfrac{7}{3} < k < \dfrac{7}{2}\) or \(\left(-\dfrac{7}{3}, \dfrac{7}{2}\right)\) | M1 | Attempt inside region for their critical values. Do not award for diagram or table alone |
| \(k > -\dfrac{7}{3}\) and \(k < \dfrac{7}{2}\) | A1 | cao. Must see "and" if regions given separately. \(k > -\dfrac{7}{3},\ k < \dfrac{7}{2}\) is A0 |
## Question 11:
### Part (a):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $7x^2 + 2kx + k^2 - k - 7(=0)$ or $a=7, b=2k, c=k^2-k-7$ | M1 | Attempts to collect terms to one side; look for $7x^2 + 2kx + k^2 \pm k \pm 7(=0)$. May write values of $a$, $b$, $c$ where $a=7$, $b=2k$, $c=k^2 \pm k \pm 7$ |
| $(2k)^2 - 4 \times 7 \times (k^2-k-7) > 0$ | A1 | Correct quadratic inequality (not the printed answer). Allow missing brackets around $2k$ or $k^2-k-7$. Do not allow if incorrect rearrangement earlier |
| $6k^2 - 7k - 49 < 0$ * | A1* | Fully correct proof with no errors, including bracketing, sign errors. Starting with $7x^2+2kx+k^2-k-7>0$ or $<0$ would be an error |
### Part (b):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $6k^2 - 7k - 49 = 0 \Rightarrow k = \ldots$ | M1 | Attempt to solve the 3TQ from part (a) to obtain 2 values for $k$. If no working shown and roots are incorrect, score M0 |
| $k = -\dfrac{7}{3},\ \dfrac{7}{2}$ | A1 | Correct values. Allow $k = \dfrac{7 \pm 35}{12}$. May be seen as part of inequalities |
| $-\dfrac{7}{3} < k < \dfrac{7}{2}$ or $\left(-\dfrac{7}{3}, \dfrac{7}{2}\right)$ | M1 | Attempt inside region for their critical values. Do not award for diagram or table alone |
| $k > -\dfrac{7}{3}$ **and** $k < \dfrac{7}{2}$ | A1 | cao. Must see "and" if regions given separately. $k > -\dfrac{7}{3},\ k < \dfrac{7}{2}$ is A0 |
---11. The equation $7 x ^ { 2 } + 2 k x + k ^ { 2 } = k + 7$, where $k$ is a constant, has two distinct real roots.
\begin{enumerate}[label=(\alph*)]
\item Show that $k$ satisfies the inequality
$$6 k ^ { 2 } - 7 k - 49 < 0$$
\item Find the range of possible values for $k$.
\end{enumerate}
\hfill \mbox{\textit{Edexcel C12 2018 Q11 [8]}}