| Exam Board | Edexcel |
|---|---|
| Module | C12 (Core Mathematics 1 & 2) |
| Year | 2016 |
| Session | January |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Discriminant and conditions for roots |
| Type | Show discriminant inequality, then solve |
| Difficulty | Standard +0.3 This is a standard discriminant problem requiring rearrangement to standard form, applying b²-4ac < 0 for no real roots, and solving a quadratic inequality. While it involves multiple steps, each is routine for C1/C2 level 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 |
| HJUV SIUI NITIAUM ION OC | VIUV SIHI NI JYHMI IONOO | VJ4V SIHI NI JIIIM ION OC |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(3kx^2 + (8k+6)x + 9k - 2 = 0\) | B1 | Multiplies by \(k\), collects terms to one side |
| Uses \(b^2 - 4ac\) with \(a=3k\), \(b=8k\pm6\), \(c=9k\pm2\) | M1 | Must use discriminant condition |
| \(-44k^2 + 120k + 36 < 0\) or \(36 < 44k^2 - 120k\) | A1 | Correct three term quadratic inequality, no errors |
| \(11k^2 - 30k - 9 > 0\) | A1* | Reached with no errors |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Attempts to solve \(11k^2 - 30k - 9 = 0\) | M1 | Factorisation, formula, or completing the square to find two values |
| Critical values \(k = 3\), \(-\frac{3}{11}\) | A1 | Accept awrt \(-0.272\) |
| \(k > 3\) or \(k < -\frac{3}{11}\) | M1 A1cao | M1: chooses outside region; A1: must be exact; \(\leq\), \(\geq\) loses final mark |
# Question 13:
## Part (a):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $3kx^2 + (8k+6)x + 9k - 2 = 0$ | B1 | Multiplies by $k$, collects terms to one side |
| Uses $b^2 - 4ac$ with $a=3k$, $b=8k\pm6$, $c=9k\pm2$ | M1 | Must use discriminant condition |
| $-44k^2 + 120k + 36 < 0$ or $36 < 44k^2 - 120k$ | A1 | Correct three term quadratic inequality, no errors |
| $11k^2 - 30k - 9 > 0$ | A1* | Reached with no errors |
## Part (b):
| Answer/Working | Mark | Guidance |
|---|---|---|
| Attempts to solve $11k^2 - 30k - 9 = 0$ | M1 | Factorisation, formula, or completing the square to find two values |
| Critical values $k = 3$, $-\frac{3}{11}$ | A1 | Accept awrt $-0.272$ |
| $k > 3$ or $k < -\frac{3}{11}$ | M1 A1cao | M1: chooses outside region; A1: must be exact; $\leq$, $\geq$ loses final mark |
---13. The equation $k \left( 3 x ^ { 2 } + 8 x + 9 \right) = 2 - 6 x$, where $k$ is a real constant, has no real roots.
\begin{enumerate}[label=(\alph*)]
\item Show that $k$ satisfies the inequality
$$11 k ^ { 2 } - 30 k - 9 > 0$$
\item Find the range of possible values for $k$.\\
\begin{center}
\begin{tabular}{|l|l|l|}
\hline
HJUV SIUI NITIAUM ION OC & VIUV SIHI NI JYHMI IONOO & VJ4V SIHI NI JIIIM ION OC \\
\hline
\end{tabular}
\end{center}
\end{enumerate}
\hfill \mbox{\textit{Edexcel C12 2016 Q13 [8]}}