| Exam Board | SPS |
|---|---|
| Module | SPS SM (SPS SM) |
| Year | 2022 |
| Session | October |
| Marks | 8 |
| 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 the condition b²-4ac < 0 for no real roots, and solving a quadratic inequality. While it involves multiple steps and careful algebraic manipulation, it follows a well-established procedure taught in C1/C2 with no novel insight required. The 8 marks reflect the working needed rather than conceptual difficulty. |
| Spec | 1.02d Quadratic functions: graphs and discriminant conditions1.02g Inequalities: linear and quadratic in single variable |
The equation $k(3x^2 + 8x + 9) = 2 - 6x$, where $k$ is a real constant, has no real roots.
\begin{enumerate}[label=(\alph*)]
\item Show that $k$ satisfies the inequality
$$11k^2 - 30k - 9 > 0$$ [4]
\item Find the range of possible values for $k$. [4]
\end{enumerate}
\hfill \mbox{\textit{SPS SPS SM 2022 Q8 [8]}}