| Exam Board | SPS |
|---|---|
| Module | SPS SM (SPS SM) |
| Year | 2023 |
| Session | October |
| Marks | 7 |
| Topic | Polynomial Division & Manipulation |
| Type | Division then Solve Polynomial Equation |
| Difficulty | Standard +0.3 This is a straightforward intersection problem requiring substitution to verify x=1, then solving a cubic equation that factors nicely. The algebra is routine (equating curves gives a cubic that factors as a quadratic times linear factor), and finding k<0 involves standard factorization techniques. Slightly easier than average due to the verification step reducing the problem complexity and the clean factorization. |
| Spec | 1.02j Manipulate polynomials: expanding, factorising, division, factor theorem |
In this question you must show detailed reasoning.
The curve $C_1$ has equation $y = 8 - 10x + 6x^2 - x^3$
The curve $C_2$ has equation $y = x^2 - 12x + 14$
\begin{enumerate}[label=(\alph*)]
\item Verify that when $x = 1$ the curves $C_1$ and $C_2$ intersect. [2]
\end{enumerate}
The curves also intersect when $x = k$.
Given that $k < 0$
\begin{enumerate}[label=(\alph*)]
\setcounter{enumi}{1}
\item use algebra to find the exact value of $k$. [5]
\end{enumerate}
\hfill \mbox{\textit{SPS SPS SM 2023 Q8 [7]}}