| Exam Board | Edexcel |
|---|---|
| Module | PURE |
| Year | 2024 |
| Session | October |
| Paper | Download PDF ↗ |
| Topic | Modulus function |
| Type | Find range of k for number of roots |
| Difficulty | Standard +0.8 This is a Further Maths Pure question requiring curve sketching of a reciprocal function, then finding conditions for intersection by forming a quadratic inequality and solving it. The algebraic manipulation to derive the discriminant condition is non-trivial, and interpreting 'at least once' as a discriminant constraint requires solid understanding. More demanding than standard A-level but routine for Further Maths. |
| Spec | 1.02c Simultaneous equations: two variables by elimination and substitution1.02g Inequalities: linear and quadratic in single variable1.02m Graphs of functions: difference between plotting and sketching1.02o Sketch reciprocal curves: y=a/x and y=a/x^2 |
\begin{enumerate}
\item In this question you must show all stages of your working.
\end{enumerate}
\section*{Solutions relying on calculator technology are not acceptable.}
(a) Sketch the curve $C$ with equation
$$y = \frac { 1 } { 2 - x } \quad x \neq 2$$
State on your sketch
\begin{itemize}
\item the equation of the vertical asymptote
\item the coordinates of the intersection of $C$ with the $y$-axis
\end{itemize}
The straight line $l$ has equation $y = k x - 4$, where $k$ is a constant.\\
Given that $l$ cuts $C$ at least once,\\
(b) (i) show that
$$k ^ { 2 } - 5 k + 4 \geqslant 0$$
(ii) find the range of possible values for $k$.
\hfill \mbox{\textit{Edexcel PURE 2024 Q6}}