| Exam Board | SPS |
|---|---|
| Module | SPS SM (SPS SM) |
| Year | 2020 |
| Session | October |
| Marks | 11 |
| Topic | Curve Sketching |
| Type | Rational curve intersections |
| Difficulty | Moderate -0.3 This is a straightforward multi-part question requiring standard techniques: sketching familiar curves (reciprocal squared and quadratic), solving x^4 - 2x^2 - 3 = 0 by substitution, and interpreting intersection points for an inequality. While it requires multiple steps and careful algebra, all techniques are routine A-level material with no novel insight needed, making it slightly easier than average. |
| Spec | 1.02c Simultaneous equations: two variables by elimination and substitution1.02n Sketch curves: simple equations including polynomials1.02o Sketch reciprocal curves: y=a/x and y=a/x^2 |
\begin{enumerate}[label=(\roman*)]
\item Sketch the curves $y = \frac{3}{x^2}$ and $y = x^2 - 2$ on the axes provided below.
\includegraphics{figure_1} [3]
\item In this question you must show detailed reasoning.
Find the exact coordinates of the points of interception of the curves
$y = \frac{3}{x^2}$ and $y = x^2 - 2$. [6]
\item Hence, solve the inequality $\frac{3}{x^2} \leq x^2 - 2$, giving your answer in interval notation. [2]
\end{enumerate}
\hfill \mbox{\textit{SPS SPS SM 2020 Q7 [11]}}