| Exam Board | OCR MEI |
|---|---|
| Module | C1 (Core Mathematics 1) |
| Year | 2011 |
| Session | January |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Topic | Discriminant and conditions for roots |
| Type | Prove/show always positive |
| Difficulty | Standard +0.8 Part (i) is routine substitution. Part (ii) requires solving a quartic that factors nicely. Part (iii) is the challenging element: students must form x^4 - kx^2 - 2 = 0, treat it as a quadratic in x^2, and show the discriminant is always positive (k^2 + 8 > 0). This requires recognizing the substitution strategy and understanding what 'intersect for all k' means discriminant-wise, which goes beyond standard C1 exercises. |
| Spec | 1.02n Sketch curves: simple equations including polynomials1.02q Use intersection points: of graphs to solve equations |
13
\begin{figure}[h]
\begin{center}
\includegraphics[alt={},max width=\textwidth]{754d34e4-2f47-48b7-9fbb-6caa7ac21eb7-4_686_878_936_632}
\captionsetup{labelformat=empty}
\caption{Fig. 13}
\end{center}
\end{figure}
Fig. 13 shows the curve $y = x ^ { 4 } - 2$.\\
(i) Find the exact coordinates of the points of intersection of this curve with the axes.\\
(ii) Find the exact coordinates of the points of intersection of the curve $y = x ^ { 4 } - 2$ with the curve $y = x ^ { 2 }$.\\
(iii) Show that the curves $y = x ^ { 4 } - 2$ and $y = k x ^ { 2 }$ intersect for all values of $k$.
\hfill \mbox{\textit{OCR MEI C1 2011 Q13 [10]}}