Moderate -0.8 This is a straightforward substitution problem requiring students to substitute y=3x into the circle equation, solve the resulting quadratic, then find corresponding y-values. It's a standard C1 exercise with clear method and no conceptual challenges, making it easier than average but not trivial due to the algebraic manipulation and surd simplification required.
for subst for \(x\) or \(y\) attempted or \(x^2 = 2.5\) o.e.; condone one error from start [allow \(10x^2 - 25 = 0 +\) correct substn in correct formula]
\(10x^2 = 25\)
M1 mark
\(x= \pm\frac{\sqrt{10}}{2}\) or \(\pm\frac{\sqrt{5}}{2}\) or \(\pm\frac{\sqrt{10}}{5}\) o.e \(y = [\pm] 3\frac{\sqrt{5}}{2}\) o.e. eg \(y = [\pm]\sqrt{22.5}\)
A2 mark
allow \(\pm 2.5\); A1 for one value; ft \(3 \times\) their \(y\) value(s) if irrational; condone not written as coords.
5 marks
$x^2 + 9x^2 = 25$ | M1 mark | for subst for $x$ or $y$ attempted or $x^2 = 2.5$ o.e.; condone one error from start [allow $10x^2 - 25 = 0 +$ correct substn in correct formula]
$10x^2 = 25$ | M1 mark |
$x= \pm\frac{\sqrt{10}}{2}$ or $\pm\frac{\sqrt{5}}{2}$ or $\pm\frac{\sqrt{10}}{5}$ o.e $y = [\pm] 3\frac{\sqrt{5}}{2}$ o.e. eg $y = [\pm]\sqrt{22.5}$ | A2 mark | allow $\pm 2.5$; A1 for one value; ft $3 \times$ their $y$ value(s) if irrational; condone not written as coords.
| | 5 marks |
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Find the coordinates of the points of intersection of the circle $x^2 + y^2 = 25$ and the line $y = 3x$. Give your answers in surd form. [5]
\hfill \mbox{\textit{OCR MEI C1 2006 Q10 [5]}}