OCR A Level: Further Pure 2 - Rational Graphs [June 2012 Q8]

The curve $C_1$ has equation $y=\dfrac{p(x)}{q(x)}$ , where $p(x)$ and $q(x)$ are polynomials of degree 2 and 1 respectively. The asymptotes of the curve are $x=-2$ and $y=\dfrac{1}{2}x+1$ , and the curve passes through the point $\left (-1, \dfrac{17}{2} \right )$ .

$(\text{i})$ Express the equation of $C_1$ in the form $y=\dfrac{p(x)}{q(x)}$ .

$(\text{ii})$ For the curve $C_1$ , find the range of values that $y$ can take.

Another curve, $C_2$ , has equation $y^2= \dfrac{p(x)}{q(x)}$ , where $p(x)$ and $q(x)$ are the polynoimals found in part $(\text{i})$ .

$(\text{iii})$ It is given that $C_2$ intersects the line $y=\dfrac{1}{2}x+1$ exactly once. Find the coordinates of the point of intersection.

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If the coordinates of the point of intersection are $(m,n)$ , input $m+n$ as your answer.

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There are 4 marks available for part (i), 4 marks for part (ii) and 4 marks for part (iii).

In total, this question is worth 16.7% of all available marks in the paper.

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This is part of the set OCR A Level Problems .