AS June 2019 Paper 1 Q11
11
(a) Curve \(C\) has equation\[y = \frac{x^2 + px - q}{x^2 - r}\]
where \(p\), \(q\) and \(r\) are positive constants.
Write down the equations of its asymptotes. [2 marks]
(b) Find the set of possible \(y\)-coordinates for the graph of\[y = \frac{x^2 + x - 6}{x^2 - 1}, \quad x \neq \pm 1\]
giving your answer in exact form.
No credit will be given for solutions based on differentiation. [6 marks]
| Scheme | Marks | AO |
|---|---|---|
| Gives \(x = \sqrt{r}\) or \(x = -\sqrt{r}\) or \(y = 1\) as an asymptote. Condone other incorrect asymptotes. | B1 | 1.1b |
| Gives \(x = \pm\sqrt{r}\) and \(y = 1\) as asymptotes, with no incorrect asymptotes given. | B1 | 1.1b |
Typical solution
\[x^2 - r = 0\]\[x^2 = r\]\[x = \pm\sqrt{r}\]\[y = 1\]| Scheme | Marks | AO |
|---|---|---|
| Rearranges \(k = \dfrac{x^2 + x - 6}{x^2 - 1}\) into a non-fractional form. Accept any sensible alternative for \(k\), e.g. \(y\) or f | M1 | 3.1a |
| Rearranges their equation into a correct three-term quadratic equation in \(x\) Condone missing \(= 0\) Possibly implied by a correct discriminant. | A1 | 1.1b |
| Correctly substitutes their coefficients into \(b^2 - 4ac\) to obtain an expression in \(k\) only. Accept any sensible alternative for \(k\), e.g. \(y\) or f | M1 | 3.1a |
| Obtains a correct quadratic equation/inequality in \(k\) – may be unsimplified. Or obtains the correct critical values of \(k\). Accept any sensible alternative for \(k\), e.g. \(y\) or f | A1 | 1.1b |
| Obtains the correct critical values. Accept non-exact values to at least 3 sig figs, e.g. 1.05 and 5.95 | A1 | 1.1b |
| Gives a correct range in terms of \(y\) using exact values. Condone ‘and’. Follow through their critical values if M2 scored and quadratic inequality seen. Do not accept an alternative for \(y\) Accept any equivalent expressions for \(\frac{7 - 2\sqrt{6}}{2}\) and \(\frac{7 + 2\sqrt{6}}{2}\) NMS scores 0/6 | A1F | 2.2a |
| (8 marks) |
Typical solution
Let \(k = \dfrac{x^2 + x - 6}{x^2 - 1}\)
\[k(x^2 - 1) = x^2 + x - 6\]\[(k - 1)x^2 - x + 6 - k = 0\]\[1 - 4(k - 1)(6 - k) \geqslant 0\]\[1 - 4(6k - k^2 - 6 + k) \geqslant 0\]\[4k^2 - 28k + 25 \geqslant 0\]\[y \leqslant \frac{7 - 2\sqrt{6}}{2}, \quad y \geqslant \frac{7 + 2\sqrt{6}}{2}\]