AS June 2023 Paper 1 Q10
10 The curve \(C\) has equation
\[y = \frac{3x^2 + mx + p}{x^2 + px + m}\]where \(m\) and \(p\) are integers.
The vertical asymptotes of \(C\) are \(x = -4\) and \(x = -1\)
The curve \(C\) is shown in the diagram below.
(a) Write down the equation of the horizontal asymptote of \(C\) [1 mark]
(b) Find the value of \(m\) and the value of \(p\) [2 marks]
(c) Hence, or otherwise, write down the coordinates of the \(y\)-intercept of \(C\) [1 mark]
(d) Without using calculus, show that the line \(y = -1\) does not intersect \(C\) [5 marks]
| Scheme | Marks | AO |
|---|---|---|
| States \(y = 3\) | B1 | 1.1b |
| (1) |
Typical solution
\[y = 3\]| Scheme | Marks | AO |
|---|---|---|
| Identifies the correct factors of the denominator PI | M1 | 1.1a |
| Obtains the correct values. | A1 | 1.1b |
| (2) |
Typical solution
The denominator is \((x + 4)(x + 1)\)
\[= x^2 + 5x + 4\]\[\therefore \ m = 4 \text{ and } p = 5\]| Scheme | Marks | AO |
|---|---|---|
| Obtains the correct \(y\)-coordinate of the intercept. Follow through their \(\dfrac{p}{m}\) | B1F | 1.1b |
| (1) |
Typical solution
When \(x = 0\), then \(y = \dfrac{p}{m} = \dfrac{5}{4}\)
\(\therefore\) \(y\)-intercept is \(\left(0, \dfrac{5}{4}\right)\)
| Scheme | Marks | AO |
|---|---|---|
| Forms an equation to find the intersection point(s) if they exist. Could equate to a letter, eg \(k\) instead of \(-1\) | M1 | 1.1a |
| Rearranges into a three-term quadratic equation. Allow one arithmetic error. Could be in terms of \(k\) | M1 | 1.1a |
| Obtains a correct quadratic equation. Could be in terms of \(k\) \((k - 3)x^2 + (5k - 4)x + 4k - 5 = 0\) | A1 | 1.1b |
| Uses a correct method to deduce that their quadratic equation has no real roots. or Considers the sign of the discriminant in terms of \(k\) \(\Delta = 9k^2 + 28k - 44\) | M1 | 1.1a |
| Completes a reasoned argument to conclude that the line \(y = -1\) does not intersect \(C\) | R1 | 2.1 |
| (5) | ||
| (9 marks) |
Typical solution
\(y = -1\) intersects \(C\) when
\[\frac{3x^2 + 4x + 5}{x^2 + 5x + 4} = -1\]\[\Rightarrow 3x^2 + 4x + 5 = -(x^2 + 5x + 4)\]\[\Rightarrow 4x^2 + 9x + 9 = 0\]\[b^2 - 4ac = 9^2 - 4 \times 4 \times 9 = -63 \lt 0\]\(\therefore\) there are no real roots
\(\therefore\) \(y = -1\) does not intersect \(C\)