Higher November 2024 Paper 1 Q23
23 The curve C has equation \(y = x^2 - 8x - 9\)
The straight line L has equation \(y = k\) where \(k\) is an integer.
C and L intersect at the points \(A\) and \(B\)
The coordinates of point \(A\) are \((p, k)\)
The coordinates of point \(B\) are \((q, k)\)
Given that \(p - q = 14\)
find the value of \(k\)
Show clear algebraic working.
(5)
| Scheme | Marks |
|---|---|
| \(\left(\dfrac{\mathrm{d}y}{\mathrm{d}x} =\right) 2x - 8\) or \((x - 4)^2\ldots\ldots\) or \((x =)\;\dfrac{-1 + 9}{2}\) | M1 |
| \(x = 4\) or \((4, -25)\) or \((4, \ldots)\) or \(\dfrac{p + q}{2} = 4\) oe | M1 |
| \(q = -3\) or \(p = 11\) | A1 |
| \((k =)\;(-3)^2 - 8(-3) - 9\) or \((k =)\;(11)^2 - 8(11) - 9\) | M1 |
| Working required Answer: 24 | A1 |
| (5) | |
| (5 marks) |
Notes
M1: for using differentiation or completing the square or by symmetry
A1: dep on M2
| Scheme | Marks |
|---|---|
| \(p^2 - 8p - 9 = k\) and \(q^2 - 8q - 9 = k\) or \((p - 9)(p + 1) = k\) and \((q - 9)(q + 1) = k\) | M1 |
| \((q + 14)^2 - 8(q + 14) - 9 = q^2 - 8q - 9\) or \((p - 14)^2 - 8(p - 14) - 9 = p^2 - 8p - 9\) or \((q + 14 - 9)(q + 14 + 1) = (q - 9)(q + 1)\) or \((p - 9)(p + 1) = (p - 14 - 9)(p - 14 + 1)\) oe | M1 |
| \(q = -3\) or \(p = 11\) | A1 |
| \((k =)\;(-3)^2 - 8(-3) - 9\) or \((k =)\;(11)^2 - 8(11) - 9\) | M1 |
| Working required Answer: 24 | A1 |
Notes
M1: for a correct equation in one variable
A1: dep on M2
| Scheme | Marks |
|---|---|
| \((q + 14)^2 - 8(q + 14) - 9\;(= k)\) or \((p - 14)^2 - 8(p - 14) - 9\;(= k)\) | M1 |
| \((q + 14)^2 - 8(q + 14) - 9 = q^2 - 8q - 9\) or \((p - 14)^2 - 8(p - 14) - 9 = p^2 - 8p - 9\) | M1 |
| \(q = -3\) or \(p = 11\) | A1 |
| \((k =)\;(-3)^2 - 8(-3) - 9\) or \((k =)\;(11)^2 - 8(11) - 9\) | M1 |
| Working required Answer: 24 | A1 |
Notes
M1: for a correct equation in one variable
A1: dep on M2
| Scheme | Marks |
|---|---|
| \(y = (x - 4)^2 - 16 - 9\) oe or \(k = (x - 4)^2 - 25\) oe or \(x^2 - 8x - 9 - k = 0\) | M1 |
\((x =)\;4 \pm \sqrt{k + 25}\) or \((x =)\;\dfrac{--8 \pm \sqrt{(-8)^2 - (4 \times 1 \times (-9 - k))}}{2 \times 1}\) oe or \((x =)\;\dfrac{8 \pm \sqrt{100 + 4k}}{2}\) oe | M1 |
\(4 + \sqrt{k + 25} - \left(4 - \sqrt{k + 25}\right) = 14\) oe or \(\dfrac{8 + \sqrt{100 + 4k}}{2} - \left(\dfrac{8 - \sqrt{100 + 4k}}{2}\right) = 14\) oe | A1 |
| \(25 + k = \left(\dfrac{14}{2}\right)^2\) oe | M1 |
| Working required Answer: 24 | A1 |
Notes
A1: for a correct equation in one variable
A1: dep on M2