4MA0 Higher January 2015 Paper 4R Q19
19
The diagram shows the straight line with equation \(y = kx\) intersecting the curve with equation \(y = \dfrac{N}{x}\) at the point \(A(p, q)\).
(a) Find \(p\) and find \(q\).
Give each answer in its simplest form, in terms of \(k\) and \(N\). (3)
Give each answer in its simplest form, in terms of \(k\) and \(N\). (3)
Given that \(p = 2q\)
(b) find the value of \(k\). (2)
| Scheme | Marks |
|---|---|
| \(kx = \dfrac{N}{x}\) oe or \(kp = \dfrac{N}{p}\) oe | M1 |
| \(p = \sqrt{\dfrac{N}{k}}\) oe | A1 |
| \(q = \sqrt{Nk}\) oe | A1 |
| (3) |
Notes
A1: \(p = \sqrt{\dfrac{N}{k}}\) or \(p = \pm\sqrt{\dfrac{N}{k}}\)
A1: \(q = \sqrt{Nk}\) or \(q = \pm\sqrt{Nk}\) oe eg. \(q = k\sqrt{\dfrac{N}{k}}\)
| Scheme | Marks |
|---|---|
\(\sqrt{\dfrac{N}{k}} = 2\sqrt{Nk}\) or \(q = k \times 2q\) or \(\dfrac{1}{2}p = kp\) | M1 |
| \(\dfrac{1}{2}\) | A1 |
| (2) | |
| (5 marks) |
Notes
M1: ft from (a)
A1: \(k = \dfrac{1}{2}\), 0.5