Higher June 2018 Paper 2R Q24
24
(a) Express \(7 - 4x - x^2\) in the form \(p - (x + q)^2\) where \(p\) and \(q\) are constants. (2)
(b) Use your answer to part (a) to solve the equation \(7 - 4(y + 3) - (y + 3)^2 = 0\)
Give your solutions in the form \(e \pm \sqrt{f}\) where \(e\) and \(f\) are integers. (3)
Give your solutions in the form \(e \pm \sqrt{f}\) where \(e\) and \(f\) are integers. (3)
The curve \(C\) has equation \(y = 3 - 5(x + 1)^2\)
The point \(A\) is the maximum point on \(C\).
(c) Write down the coordinates of \(A\). (1)
| Scheme | Marks |
|---|---|
| M1 | |
| \(11 - (x + 2)^2\) | A1 |
| (2) |
Notes
M1: For \(11 - (x + q)^2\) or \(p - (x + 2)^2\)
A1: fully correct, accept \(p = 11\), \(q = 2\)
| Scheme | Marks |
|---|---|
| \((y + 3 + 2)^2 = 11\) or \(11 - (y + 3 + 2)^2\) | M1 |
| \(y + 3 + 2 = \pm\sqrt{11}\) | M1 |
| \(-5 \pm \sqrt{11}\) | A1 |
| (3) |
Notes
M1: substituting \(x = y + 3\) into their \(p - (x + q)^2\)
A1: Both answers correct, ft their answer from (a) eg \(-(3 + \text{``}{q}\text{''}) \pm \sqrt{\text{``}{p}\text{''}}\)
| Scheme | Marks |
|---|---|
| M2 | |
| \(-5 \pm \sqrt{11}\) | A1 |
Notes
M2: for \(-y^2 - 10y - 14 = 0\) or \(y^2 + 10y + 14 = 0\)
A1: cao, both values correct
| Scheme | Marks |
|---|---|
| (−1, 3) | B1 |
| (1) | |
| (6 marks) |
Notes
B1: cao