Higher November 2020 Paper 2R Q24
24
(a) Write \(7 + 12x - 3x^2\) in the form \(a + b(x + c)^2\) where \(a\), \(b\) and \(c\) are integers. (4)
The curve C has equation \(y = 7 + 12x - 3x^2\)
The point \(A\) is the turning point on C.
(b) Using your answer to part (a), write down the coordinates of \(A\). (1)
| Scheme | Marks |
|---|---|
| \(-3(x^2 - 4x) + 7\) or \(-3\left(x^2 - 4x - \dfrac{7}{3}\right)\) | M1 |
\(-3\left[(x - 2)^2 \ldots\ldots\right]\) or c = –2 \(-3\left[(x - 2)^2 - 4\right] + 7\) or \(-3\left[(x - 2)^2 - 4 - \dfrac{7}{3}\right]\) | M1 |
| \(-3(x - 2)^2 + 12 + 7\) or \(-3\left[(x - 2)^2 - \dfrac{19}{3}\right]\) | M1 |
| \(19 - 3(x - 2)^2\) | A1 |
| (4) |
Notes
M1: for factorising the expression to find \(b\) or \(b = -3\) stated or shown clearly in answer.
M1: or for \(c\) shown clearly in answer.
M1: fully correct method.
A1: for \(19 - 3(x - 2)^2\) oe
Alternative mark scheme for 24 (a)
| Scheme | Marks |
|---|---|
| \(a + bx^2 + 2bcx + bc^2\) | M1 |
| \(b = -3\) or \(2bc = 12\) or \(a + bc^2 = 7\) oe | M1 |
| \(b = -3\) and \(c = -2\) | M1 |
| \(a = 19\) Answer: \(19 - 3(x - 2)^2\) | A1 |
Notes
M1: for multiplying out \(a + b(x + c)^2\) to obtain \(a + bx^2 + 2bcx + bc^2\) oe
M1: for equating coefficients
M1: for correctly finding \(b\) and \(c\)
A1: for \(19 - 3(x - 2)^2\) oe
| Scheme | Marks |
|---|---|
| (2, 19) | B1 ft |
| (1) | |
| (5 marks) |
Notes
B1 ft: dep on M1 in part (a)
answer must follow answer from (a) if given