Higher June 2025 Paper 2 Q25
25
(a) Write \(28 + 24x - 6x^2\) in the form \(a - b(x - c)^2\) where \(a\), \(b\) and \(c\) are integers. (3)
(b) On the axes below, sketch the graph of \(y = 28 + 24x - 6x^2\)
Show clearly the coordinates of the turning point and the coordinates of the point of intersection of the graph with the \(y\)-axis. (3)
Show clearly the coordinates of the turning point and the coordinates of the point of intersection of the graph with the \(y\)-axis. (3)
| Scheme | Marks |
|---|---|
\(\pm 6\left(x \pm \dfrac{4}{2}\right)^2\ldots\ldots\) or \(\pm 6(x \pm 2)^2\ldots\ldots\) or \(\pm 6\left[(x \pm 2)^2\ldots\ldots\right]\) or \(\pm 6\left[\left(x \pm \dfrac{4}{2}\right)^2\ldots\ldots\right]\) or \(\pm 6\left(x \pm \dfrac{24}{2 \times -6}\right)^2\ldots\) | M1 |
\(-6\left[\left(x - \dfrac{4}{2}\right)^2 - \left(\dfrac{4}{2}\right)^2\right]\ldots\ldots\) or \(-6\left[(x - 2)^2 - 2^2\right]\ldots\ldots\) or \(-6\left[\left(x - \dfrac{4}{2}\right)^2 - \left(\dfrac{4}{2}\right)^2\ldots\ldots\right]\) or \(-6\left[(x - 2)^2 - 2^2\ldots\ldots\right]\) or \(-6\left(x + \dfrac{24}{2 \times -6}\right)^2 - \dfrac{24^2}{4 \times -6}\ldots\ldots\) | M1 |
| Working not required, so correct answer scores full marks (unless from obvious incorrect working) Answer: \(52 - 6(x - 2)^2\) | A1 |
| (3) |
Notes
M1: for a start to completing the square
or
correct substitution into \(a\left(x + \dfrac{b}{2a}\right)^2 + \ldots\) from the formula \(a\left(x + \dfrac{b}{2a}\right)^2 - \dfrac{(b)^2}{4a} + c\)
M1: for correctly completing the square but terms do not need to be simplified and 28 may or may not be present
or
correct simplification of the first two parts of \(a\left(x + \dfrac{b}{2a}\right)^2 - \dfrac{(b)^2}{4a}(+c)\)
NB: Please refer to ALT mark scheme after (b) for comparison of coefficients method
A1: oe eg \(-6(x - 2)^2 + 52\)
25(a) ALT
| Scheme | Marks |
|---|---|
| \(-bx^2 + 2bcx - bc^2 + a\) and \(b = 6\) or \(b = -6\) | M1 |
| \(2bc = 24\) or \(-bc^2 + a = 28\) | M1 |
| Working not required, so correct answer scores full marks (unless from obvious incorrect working) Answer: \(52 - 6(x - 2)^2\) | A1 |
Notes
M1: for multiplying out \(a - b(x - c)^2\) and \(b = 6\) or \(b = -6\)
M1: for equating coefficients
A1: oe eg \(-6(x - 2)^2 + 52\)
| Scheme | Marks |
|---|---|
| ‘∩’ or ‘∪’ shaped symmetrical quadratic curve | B1 |
 Answer: Turning point marked as (2, 52) | B1 |
 Answer: Intersection with \(y\)-axis marked as (0, 28) or crossing at 28 marked | B1 |
| (3) | |
| (6 marks) |
Notes
B1: for drawing a ‘∩’ or ‘∪’ shaped symmetrical quadratic curve with the turning point in any quadrant
B1: for drawing a ‘∩’ shaped symmetrical quadratic curve in the correct quadrant with a turning point at (2, 52)
B1: for drawing a ‘∩’ shaped symmetrical quadratic curve in the correct quadrant with an intersection on the \(y\)-axis marked as (0, 28) or marked as 28 on the \(y\)-axis