June 2024 Paper 2 Q21
21
\[\mathrm{f}(x) = \frac{2 - x}{3}\]\[\mathrm{g}(x) = 18x^2 + 15x\]\[\mathrm{f}^{-1}(x) + \mathrm{gf}(x) \quad \text{simplifies to} \quad ax^2 + bx + c\]Work out the values of \(a\), \(b\) and \(c\). [6 marks]
| Answer | Mark | Comments |
|---|---|---|
| \(2 - 3x\) or \(2 - 3y\) | B1 | oe accept \(2 - 3\mathrm{f}\) or \(2 - 3\mathrm{f}(x)\) |
| \(15\left(\dfrac{2 - x}{3}\right)\) or \(18\left(\dfrac{2 - x}{3}\right)^2\) | M1 | may be implied eg \(5(2 - x)\) or \(2(2 - x)^2\) |
| \(\dfrac{30 - 15x}{3}\) or \(10 - 5x\) | M1 | expansion of \(15\left(\dfrac{2 - x}{3}\right)\) implies 1st M1 |
| \(18\left(\dfrac{4 - 2x - 2x + x^2}{9}\right)\) or \(18\left(\dfrac{4 - 4x + x^2}{9}\right)\) or \(8 - 8x + 2x^2\) | M1 | oe squaring of bracket in \(18\left(\dfrac{2 - x}{3}\right)^2\) eg \(2(4 - 2x - 2x + x^2)\) or \(2(4 - 4x + x^2)\) implies 1st M1 |
| \(8 - 4x - 4x + 2x^2 + 10 - 5x\) or \(2x^2 - 13x + 18\) | M1dep | oe \(\mathrm{gf}(x)\) with elimination of brackets and fractions eg \(8 - 8x + 2x^2 + 10 - 5x\) dep on M3 \(2x^2 - 16x + 20\) is B1M4 |
| \(a = 2 \quad b = -16 \quad c = 20\) | A1ft | ft their \(2 - 3x\) if B0M4 their \(2 - 3x\) must be of the form \(px + q\) with \(p\) and \(q\) non-zero their \(2 - 3x\) cannot be \(\dfrac{2 - x}{3}\) |
Additional guidance
| Missing brackets must be recovered | |
| Ignore irrelevant brackets eg \((8 - 4x - 4x + 2x^2) + (10 - 5x)\) | M4 |
| \(3x - 2\) \(8 - 4x - 4x + 2x^2 + 10 - 5x\) (\(3x - 2\) added to the above expression is \(2x^2 - 10x + 16\)) \(a = 2 \quad b = -10 \quad c = 16\) | B0 M4 A1ft |