| Exam Board | OCR |
|---|---|
| Module | C3 (Core Mathematics 3) |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Composite & Inverse Functions |
| Type | Solve equation involving composites |
| Difficulty | Standard +0.3 This is a straightforward multi-part question on composite and inverse functions. Part (i) requires routine inversion of a linear function, part (ii) involves completing the square to find the range of a quadratic (standard technique), and part (iii) requires computing gf(3) and solving a quadratic equation. All steps are standard C3 techniques with no novel insight required, making it slightly easier than average. |
| Spec | 1.02e Complete the square: quadratic polynomials and turning points1.02v Inverse and composite functions: graphs and conditions for existence |
| Answer | Marks | Guidance |
|---|---|---|
| (i) \(y = 1 - ax\), \(x = \frac{1-y}{a}\) | M1 | |
| \(f^{-1}(x) = \frac{1-x}{a}\) | A1 | |
| (ii) \(g(x) = (x+a)^2 - a^2 + 2\) | M1, A1 | |
| \(\therefore g(x) \geq 2 - a^2\) | A1 | |
| (iii) \(g(3) = g(1-3a) = (1-3a)^2 + 2a(1-3a) + 2\) | M1 | |
| \(\therefore 1 - 6a + 9a^2 + 2a - 6a^2 + 2 = 7\) | A1 | |
| \(3a^2 - 4a - 4 = 0\) | A1 | |
| \((3a + 2)(a - 2) = 0\) | M1 | |
| \(a = -\frac{2}{3}, 2\) | A1 | (9 marks) |
**(i)** $y = 1 - ax$, $x = \frac{1-y}{a}$ | M1 |
$f^{-1}(x) = \frac{1-x}{a}$ | A1 |
**(ii)** $g(x) = (x+a)^2 - a^2 + 2$ | M1, A1 |
$\therefore g(x) \geq 2 - a^2$ | A1 |
**(iii)** $g(3) = g(1-3a) = (1-3a)^2 + 2a(1-3a) + 2$ | M1 |
$\therefore 1 - 6a + 9a^2 + 2a - 6a^2 + 2 = 7$ | A1 |
$3a^2 - 4a - 4 = 0$ | A1 |
$(3a + 2)(a - 2) = 0$ | M1 |
$a = -\frac{2}{3}, 2$ | A1 | **(9 marks)** |
---6. The functions $f$ and $g$ are defined by
$$\begin{aligned}
& \mathrm { f } : x \rightarrow 1 - a x , \quad x \in \mathbb { R } \\
& \mathrm {~g} : x \rightarrow x ^ { 2 } + 2 a x + 2 , \quad x \in \mathbb { R }
\end{aligned}$$
where $a$ is a constant.\\
Find, in terms of $a$,\\
(i) an expression for $\mathrm { f } ^ { - 1 } ( x )$,\\
(ii) the range of g .
Given that $g f ( 3 ) = 7$,\\
(iii) find the two possible values of $a$.\\
\hfill \mbox{\textit{OCR C3 Q6 [9]}}