| Exam Board | CAIE |
|---|---|
| Module | P1 (Pure Mathematics 1) |
| Year | 2010 |
| Session | June |
| Marks | 5 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Composite & Inverse Functions |
| Type | State domain or range |
| Difficulty | Moderate -0.3 This is a slightly below-average A-level question. Part (i) requires completing the square to find the range of a quadratic (standard technique), and part (ii) involves forming a composite function and using the discriminant condition for equal roots. Both are routine procedures with no novel insight required, though part (ii) requires careful algebraic manipulation across multiple steps. |
| Spec | 1.02e Complete the square: quadratic polynomials and turning points1.02u Functions: definition and vocabulary (domain, range, mapping)1.02v Inverse and composite functions: graphs and conditions for existence |
| Answer | Marks | Guidance |
|---|---|---|
| (i) Turning point at \(x = 1\). Range is \(≤ 2\). | M1, A1 [2] | Calculus or completing the square etc. Condone \(<\) instead of \(≤\). |
| (ii) \(gf(x) = 5(4x - 2x^2) + 3 = k\) and use of \(b^2 - 4ac \rightarrow k = 13\) | B1, M1, A1 [3] | For putting f into g. Setting to k, using \(b^2 - 4ac\) co. |
$f: x \mapsto 4x - 2x^2$, $g: x \mapsto 5x + 3$.
(i) Turning point at $x = 1$. Range is $≤ 2$. | M1, A1 [2] | Calculus or completing the square etc. Condone $<$ instead of $≤$.
(ii) $gf(x) = 5(4x - 2x^2) + 3 = k$ and use of $b^2 - 4ac \rightarrow k = 13$ | B1, M1, A1 [3] | For putting f into g. Setting to k, using $b^2 - 4ac$ co.The functions f and g are defined for $x \in \mathbb{R}$ by
$$f : x \mapsto 4x - 2x^2,$$
$$g : x \mapsto 5x + 3.$$
\begin{enumerate}[label=(\roman*)]
\item Find the range of f. [2]
\item Find the value of the constant $k$ for which the equation $gf(x) = k$ has equal roots. [3]
\end{enumerate}
\hfill \mbox{\textit{CAIE P1 2010 Q3 [5]}}