| Exam Board | CAIE |
|---|---|
| Module | P1 (Pure Mathematics 1) |
| Year | 2015 |
| Session | June |
| Marks | 12 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Composite & Inverse Functions |
| Type | Complete the square |
| Difficulty | Moderate -0.3 This is a standard multi-part question on quadratic functions covering discriminants, completing the square, range finding, and inverse functions. All parts use routine techniques taught in P1 with no novel problem-solving required. The completing the square and inverse function parts are textbook exercises, making this slightly easier than average despite having multiple parts. |
| Spec | 1.02d Quadratic functions: graphs and discriminant conditions1.02e Complete the square: quadratic polynomials and turning points1.02v Inverse and composite functions: graphs and conditions for existence |
The function f is defined by $\mathrm{f} : x \mapsto 2x^2 - 6x + 5$ for $x \in \mathbb{R}$.
\begin{enumerate}[label=(\roman*)]
\item Find the set of values of $p$ for which the equation $\mathrm{f}(x) = p$ has no real roots. [3]
\end{enumerate}
The function g is defined by $\mathrm{g} : x \mapsto 2x^2 - 6x + 5$ for $0 \leqslant x \leqslant 4$.
\begin{enumerate}[label=(\roman*)]
\setcounter{enumi}{1}
\item Express $\mathrm{g}(x)$ in the form $a(x + b)^2 + c$, where $a$, $b$ and $c$ are constants. [3]
\item Find the range of g. [2]
\end{enumerate}
The function h is defined by $\mathrm{h} : x \mapsto 2x^2 - 6x + 5$ for $k \leqslant x \leqslant 4$, where $k$ is a constant.
\begin{enumerate}[label=(\roman*)]
\setcounter{enumi}{3}
\item State the smallest value of $k$ for which h has an inverse. [1]
\item For this value of $k$, find an expression for $\mathrm{h}^{-1}(x)$. [3]
\end{enumerate}
\hfill \mbox{\textit{CAIE P1 2015 Q11 [12]}}