| Exam Board | CAIE |
|---|---|
| Module | P1 (Pure Mathematics 1) |
| Year | 2021 |
| Session | March |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Composite & Inverse Functions |
| Type | Complete the square |
| Difficulty | Moderate -0.3 This is a slightly below-average A-level question. Part (a) is routine completing the square with straightforward range determination. Part (b) requires finding an inverse function, which is standard P1 content. Part (c) involves composing two functions and solving a quadratic equation. All parts use well-practiced techniques with no novel insights required, though the restricted domain adds minor complexity. |
| 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 |
|---|---|---|
| Answer | Marks | Guidance |
| (a) \([f(x) =]\ (x+1)^2 + 2\) | B1 B1 | Accept \(a = 1, b = 2\) |
| (a) Range of f is \((y) \geqslant 2\) | B1 FT | OE. Do not allow \(x \geqslant 2\). FT on *their* \(b\) |
| (b) \(y = (x+1)^2 + 2\) leading to \(x = [\pm]\sqrt{y-2} - 1\) | M1 | Or by using the formula. Allow one sign error |
| (b) \(f^{-1}(x) = -\sqrt{x-2} - 1\) | A1 | |
| (c) \(2(x^2 + 2x + 3) + 1 = 13\) | B1 | Or using a correct completed square form of \(f(x)\) |
| (c) \(2x^2 + 4x - 6[= 0]\) leading to \((2)(x-1)(x+3)[= 0]\) | B1 | Or \(x = 1, x = -3\) using formula or completing the square. Must reach 2 solutions |
| (c) \(x = -3\) only | B1 |
## Question 7:
| Answer | Marks | Guidance |
|--------|-------|----------|
| (a) $[f(x) =]\ (x+1)^2 + 2$ | B1 B1 | Accept $a = 1, b = 2$ |
| (a) Range of f is $(y) \geqslant 2$ | B1 FT | OE. Do not allow $x \geqslant 2$. FT on *their* $b$ |
| (b) $y = (x+1)^2 + 2$ leading to $x = [\pm]\sqrt{y-2} - 1$ | M1 | Or by using the formula. Allow one sign error |
| (b) $f^{-1}(x) = -\sqrt{x-2} - 1$ | A1 | |
| (c) $2(x^2 + 2x + 3) + 1 = 13$ | B1 | Or using a correct completed square form of $f(x)$ |
| (c) $2x^2 + 4x - 6[= 0]$ leading to $(2)(x-1)(x+3)[= 0]$ | B1 | Or $x = 1, x = -3$ using formula or completing the square. Must reach 2 solutions |
| (c) $x = -3$ only | B1 | |
---7 Functions f and g are defined as follows:
$$\begin{aligned}
& \mathrm { f } : x \mapsto x ^ { 2 } + 2 x + 3 \text { for } x \leqslant - 1 , \\
& \mathrm {~g} : x \mapsto 2 x + 1 \text { for } x \geqslant - 1 .
\end{aligned}$$
\begin{enumerate}[label=(\alph*)]
\item Express $\mathrm { f } ( x )$ in the form $( x + a ) ^ { 2 } + b$ and state the range of f .
\item Find an expression for $\mathrm { f } ^ { - 1 } ( x )$.
\item Solve the equation $\operatorname { gf } ( x ) = 13$.
\end{enumerate}
\hfill \mbox{\textit{CAIE P1 2021 Q7 [8]}}