| Exam Board | CAIE |
|---|---|
| Module | P1 (Pure Mathematics 1) |
| Year | 2021 |
| Session | June |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Composite & Inverse Functions |
| Type | Find composite function expression |
| Difficulty | Moderate -0.3 Part (a) requires straightforward substitution of f(x) into itself and algebraic simplification. Part (b) involves solving a quadratic equation after the substitution, which is routine. Both parts are standard textbook exercises requiring no novel insight, making this slightly easier than average. |
| Spec | 1.02u Functions: definition and vocabulary (domain, range, mapping)1.02v Inverse and composite functions: graphs and conditions for existence |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(ff(x)=2(2x^2+3)^2+3\) | M1 | Condone \(=0\) |
| \(8x^4+24x^2+21\) | A1 | ISW if correct answer seen. Condone \(=0\) |
| 2 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(8x^4+24x^2+21=34x^2+19\Rightarrow 8x^4+24x^2-34x^2+21-19\ [=0]\) | M1 | Equating \(34x^2+19\) to *their* 3-term \(ff(x)\) and collect all terms on one side, condone \(\pm\) sign errors |
| \(8x^4-10x^2+2\ [=0]\) | A1 | |
| \([2](x^2-1)(4x^2-1)\) | M1 | Attempt to solve 3-term quartic or 3-term quadratic by factorisation, formula or completing the square or factor theorem |
| \(x^2=1\) or \(\frac{1}{4}\) leading to \(x=1\) or \(x=\frac{1}{2}\) | A1 | If factorising, factors must expand to give \(8x^4\) or \(4x^4\) or *their* \(ax^4\), otherwise M0A0 due to calculator use. Condone \(\pm 1\), \(\pm\frac{1}{2}\) but not \(\sqrt{\frac{1}{4}}\) or \(\sqrt{1}\) |
| 4 |
## Question 5(a):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $ff(x)=2(2x^2+3)^2+3$ | M1 | Condone $=0$ |
| $8x^4+24x^2+21$ | A1 | ISW if correct answer seen. Condone $=0$ |
| | **2** | |
## Question 5(b):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $8x^4+24x^2+21=34x^2+19\Rightarrow 8x^4+24x^2-34x^2+21-19\ [=0]$ | M1 | Equating $34x^2+19$ to *their* 3-term $ff(x)$ and collect all terms on one side, condone $\pm$ sign errors |
| $8x^4-10x^2+2\ [=0]$ | A1 | |
| $[2](x^2-1)(4x^2-1)$ | M1 | Attempt to solve 3-term quartic or 3-term quadratic by factorisation, formula or completing the square or factor theorem |
| $x^2=1$ or $\frac{1}{4}$ leading to $x=1$ or $x=\frac{1}{2}$ | A1 | If factorising, factors must expand to give $8x^4$ or $4x^4$ or *their* $ax^4$, otherwise M0A0 due to calculator use. Condone $\pm 1$, $\pm\frac{1}{2}$ but not $\sqrt{\frac{1}{4}}$ or $\sqrt{1}$ |
| | **4** | |5 The function f is defined by $\mathrm { f } ( x ) = 2 x ^ { 2 } + 3$ for $x \geqslant 0$.
\begin{enumerate}[label=(\alph*)]
\item Find and simplify an expression for $\mathrm { ff } ( x )$.
\item Solve the equation $\mathrm { ff } ( x ) = 34 x ^ { 2 } + 19$.
\end{enumerate}
\hfill \mbox{\textit{CAIE P1 2021 Q5 [6]}}