| Exam Board | CAIE |
|---|---|
| Module | P1 (Pure Mathematics 1) |
| Year | 2017 |
| Session | November |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Composite & Inverse Functions |
| Type | Evaluate composite at point |
| Difficulty | Moderate -0.8 This question involves straightforward algebraic manipulation to find constants using given function values, then evaluating a simple composite function. Part (a)(i) requires solving two simultaneous equations with known sine values, (a)(ii) is direct substitution, and part (b) uses basic understanding of sine function range. All techniques are routine for P1 level with no problem-solving insight required. |
| Spec | 1.02v Inverse and composite functions: graphs and conditions for existence1.05f Trigonometric function graphs: symmetries and periodicities |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(4 = a + \frac{1}{2}b\); \(3 = a + b\) | M1 | Forming simultaneous equations and eliminating one of the variables — probably \(a\). May still include \(\sin\frac{\pi}{2}\) and/or \(\sin\frac{\pi}{6}\) |
| \(\rightarrow a = 5,\ b = -2\) | A1 A1 | |
| Total: 3 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(ff(x) = a + b\sin(a + b\sin x)\) | M1 | Valid method for \(ff\). Could be \(f(0) = N\) followed by \(f(N) = M\). |
| \(ff(0) = 5 - 2\sin 5 = 6.92\) | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| *EITHER:* \(10 = c + d\) and \(-4 = c - d\); \(10 = c - d\) and \(-4 = c + d\) | (M1 | Either pair of equations stated. |
| \(c = 3,\ d = 7,\ {-7}\) or \(\pm 7\) | A1 A1) | Either pair solved ISW. Alternately \(c=3\) B1, range \(= 14\) M1 \(\rightarrow d = 7, -7\) or \(\pm 7\) A1 |
| *OR:* [diagrams of \(y = 3 + 7\sin(x)\) and \(y = 3 - 7\sin(x)\)] | (M1 A1 A1) | Either of these diagrams can be awarded M1. Correct values of \(c\) and/or \(d\) can be awarded the A1, A1 |
| Total: 3 |
## Question 6(a)(i):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $4 = a + \frac{1}{2}b$; $3 = a + b$ | M1 | Forming simultaneous equations and eliminating one of the variables — probably $a$. May still include $\sin\frac{\pi}{2}$ and/or $\sin\frac{\pi}{6}$ |
| $\rightarrow a = 5,\ b = -2$ | A1 A1 | |
| **Total: 3** | | |
## Question 6(a)(ii):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $ff(x) = a + b\sin(a + b\sin x)$ | M1 | Valid method for $ff$. Could be $f(0) = N$ followed by $f(N) = M$. |
| $ff(0) = 5 - 2\sin 5 = 6.92$ | A1 | |
## Question 6(b):
| Answer | Marks | Guidance |
|--------|-------|----------|
| *EITHER:* $10 = c + d$ and $-4 = c - d$; $10 = c - d$ and $-4 = c + d$ | (M1 | Either pair of equations stated. |
| $c = 3,\ d = 7,\ {-7}$ or $\pm 7$ | A1 A1) | Either pair solved ISW. **Alternately** $c=3$ **B1**, range $= 14$ **M1** $\rightarrow d = 7, -7$ or $\pm 7$ **A1** |
| *OR:* [diagrams of $y = 3 + 7\sin(x)$ and $y = 3 - 7\sin(x)$] | (M1 A1 A1) | Either of these diagrams can be awarded M1. Correct values of $c$ and/or $d$ can be awarded the A1, A1 |
| **Total: 3** | | |6
\begin{enumerate}[label=(\alph*)]
\item The function f , defined by $\mathrm { f } : x \mapsto a + b \sin x$ for $x \in \mathbb { R }$, is such that $\mathrm { f } \left( \frac { 1 } { 6 } \pi \right) = 4$ and $\mathrm { f } \left( \frac { 1 } { 2 } \pi \right) = 3$.
\begin{enumerate}[label=(\roman*)]
\item Find the values of the constants $a$ and $b$.
\item Evaluate $\mathrm { ff } ( 0 )$.
\end{enumerate}\item The function g is defined by $\mathrm { g } : x \mapsto c + d \sin x$ for $x \in \mathbb { R }$. The range of g is given by $- 4 \leqslant \mathrm {~g} ( x ) \leqslant 10$. Find the values of the constants $c$ and $d$.
\end{enumerate}
\hfill \mbox{\textit{CAIE P1 2017 Q6 [8]}}