| Exam Board | CAIE |
|---|---|
| Module | P1 (Pure Mathematics 1) |
| Year | 2017 |
| Session | June |
| Marks | 11 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Trigonometric equations in context |
| Type | Solve shifted trig equation |
| Difficulty | Standard +0.2 This is a straightforward multi-part question on function properties requiring standard techniques: solving a simple trig equation, finding an inverse function by swapping and rearranging, and sketching related graphs. All parts are routine A-level exercises with no novel problem-solving required, making it slightly easier than average. |
| Spec | 1.02m Graphs of functions: difference between plotting and sketching1.02v Inverse and composite functions: graphs and conditions for existence1.05a Sine, cosine, tangent: definitions for all arguments1.05k Further identities: sec^2=1+tan^2 and cosec^2=1+cot^2 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(3\tan\!\left(\frac{1}{2}x\right) = -2 \rightarrow \tan\!\left(\frac{1}{2}x\right) = -\frac{2}{3}\) | M1 | Attempt to obtain \(\tan\!\left(\frac{1}{2}x\right)=k\) from \(3\tan\!\left(\frac{1}{2}x\right)+2=0\) |
| \(\frac{1}{2}x = -0.6\ (-0.588) \rightarrow x = -1.2\) | M1A1 | \(\tan^{-1}k\). Seeing \(\frac{1}{2}x = -33.69°\) or \(x=-67.4°\) implies M1M1. Extra answers between \(-1.57\) & \(1.57\) lose the A1. Multiples of \(\pi\) acceptable (e.g. \(-0.374\pi\)) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(\frac{y+2}{3} = \tan\!\left(\frac{1}{2}x\right)\) | M1 | Attempt at isolating \(\tan(\frac{1}{2}x)\) |
| \(\rightarrow f^{-1}(x) = 2\tan^{-1}\!\left(\frac{x+2}{3}\right)\) | M1A1 | Inverse tan followed by \(\times 2\). Must be function of \(x\) for A1 |
| \(-5,\ 1\) | B1 B1 | Values stated B1 for \(-5\), B1 for \(1\) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| [Graph showing tan curve and invtan curve] | B1 B1 B1 | A tan graph through the first, third and fourth quadrants (B1); An invtan graph through the first, second and third quadrants (B1); Two curves clearly symmetrical about \(y=x\) either by sight or by exact end points. Line not required. Approximately in correct domain and range. Not intersecting (B1). Labels on axes not required |
## Question 10(i):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $3\tan\!\left(\frac{1}{2}x\right) = -2 \rightarrow \tan\!\left(\frac{1}{2}x\right) = -\frac{2}{3}$ | M1 | Attempt to obtain $\tan\!\left(\frac{1}{2}x\right)=k$ from $3\tan\!\left(\frac{1}{2}x\right)+2=0$ |
| $\frac{1}{2}x = -0.6\ (-0.588) \rightarrow x = -1.2$ | M1A1 | $\tan^{-1}k$. Seeing $\frac{1}{2}x = -33.69°$ or $x=-67.4°$ implies **M1M1**. Extra answers between $-1.57$ & $1.57$ lose the **A1**. Multiples of $\pi$ acceptable (e.g. $-0.374\pi$) |
---
## Question 10(ii):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $\frac{y+2}{3} = \tan\!\left(\frac{1}{2}x\right)$ | M1 | Attempt at isolating $\tan(\frac{1}{2}x)$ |
| $\rightarrow f^{-1}(x) = 2\tan^{-1}\!\left(\frac{x+2}{3}\right)$ | M1A1 | Inverse tan followed by $\times 2$. Must be function of $x$ for **A1** |
| $-5,\ 1$ | B1 B1 | Values stated **B1** for $-5$, **B1** for $1$ |
---
## Question 10(iii):
| Answer | Marks | Guidance |
|--------|-------|----------|
| [Graph showing tan curve and invtan curve] | B1 B1 B1 | A tan graph through the first, third and fourth quadrants (**B1**); An invtan graph through the first, second and third quadrants (**B1**); Two curves clearly symmetrical about $y=x$ either by sight or by exact end points. Line not required. Approximately in correct domain and range. Not intersecting (**B1**). Labels on axes not required |10 The function f is defined by $\mathrm { f } ( x ) = 3 \tan \left( \frac { 1 } { 2 } x \right) - 2$, for $- \frac { 1 } { 2 } \pi \leqslant x \leqslant \frac { 1 } { 2 } \pi$.\\
(i) Solve the equation $\mathrm { f } ( x ) + 4 = 0$, giving your answer correct to 1 decimal place.\\
(ii) Find an expression for $\mathrm { f } ^ { - 1 } ( x )$ and find the domain of $\mathrm { f } ^ { - 1 }$.\\
(iii) Sketch, on the same diagram, the graphs of $y = \mathrm { f } ( x )$ and $y = \mathrm { f } ^ { - 1 } ( x )$.
\hfill \mbox{\textit{CAIE P1 2017 Q10 [11]}}