| Exam Board | CAIE |
|---|---|
| Module | P1 (Pure Mathematics 1) |
| Year | 2019 |
| Session | November |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Trigonometric equations in context |
| Type | Solve shifted trig equation |
| Difficulty | Moderate -0.3 This is a straightforward trigonometry question requiring standard techniques: solving a basic tan equation with a linear argument, using the identity cos²x + sin²x = 1 to rewrite an expression, and finding a range from the rewritten form. All steps are routine A-level procedures with no novel problem-solving required, making it slightly easier than average. |
| Spec | 1.05j Trigonometric identities: tan=sin/cos and sin^2+cos^2=11.05o Trigonometric equations: solve in given intervals |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \((2x+1) = \tan^{-1}(\frac{1}{3})\) (\(= 0.322\) or \(18.4°\) OR \(-0.339\) or \(8.7°\)) | \*M1 | Correct order of operations. Allow degrees. |
| Either their \(0.322 + \pi\) or \(2\pi\); Or their \(-0.339 + \frac{\pi}{2}\) or \(\pi\) | DM1 | Must be in radians |
| \(x = 1.23\) or \(x = 2.80\) | A1 | AWRT for either correct answer, accept \(0.39\pi\) or \(0.89\pi\) |
| A1 | For the second answer with no other answers between 0 and 2.8. SC1 For both 1.2 and 2.8 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(5\cos^2 x - 2\) | B1 | Allow \(a = 5,\ b = -2\) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(-2\) | B1FT | FT for sight of their \(b\) |
| \(3\) | B1FT | FT for sight of their \(a + b\) |
## Question 6(a):
| Answer | Mark | Guidance |
|--------|------|----------|
| $(2x+1) = \tan^{-1}(\frac{1}{3})$ ($= 0.322$ or $18.4°$ OR $-0.339$ or $8.7°$) | \*M1 | Correct order of operations. Allow degrees. |
| Either their $0.322 + \pi$ or $2\pi$; Or their $-0.339 + \frac{\pi}{2}$ or $\pi$ | DM1 | Must be in radians |
| $x = 1.23$ **or** $x = 2.80$ | A1 | AWRT for either correct answer, accept $0.39\pi$ or $0.89\pi$ |
| | A1 | For the second answer with no other answers between 0 and 2.8. **SC1** For both 1.2 and 2.8 |
---
## Question 6(b)(i):
| Answer | Mark | Guidance |
|--------|------|----------|
| $5\cos^2 x - 2$ | B1 | Allow $a = 5,\ b = -2$ |
---
## Question 6(b)(ii):
| Answer | Mark | Guidance |
|--------|------|----------|
| $-2$ | B1FT | FT for sight of their $b$ |
| $3$ | B1FT | FT for sight of their $a + b$ |
---6
\begin{enumerate}[label=(\alph*)]
\item Given that $x > 0$, find the two smallest values of $x$, in radians, for which $3 \tan ( 2 x + 1 ) = 1$. Show all necessary working.
\item The function f : $x \mapsto 3 \cos ^ { 2 } x - 2 \sin ^ { 2 } x$ is defined for $0 \leqslant x \leqslant \pi$.
\begin{enumerate}[label=(\roman*)]
\item Express $\mathrm { f } ( x )$ in the form $a \cos ^ { 2 } x + b$, where $a$ and $b$ are constants.
\item Find the range of $f$.
\end{enumerate}\end{enumerate}
\hfill \mbox{\textit{CAIE P1 2019 Q6 [7]}}