| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2018 |
| Session | June |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Trig Proofs |
| Type | Solve equation using proven identity |
| Difficulty | Moderate -0.3 This is a straightforward trigonometric question requiring standard identities and routine algebraic manipulation. Part (a)(i) uses the identity tan²θ = sin²θ/cos²θ and simplifies to -cos(2θ), which is a textbook exercise. Part (a)(ii) is direct substitution and solving. Part (b) involves solving sin x = 2cos x using tan x = 2, which is standard. All parts are routine applications of A-level techniques with no novel problem-solving required, making it slightly easier than average. |
| Spec | 1.05a Sine, cosine, tangent: definitions for all arguments1.05j Trigonometric identities: tan=sin/cos and sin^2+cos^2=11.05o Trigonometric equations: solve in given intervals |
| Answer | Marks | Guidance |
|---|---|---|
| 7(i) | State answer R = 5 | B1 |
| Use trig formulae to find tan α | M1 | |
| Obtain tanα=2 | A1 | |
| Total: | 3 |
| Answer | Marks |
|---|---|
| 7(ii) | ( ) |
| State that the integrand is 3sec2 θ−α | B1FT |
| Answer | Marks | Guidance |
|---|---|---|
| State correct indefinite integral 3tan θ−α | B1FT | |
| Substitute limits correctly | M1 | |
| Use tan(A ± B) formula | M1 | |
| Obtain the given exact answer correctly | A1 | |
| Total: | 5 | |
| Question | Answer | Marks |
Question 7:
--- 7(i) ---
7(i) | State answer R = 5 | B1
Use trig formulae to find tan α | M1
Obtain tanα=2 | A1
Total: | 3
--- 7(ii) ---
7(ii) | ( )
State that the integrand is 3sec2 θ−α | B1FT
( )
State correct indefinite integral 3tan θ−α | B1FT
Substitute limits correctly | M1
Use tan(A ± B) formula | M1
Obtain the given exact answer correctly | A1
Total: | 5
Question | Answer | Marks\begin{enumerate}[label=(\alph*)]
\item
\begin{enumerate}[label=(\roman*)]
\item Express $\frac{\tan^2 \theta - 1}{\tan^2 \theta + 1}$ in the form $a \sin^2 \theta + b$, where $a$ and $b$ are constants to be found. [3]
\item Hence, or otherwise, and showing all necessary working, solve the equation
$$\frac{\tan^2 \theta - 1}{\tan^2 \theta + 1} = \frac{1}{4}$$
for $-90° \leqslant \theta \leqslant 0°$. [2]
\end{enumerate}
\item
\includegraphics{figure_7b}
The diagram shows the graphs of $y = \sin x$ and $y = 2 \cos x$ for $-\pi \leqslant x \leqslant \pi$. The graphs intersect at the points $A$ and $B$.
\begin{enumerate}[label=(\roman*)]
\item Find the $x$-coordinate of $A$. [2]
\item Find the $y$-coordinate of $B$. [2]
\end{enumerate}
\end{enumerate}
\hfill \mbox{\textit{CAIE P3 2018 Q7 [9]}}