| Exam Board | Edexcel |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2023 |
| Session | June |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Reciprocal Trig & Identities |
| Type | Solve equation with reciprocal functions |
| Difficulty | Moderate -0.3 Part (i) is trivial algebra followed by basic sec x = cos x conversion. Part (ii) requires the double angle formula and solving a quadratic in sin θ, but this is a standard textbook exercise with no novel insight needed. The multi-part structure and use of reciprocal functions places it slightly below average difficulty overall. |
| Spec | 1.05h Reciprocal trig functions: sec, cosec, cot definitions and graphs1.05l Double angle formulae: and compound angle formulae1.05o Trigonometric equations: solve in given intervals |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| States \(x = 2\) | B1 | May be seen anywhere in (i) |
| \(\sqrt{3}\sec x + 2 = 0 \Rightarrow \cos x = -\frac{\sqrt{3}}{2} \Rightarrow x = ...\) | M1 | Correct process to solve; allow slips in rearranging but must attempt \(\cos x = k\), \( |
| \(x = \frac{5\pi}{6}\) | A1 | No other extra solutions in range; accept awrt 2.62 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Attempts to use \(\cos 2\theta = 1 - 2\sin^2\theta\) | M1 | To form quadratic in \(\sin\theta\); if using alternative forms must also use \(\cos^2\theta = 1-\sin^2\theta\) |
| \(6\sin^2\theta + 10\sin\theta - 3 = 0\) | A1 | Correct 3-term quadratic or multiple of this |
| \(\sin\theta = \frac{-5 \pm \sqrt{43}}{6}\) \((= -1.926..., 0.2595...) \Rightarrow \theta = \arcsin(...)\) | M1 | Full attempt to find one value of \(\theta\) from quadratic in \(\sin\theta\); correct method to solve quadratic and use of arcsin |
| \(\theta = 15.0°, 165°\) | A1 | Both values, no other solutions in range; accept just \(15°\) for \(15.0°\) but not awrt \(15°\) if it doesn't round to \(15.0°\) |
# Question 5:
## Part (i):
| Answer/Working | Mark | Guidance |
|---|---|---|
| States $x = 2$ | B1 | May be seen anywhere in (i) |
| $\sqrt{3}\sec x + 2 = 0 \Rightarrow \cos x = -\frac{\sqrt{3}}{2} \Rightarrow x = ...$ | M1 | Correct process to solve; allow slips in rearranging but must attempt $\cos x = k$, $|k|<1$ or $\sec x = k$, $|k|>1$ |
| $x = \frac{5\pi}{6}$ | A1 | No other extra solutions in range; accept awrt 2.62 |
## Part (ii):
| Answer/Working | Mark | Guidance |
|---|---|---|
| Attempts to use $\cos 2\theta = 1 - 2\sin^2\theta$ | M1 | To form quadratic in $\sin\theta$; if using alternative forms must also use $\cos^2\theta = 1-\sin^2\theta$ |
| $6\sin^2\theta + 10\sin\theta - 3 = 0$ | A1 | Correct 3-term quadratic or multiple of this |
| $\sin\theta = \frac{-5 \pm \sqrt{43}}{6}$ $(= -1.926..., 0.2595...) \Rightarrow \theta = \arcsin(...)$ | M1 | Full attempt to find one value of $\theta$ from quadratic in $\sin\theta$; correct method to solve quadratic and use of arcsin |
| $\theta = 15.0°, 165°$ | A1 | Both values, no other solutions in range; accept just $15°$ for $15.0°$ but not awrt $15°$ if it doesn't round to $15.0°$ |
---\begin{enumerate}
\item In this question you must show all stages of your working. Solutions relying entirely on calculator technology are not acceptable.\\
(i) Solve, for $0 < x < \pi$
\end{enumerate}
$$( x - 2 ) ( \sqrt { 3 } \sec x + 2 ) = 0$$
(ii) Solve, for $0 < \theta < 360 ^ { \circ }$
$$10 \sin \theta = 3 \cos 2 \theta$$
\hfill \mbox{\textit{Edexcel P3 2023 Q5 [7]}}