| Exam Board | Edexcel |
|---|---|
| Module | AS Paper 1 (AS Paper 1) |
| Year | 2024 |
| Session | June |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Standard trigonometric equations |
| Type | Deduce related solution |
| Difficulty | Standard +0.3 This is a standard A-level technique of converting a trigonometric equation to a quadratic in tan θ using the identity sin²θ + cos²θ = 1 and dividing by cos²θ. Part (a) is routine algebraic manipulation, part (b) requires solving a quadratic and finding angles in the given range, and part (c) applies the same solution with a simple transformation (4α substitution). While multi-part, each step follows well-established procedures with no novel insight 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/Working | Mark | Guidance |
| \(7\sin^2\theta - 4\sin\theta\cos\theta = 4 \Rightarrow 7\tan^2\theta - 4\tan\theta = \frac{4}{\cos^2\theta}\) | M1 | 1.1b – Divides all terms by \(\cos^2\theta\), or uses \(\tan\theta = \frac{\sin\theta}{\cos\theta}\), or uses \(\pm\sin^2\theta \pm \cos^2\theta = \pm 1\) |
| \(\Rightarrow 3\tan^2\theta - 4\tan\theta + \frac{4\sin^2\theta}{\cos^2\theta} - \frac{4}{\cos^2\theta} = 0\) | dM1 | 1.1b – Dependent on M1; further manipulation using same techniques |
| \(\frac{4\sin^2\theta - 4}{\cos^2\theta} = -4\frac{\cos^2\theta}{\cos^2\theta} = -4 \Rightarrow 3\tan^2\theta - 4\tan\theta - 4 = 0\) * | ddM1, A1* | 2.1, 1.1b – Full attempt to reach required form; cso. Final answer must be in terms of \(\theta\) with correct notation |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Roots: \(-\frac{2}{3},\ 2\) | B1 | 1.1b – Both correct roots |
| \(\tan^{-1}\!\left(-\frac{2}{3}\right) = \ldots\) or \(\tan^{-1}(2) = \ldots\) | M1 | 1.1b – Attempts to find at least one angle for one root. Answers with no working score 0 |
| Two of: awrt 63, awrt 146, awrt 243, awrt 326 | A1 | 1.1b – Two correct values, must come from correct root |
| All four of: awrt 63.4, awrt 146.3, awrt 243.4, awrt 326.3 (and no others in range) | A1 | 1.1b – Withhold final mark if they solve for \(4\alpha\) and divide by 4 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| awrt 735.9 | B1 | 2.2a |
# Question 13:
## Part (a):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $7\sin^2\theta - 4\sin\theta\cos\theta = 4 \Rightarrow 7\tan^2\theta - 4\tan\theta = \frac{4}{\cos^2\theta}$ | M1 | 1.1b – Divides all terms by $\cos^2\theta$, or uses $\tan\theta = \frac{\sin\theta}{\cos\theta}$, or uses $\pm\sin^2\theta \pm \cos^2\theta = \pm 1$ |
| $\Rightarrow 3\tan^2\theta - 4\tan\theta + \frac{4\sin^2\theta}{\cos^2\theta} - \frac{4}{\cos^2\theta} = 0$ | dM1 | 1.1b – Dependent on M1; further manipulation using same techniques |
| $\frac{4\sin^2\theta - 4}{\cos^2\theta} = -4\frac{\cos^2\theta}{\cos^2\theta} = -4 \Rightarrow 3\tan^2\theta - 4\tan\theta - 4 = 0$ * | ddM1, A1* | 2.1, 1.1b – Full attempt to reach required form; cso. Final answer must be in terms of $\theta$ with correct notation |
**(4 marks)**
## Part (b):
| Answer/Working | Mark | Guidance |
|---|---|---|
| Roots: $-\frac{2}{3},\ 2$ | B1 | 1.1b – Both correct roots |
| $\tan^{-1}\!\left(-\frac{2}{3}\right) = \ldots$ or $\tan^{-1}(2) = \ldots$ | M1 | 1.1b – Attempts to find at least one angle for one root. Answers with no working score 0 |
| Two of: awrt 63, awrt 146, awrt 243, awrt 326 | A1 | 1.1b – Two correct values, must come from correct root |
| All four of: awrt 63.4, awrt 146.3, awrt 243.4, awrt 326.3 (and no others in range) | A1 | 1.1b – Withhold final mark if they solve for $4\alpha$ and divide by 4 |
**(4 marks)**
## Part (c):
| Answer/Working | Mark | Guidance |
|---|---|---|
| awrt 735.9 | B1 | 2.2a |
**(1 mark)**
**(9 marks total)**\begin{enumerate}
\item In this question you must show all stages of your working. Solutions relying entirely on calculator technology are not acceptable.\\
(a) Show that the equation
\end{enumerate}
$$\sin \theta ( 7 \sin \theta - 4 \cos \theta ) = 4$$
can be written as
$$3 \tan ^ { 2 } \theta - 4 \tan \theta - 4 = 0$$
(b) Hence solve, for $0 < x < 360 ^ { \circ }$
$$\sin x ( 7 \sin x - 4 \cos x ) = 4$$
giving your answers to one decimal place.\\
(c) Hence find the smallest solution of the equation
$$\sin 4 \alpha ( 7 \sin 4 \alpha - 4 \cos 4 \alpha ) = 4$$
in the range $720 ^ { \circ } < \alpha < 1080 ^ { \circ }$, giving your answer to one decimal place.
\hfill \mbox{\textit{Edexcel AS Paper 1 2024 Q13 [9]}}