| Exam Board | Edexcel |
|---|---|
| Module | Paper 2 (Paper 2) |
| Session | Specimen |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Trigonometric equations in context |
| Type | Deduce solutions from earlier result |
| Difficulty | Standard +0.3 This is a standard A-level trigonometric equation requiring substitution of cos²x = 1 - sin²x to form a quadratic, then solving for sin x. Part (b) is a routine substitution exercise using the solution from part (a). The multi-step nature and decimal answers add slight complexity, but this follows a well-practiced procedure with no novel insight required. |
| 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 |
| Uses \(\cos^2 x = 1-\sin^2 x \Rightarrow 3\sin^2 x+\sin x+8=9(1-\sin^2 x)\) | M1 | Substitutes \(\cos^2 x=1-\sin^2 x\) to create quadratic in \(\sin x\) |
| \(\Rightarrow 12\sin^2 x+\sin x-1=0\) | A1 | Or exact equivalent |
| \(\Rightarrow (4\sin x-1)(3\sin x+1)=0\) | M1 | Attempts to solve quadratic by factorisation, formula or completing the square |
| \(\Rightarrow \sin x = \frac{1}{4}, -\frac{1}{3}\) | A1 | |
| Uses arcsin to obtain two correct values | M1 | Obtains two correct values for their \(\sin x = k\) |
| All four of \(x=14.48°, 165.52°, -19.47°, -160.53°\) | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Attempts \(2\theta - 30° = -19.47°\) | M1 | Sets \(2\theta-30°=\) their \(-19.47°\) |
| \(\Rightarrow \theta = 5.26°\) | A1ft | Follow through on their \(-19.47°\) |
# Question 12(a):
| Answer/Working | Mark | Guidance |
|---|---|---|
| Uses $\cos^2 x = 1-\sin^2 x \Rightarrow 3\sin^2 x+\sin x+8=9(1-\sin^2 x)$ | M1 | Substitutes $\cos^2 x=1-\sin^2 x$ to create quadratic in $\sin x$ |
| $\Rightarrow 12\sin^2 x+\sin x-1=0$ | A1 | Or exact equivalent |
| $\Rightarrow (4\sin x-1)(3\sin x+1)=0$ | M1 | Attempts to solve quadratic by factorisation, formula or completing the square |
| $\Rightarrow \sin x = \frac{1}{4}, -\frac{1}{3}$ | A1 | |
| Uses arcsin to obtain two correct values | M1 | Obtains two correct values for their $\sin x = k$ |
| All four of $x=14.48°, 165.52°, -19.47°, -160.53°$ | A1 | |
# Question 12(b):
| Answer/Working | Mark | Guidance |
|---|---|---|
| Attempts $2\theta - 30° = -19.47°$ | M1 | Sets $2\theta-30°=$ their $-19.47°$ |
| $\Rightarrow \theta = 5.26°$ | A1ft | Follow through on their $-19.47°$ |
---\begin{enumerate}
\item (a) Solve, for $- 180 ^ { \circ } \leqslant x < 180 ^ { \circ }$, the equation
\end{enumerate}
$$3 \sin ^ { 2 } x + \sin x + 8 = 9 \cos ^ { 2 } x$$
giving your answers to 2 decimal places.\\
(b) Hence find the smallest positive solution of the equation
$$3 \sin ^ { 2 } \left( 2 \theta - 30 ^ { \circ } \right) + \sin \left( 2 \theta - 30 ^ { \circ } \right) + 8 = 9 \cos ^ { 2 } \left( 2 \theta - 30 ^ { \circ } \right)$$
giving your answer to 2 decimal places.
\hfill \mbox{\textit{Edexcel Paper 2 Q12 [8]}}