| Exam Board | Edexcel |
|---|---|
| Module | AS Paper 1 (AS Paper 1) |
| Session | Specimen |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Trigonometric equations in context |
| Type | Identify student error in trig solution |
| Difficulty | Standard +0.3 Part (i) is a routine inverse sine calculation with a linear transformation requiring careful attention to range. Part (ii)(a) tests understanding of common algebraic errors (dividing by sin x loses x=0 solution, and degree/radian confusion). Part (ii)(b) applies the corrected method to a transformed equation. This is slightly easier than average as it's mostly procedural with standard techniques, though the error-spotting adds modest conceptual demand. |
| Spec | 1.05j Trigonometric identities: tan=sin/cos and sin^2+cos^2=11.05o Trigonometric equations: solve in given intervals |
| Answer | Marks | Guidance |
|---|---|---|
| Working | Mark | Guidance |
| \((2\theta+10°) = \arcsin(-0.6)\) | M1 | Attempts \(\arcsin(-0.6)\); implied by any correct answer |
| \((2\theta+10°) = -143.13°, -36.87°, 216.87°, 323.13°\) (any two) | A1 | Any two correct values |
| Correct order to find \(\theta = \cdots\) | dM1 | Correct method to find any value of \(\theta\) |
| Two of \(\theta = -76.6°, -23.4°, 103.4°, 156.6°\) | A1 | Any two correct values of \(\theta\) |
| \(\theta = -76.6°, -23.4°, 103.4°, 156.6°\) only | A1 | All four correct and no extras |
| Answer | Marks | Guidance |
|---|---|---|
| Working | Mark | Guidance |
| Explains student has not considered negative value \(x(=-29.0°)\) when solving \(\cos x = \frac{7}{8}\) | B1 | See scheme |
| Explains student should check if solutions of \(\sin x = 0\) (cancelled term) are solutions of given equation; \(x=0°\) should have been included | B1 | See scheme |
| Answer | Marks | Guidance |
|---|---|---|
| Working | Mark | Guidance |
| Attempts \(4\alpha + 199° = (360-29.0)°\) | M1 | Deduces smallest positive solution occurs when \(4\alpha+199°=(360-29.0)°\) |
| \(\alpha = 33.0°\) | A1 | Correct answer |
## Question 11:
### Part (i):
| Working | Mark | Guidance |
|---------|------|----------|
| $(2\theta+10°) = \arcsin(-0.6)$ | M1 | Attempts $\arcsin(-0.6)$; implied by any correct answer |
| $(2\theta+10°) = -143.13°, -36.87°, 216.87°, 323.13°$ (any two) | A1 | Any two correct values |
| Correct order to find $\theta = \cdots$ | dM1 | Correct method to find any value of $\theta$ |
| Two of $\theta = -76.6°, -23.4°, 103.4°, 156.6°$ | A1 | Any two correct values of $\theta$ |
| $\theta = -76.6°, -23.4°, 103.4°, 156.6°$ only | A1 | All four correct and no extras |
### Part (ii)(a):
| Working | Mark | Guidance |
|---------|------|----------|
| Explains student has not considered negative value $x(=-29.0°)$ when solving $\cos x = \frac{7}{8}$ | B1 | See scheme |
| Explains student should check if solutions of $\sin x = 0$ (cancelled term) are solutions of given equation; $x=0°$ should have been included | B1 | See scheme |
### Part (ii)(b):
| Working | Mark | Guidance |
|---------|------|----------|
| Attempts $4\alpha + 199° = (360-29.0)°$ | M1 | Deduces smallest positive solution occurs when $4\alpha+199°=(360-29.0)°$ |
| $\alpha = 33.0°$ | A1 | Correct answer |\begin{enumerate}
\item (i) Solve, for $- 90 ^ { \circ } \leqslant \theta < 270 ^ { \circ }$, the equation,
\end{enumerate}
$$\sin \left( 2 \theta + 10 ^ { \circ } \right) = - 0.6$$
giving your answers to one decimal place.\\
(ii) (a) A student's attempt at the question\\
"Solve, for $- 90 ^ { \circ } < x < 90 ^ { \circ }$, the equation $7 \tan x = 8 \sin x$ " is set out below.
$$\begin{gathered}
7 \tan x = 8 \sin x \\
7 \times \frac { \sin x } { \cos x } = 8 \sin x \\
7 \sin x = 8 \sin x \cos x \\
7 = 8 \cos x \\
\cos x = \frac { 7 } { 8 } \\
x = 29.0 ^ { \circ } \text { (to } 3 \text { sf) }
\end{gathered}$$
Identify two mistakes made by this student, giving a brief explanation of each mistake.\\
(b) Find the smallest positive solution to the equation
$$7 \tan \left( 4 \alpha + 199 ^ { \circ } \right) = 8 \sin \left( 4 \alpha + 199 ^ { \circ } \right)$$
\hfill \mbox{\textit{Edexcel AS Paper 1 Q11 [9]}}