| Exam Board | AQA |
|---|---|
| Module | C3 (Core Mathematics 3) |
| Year | 2009 |
| Session | January |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Trigonometric equations in context |
| Type | Reduce to quadratic in trig |
| Difficulty | Moderate -0.3 Part (a) is straightforward recall (convert sec to cos, solve basic equation). Part (b) requires knowing the identity tan²x = sec²x - 1 to form a quadratic in sec x, then solving - this is a standard C3 technique but requires multiple steps and careful handling of the quadratic. Overall slightly easier than average due to being a well-practiced question type with clear methodology. |
| Spec | 1.05h Reciprocal trig functions: sec, cosec, cot definitions and graphs1.05k Further identities: sec^2=1+tan^2 and cosec^2=1+cot^21.05o Trigonometric equations: solve in given intervals |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(\sec x = \frac{3}{2} \Rightarrow \cos x = \frac{2}{3}\); \(x = 48, 312\) | B1 | 1 correct |
| Both correct and no extras in interval | B1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(2\tan^2 x = 10 - 5\sec x \Rightarrow 2(\sec^2 x - 1) = 10 - 5\sec x\) | M1 | Use of trig identity correctly |
| \(2\sec^2 x + 5\sec x - 12 = 0\) | A1 | |
| \((2\sec x - 3)(\sec x + 4) = 0\) | m1 | Attempt to solve or factorise; 1 slip using formula |
| \(\sec x = \frac{3}{2},\ -4\) or \(\cos x = \frac{2}{3},\ -\frac{1}{4}\) | A1 | Either of these |
| \(x = 48, 312, 104, 256\) | B1 | AWRT 3 correct; condone 105 or 255 |
| All correct and no extras in interval | B1 |
## Question 4:
### Part (a):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $\sec x = \frac{3}{2} \Rightarrow \cos x = \frac{2}{3}$; $x = 48, 312$ | B1 | 1 correct |
| Both correct and no extras in interval | B1 | |
### Part (b):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $2\tan^2 x = 10 - 5\sec x \Rightarrow 2(\sec^2 x - 1) = 10 - 5\sec x$ | M1 | Use of trig identity correctly |
| $2\sec^2 x + 5\sec x - 12 = 0$ | A1 | |
| $(2\sec x - 3)(\sec x + 4) = 0$ | m1 | Attempt to solve or factorise; 1 slip using formula |
| $\sec x = \frac{3}{2},\ -4$ or $\cos x = \frac{2}{3},\ -\frac{1}{4}$ | A1 | Either of these |
| $x = 48, 312, 104, 256$ | B1 | AWRT 3 correct; condone 105 or 255 |
| All correct and no extras in interval | B1 | |
**Total: 8 marks**
---4
\begin{enumerate}[label=(\alph*)]
\item Solve the equation $\sec x = \frac { 3 } { 2 }$, giving all values of $x$ to the nearest degree in the interval $0 ^ { \circ } < x < 360 ^ { \circ }$.
\item By using a suitable trigonometrical identity, solve the equation
$$2 \tan ^ { 2 } x = 10 - 5 \sec x$$
giving all values of $x$ to the nearest degree in the interval $0 ^ { \circ } < x < 360 ^ { \circ }$.
\end{enumerate}
\hfill \mbox{\textit{AQA C3 2009 Q4 [8]}}