| Exam Board | AQA |
|---|---|
| Module | C3 (Core Mathematics 3) |
| Year | 2010 |
| 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 cosec to sin and solve. Part (b) requires knowing the identity cot²x = cosec²x - 1, substituting to form a quadratic in cosec x, then solving. While it involves multiple steps and the identity, this is a standard C3 textbook exercise with no novel insight required—slightly easier than average due to the routine nature of the technique. |
| 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 | Mark | Guidance |
| \(\sin x = \frac{1}{3}\), or sight of \(\pm 0.34, \pm 0.11\pi\) or \(\pm 19.47\) (or better) | M1 | |
| \(x = 0.34, 2.8(0)\) | A1 | Penalise if incorrect answers in range; ignore answers outside range |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(\cosec^2 x - 1 = 11 - \cosec x\) | M1 | Correct use of \(\cot^2 x = \cosec^2 x - 1\) |
| \(\cosec^2 x + \cosec x - 12(=0)\) | A1 | |
| \((\cosec x + 4)(\cosec x - 3)(=0)\) | m1 | Attempt at factors; gives \(\cosec x\) or \(-12\) when expanded; formula one error condoned |
| \(\cosec x = -4, 3\) and \(\sin x = -\frac{1}{4}, \frac{1}{3}\) | A1 | Either line |
| \(\sin x = -\frac{1}{4} \Rightarrow x = 3.39, 6.03\) | B1F | 3 correct or their two answers from (a) and 3.39, 6.03 |
| \(0.34, 2.8(0)\) | B1 | 4 correct and no extras in range; ignore answers outside range. SC 19.47, 160.53, 194.48, 345.52 B1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(\frac{\cos^2 x}{\sin^2 x} = 11 - \frac{1}{\sin x}\), so \(\cos^2 x = 11\sin^2 x - \sin x\) | (M1) | Correct use of trig ratios and multiplying by \(\sin^2 x\) |
| \(1 - \sin^2 x = 11\sin^2 x - \sin x\), so \(0 = 12\sin^2 x - \sin x - 1\) | (A1) | |
| \(0 = (4\sin x + 1)(3\sin x - 1)\) | (m1) | Attempt at factors as above |
| \(\sin x = -\frac{1}{4}, \frac{1}{3}\) | (A1) | |
| Further answers as above | (B1F)(B1) | As above |
## Question 3(a):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $\sin x = \frac{1}{3}$, or sight of $\pm 0.34, \pm 0.11\pi$ or $\pm 19.47$ (or better) | M1 | |
| $x = 0.34, 2.8(0)$ | A1 | Penalise if incorrect answers in range; ignore answers outside range |
## Question 3(b):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $\cosec^2 x - 1 = 11 - \cosec x$ | M1 | Correct use of $\cot^2 x = \cosec^2 x - 1$ |
| $\cosec^2 x + \cosec x - 12(=0)$ | A1 | |
| $(\cosec x + 4)(\cosec x - 3)(=0)$ | m1 | Attempt at factors; gives $\cosec x$ or $-12$ when expanded; formula one error condoned |
| $\cosec x = -4, 3$ and $\sin x = -\frac{1}{4}, \frac{1}{3}$ | A1 | Either line |
| $\sin x = -\frac{1}{4} \Rightarrow x = 3.39, 6.03$ | B1F | 3 correct or their two answers from (a) and 3.39, 6.03 |
| $0.34, 2.8(0)$ | B1 | 4 correct and no extras in range; ignore answers outside range. SC 19.47, 160.53, 194.48, 345.52 B1 |
**Alternative:**
| Answer/Working | Mark | Guidance |
|---|---|---|
| $\frac{\cos^2 x}{\sin^2 x} = 11 - \frac{1}{\sin x}$, so $\cos^2 x = 11\sin^2 x - \sin x$ | (M1) | Correct use of trig ratios and multiplying by $\sin^2 x$ |
| $1 - \sin^2 x = 11\sin^2 x - \sin x$, so $0 = 12\sin^2 x - \sin x - 1$ | (A1) | |
| $0 = (4\sin x + 1)(3\sin x - 1)$ | (m1) | Attempt at factors as above |
| $\sin x = -\frac{1}{4}, \frac{1}{3}$ | (A1) | |
| Further answers as above | (B1F)(B1) | As above |
---3
\begin{enumerate}[label=(\alph*)]
\item Solve the equation
$$\operatorname { cosec } x = 3$$
giving all values of $x$ in radians to two decimal places, in the interval $0 \leqslant x \leqslant 2 \pi$.\\
(2 marks)
\item By using a suitable trigonometric identity, solve the equation
$$\cot ^ { 2 } x = 11 - \operatorname { cosec } x$$
giving all values of $x$ in radians to two decimal places, in the interval $0 \leqslant x \leqslant 2 \pi$.\\
(6 marks)
\end{enumerate}
\hfill \mbox{\textit{AQA C3 2010 Q3 [8]}}