| Exam Board | AQA |
|---|---|
| Module | C3 (Core Mathematics 3) |
| Year | 2006 |
| Session | January |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Quadratic trigonometric equations |
| Type | Show then solve: secant/cosecant/cotangent identities |
| Difficulty | Moderate -0.3 This is a structured, guided trigonometric equation question that uses the standard identity cosec²x = 1 + cot²x. Parts (a) and (b) provide scaffolding through the algebraic manipulation and quadratic factorization, while part (c) requires finding solutions in a given interval. The techniques are all standard C3 material with clear signposting, making it slightly easier than average but still requiring competent execution of multiple steps. |
| 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 |
| \(2\cosec^2 x = 5(1-\cot x)\) | ||
| \(2 + 2\cot^2 x = 5 - 5\cot x\) | M1 | Use of \(\cosec^2 x = 1+\cot^2 x\) |
| \(2\cot^2 x + 5\cot x - 3 = 0\) | A1 | AG |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \((2\cot x - 1)(\cot x + 3) = 0\) | M1 | or \(2+5t-3t^2=0\) or in \(\tan x\): \((2-t)(1+3t)=0\) |
| \(\cot x = \frac{1}{2},\ -3\) | ||
| \(\tan x = 2,\ -\frac{1}{3}\) | A1 | AG |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(x = 1.1,\ -2.0\) | B1 | Any 2 correct — In degrees: B0 |
| \(x = -0.3,\ 2.8\) | B1 | Any 3 correct — B1 |
| AWRT | B1 | 4 correct — B2 |
## Question 4:
### Part (a)
| Answer/Working | Marks | Guidance |
|---|---|---|
| $2\cosec^2 x = 5(1-\cot x)$ | | |
| $2 + 2\cot^2 x = 5 - 5\cot x$ | M1 | Use of $\cosec^2 x = 1+\cot^2 x$ |
| $2\cot^2 x + 5\cot x - 3 = 0$ | A1 | AG |
### Part (b)
| Answer/Working | Marks | Guidance |
|---|---|---|
| $(2\cot x - 1)(\cot x + 3) = 0$ | M1 | or $2+5t-3t^2=0$ or in $\tan x$: $(2-t)(1+3t)=0$ |
| $\cot x = \frac{1}{2},\ -3$ | | |
| $\tan x = 2,\ -\frac{1}{3}$ | A1 | AG |
### Part (c)
| Answer/Working | Marks | Guidance |
|---|---|---|
| $x = 1.1,\ -2.0$ | B1 | Any 2 correct — In degrees: B0 |
| $x = -0.3,\ 2.8$ | B1 | Any 3 correct — B1 |
| AWRT | B1 | 4 correct — B2 |
---4 It is given that $2 \operatorname { cosec } ^ { 2 } x = 5 - 5 \cot x$.
\begin{enumerate}[label=(\alph*)]
\item Show that the equation $2 \operatorname { cosec } ^ { 2 } x = 5 - 5 \cot x$ can be written in the form
$$2 \cot ^ { 2 } x + 5 \cot x - 3 = 0$$
\item Hence show that $\tan x = 2$ or $\tan x = - \frac { 1 } { 3 }$.
\item Hence, or otherwise, solve the equation $2 \operatorname { cosec } ^ { 2 } x = 5 - 5 \cot x$, giving all values of $x$ in radians to one decimal place in the interval $- \pi < x \leqslant \pi$.
\end{enumerate}
\hfill \mbox{\textit{AQA C3 2006 Q4 [7]}}