| Exam Board | AQA |
|---|---|
| Module | C3 (Core Mathematics 3) |
| Year | 2008 |
| Session | January |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Reciprocal Trig & Identities |
| Type | Convert equation to quadratic form |
| Difficulty | Moderate -0.3 This is a straightforward multi-part question on reciprocal trig functions. Part (a) is basic calculator work, part (b) is a guided algebraic manipulation using the standard identity cosec²x = 1 + cot²x, and part (c) applies the quadratic formula then solves simple trig equations. All steps are routine with clear signposting, making it slightly easier than average. |
| 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 |
|---|---|---|
| \(\cot x = 2 \Rightarrow \tan x = 0.5\) | M1 A1 | AWRT; no others within range |
| \(x = 0.46, 3.61\) |
| Answer | Marks | Guidance |
|---|---|---|
| \(\cosec^2 x = \frac{3\cot x + 4}{2}\) | M1 | Correct use of \(\cosec^2 x = 1 + \cot^2 x\) |
| \(2(1 + \cot^2 x) = 3\cot x + 4\) | ||
| \((2\cot^2 x - 3\cot x + 2 - 4 = 0)\) | ||
| \(2\cot^2 x - 3\cot x - 2 = 0\) | A1 | AG; correct with no slips from line with no fractions |
| Answer | Marks | Guidance |
|---|---|---|
| \((2\cot x + 1)(\cot x - 2)(= 0)\) | M1 | Attempt to solve |
| \(\cot x = -\frac{1}{2}, 2\) | A1 | |
| \(\tan x = -2, 0.5\) | ||
| \(x = 0.46, 3.61, 2.03, 5.18\) | B1 B1 | 2 correct Allow 3.6(0) / 4 correct (with no extras in range) AWRT |
| SC Degrees: \(26.57°, 206.57°, 116.57°, 296.57°\) B1 for 2 correct |
### 2(a)
$\cot x = 2 \Rightarrow \tan x = 0.5$ | M1 A1 | AWRT; no others within range | **Total: 2**
$x = 0.46, 3.61$ |
### 2(b)
$\cosec^2 x = \frac{3\cot x + 4}{2}$ | M1 | Correct use of $\cosec^2 x = 1 + \cot^2 x$ |
$2(1 + \cot^2 x) = 3\cot x + 4$ | | |
$(2\cot^2 x - 3\cot x + 2 - 4 = 0)$ | | |
$2\cot^2 x - 3\cot x - 2 = 0$ | A1 | AG; correct with no slips from line with no fractions | **Total: 2**
### 2(c)
$(2\cot x + 1)(\cot x - 2)(= 0)$ | M1 | Attempt to solve |
$\cot x = -\frac{1}{2}, 2$ | A1 | |
$\tan x = -2, 0.5$ | | |
$x = 0.46, 3.61, 2.03, 5.18$ | B1 B1 | 2 correct Allow 3.6(0) / 4 correct (with no extras in range) AWRT | **Total: 4**
| | | **SC Degrees:** $26.57°, 206.57°, 116.57°, 296.57°$ B1 for 2 correct |2
\begin{enumerate}[label=(\alph*)]
\item Solve the equation $\cot x = 2$, giving all values of $x$ in the interval $0 \leqslant x \leqslant 2 \pi$ in radians to two decimal places.
\item Show that the equation $\operatorname { cosec } ^ { 2 } x = \frac { 3 \cot x + 4 } { 2 }$ can be written as
$$2 \cot ^ { 2 } x - 3 \cot x - 2 = 0$$
\item Solve the equation $\operatorname { cosec } ^ { 2 } x = \frac { 3 \cot x + 4 } { 2 }$, giving all values of $x$ in the interval $0 \leqslant x \leqslant 2 \pi$ in radians to two decimal places.
\end{enumerate}
\hfill \mbox{\textit{AQA C3 2008 Q2 [8]}}