| Exam Board | AQA |
|---|---|
| Module | C3 (Core Mathematics 3) |
| Year | 2008 |
| Session | June |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Reciprocal Trig & Identities |
| Type | Solve equation using Pythagorean identities |
| Difficulty | Moderate -0.3 This is a straightforward multi-part question on reciprocal trig functions and Pythagorean identities. Part (a) is basic recall (sec x = 1/cos x), part (b) is a guided algebraic manipulation using tan²x = sec²x - 1, and part (c) applies the quadratic formula to solve. The question requires standard techniques with clear scaffolding, making it slightly easier than average for C3 level. |
| Spec | 1.05h Reciprocal trig functions: sec, cosec, cot definitions and graphs1.05j Trigonometric identities: tan=sin/cos and sin^2+cos^2=11.05o Trigonometric equations: solve in given intervals |
| Answer | Marks | Guidance |
|---|---|---|
| \(x = \cos^{-1}\frac{1}{3} = 1.23, 5.05\) (0.39π, 1.61π) | M1, A1, A1 | PI; AWRT (−1 for each error in range); SC 70.53, 289.47 B1 |
| Answer | Marks | Guidance |
|---|---|---|
| \(\sec^2 x - 1 = 2\sec x + 2\); \(\sec^2 x - 2\sec x - 3 = 0\) | M1, A1 | Use of \(\sec^2 x = 1 + \tan^2 x\); AG; CSO |
| Answer | Marks | Guidance |
|---|---|---|
| \(\sec^2 x - 2\sec x - 3 = 0\); \((\sec x - 3)(\sec x + 1) = 0\); \(\cos x = \frac{1}{3}\) or \(-1\); \(x = 1.23, 5.05, 3.14\) (π) | M1, A1, B1f, B1 | Attempt to solve; AG; CSO; (2 answers in range from (a)) AWRT all correct and no extras in range; SC 70.53, 289.47, 180 B1 |
**2(a)**
| $x = \cos^{-1}\frac{1}{3} = 1.23, 5.05$ (0.39π, 1.61π) | M1, A1, A1 | PI; AWRT (−1 for each error in range); SC 70.53, 289.47 B1 |
**2(b)**
| $\sec^2 x - 1 = 2\sec x + 2$; $\sec^2 x - 2\sec x - 3 = 0$ | M1, A1 | Use of $\sec^2 x = 1 + \tan^2 x$; AG; CSO |
**2(c)**
| $\sec^2 x - 2\sec x - 3 = 0$; $(\sec x - 3)(\sec x + 1) = 0$; $\cos x = \frac{1}{3}$ or $-1$; $x = 1.23, 5.05, 3.14$ (π) | M1, A1, B1f, B1 | Attempt to solve; AG; CSO; (2 answers in range from (a)) AWRT all correct and no extras in range; SC 70.53, 289.47, 180 B1 |
**Total: 9 marks**
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\begin{enumerate}[label=(\alph*)]
\item Solve the equation $\sec x = 3$, giving the values of $x$ in radians to two decimal places in the interval $0 \leqslant x < 2 \pi$.
\item Show that the equation $\tan ^ { 2 } x = 2 \sec x + 2$ can be written as $\sec ^ { 2 } x - 2 \sec x - 3 = 0$.
\item Solve the equation $\tan ^ { 2 } x = 2 \sec x + 2$, giving the values of $x$ in radians to two decimal places in the interval $0 \leqslant x < 2 \pi$.
\end{enumerate}
\hfill \mbox{\textit{AQA C3 2008 Q2 [9]}}