| Exam Board | AQA |
|---|---|
| Module | C3 (Core Mathematics 3) |
| Year | 2006 |
| 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. All steps are standard textbook exercises with no novel problem-solving required, making it slightly easier than average. |
| 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 |
|---|---|---|
| \(\sec x = 5\); \(\cos x = 0.2\); \(x = 1.37, 4.91\) AWRT | M1 A1 A1 | 3 marks total |
| Answer | Marks | Guidance |
|---|---|---|
| \(\tan^2 x = 3\sec x + 9\); \(\sec^2 x - 1 = 3\sec x + 9\); \(\sec^2 x - 3\sec x - 10 = 0\) | M1 A1 | for using \(\sec^2 x = 1 + \tan^2 x\) OE; AG; 2 marks total |
| Answer | Marks | Guidance |
|---|---|---|
| \((\sec x - 5)(\sec x + 2) = 0\); \(\sec x = 5, -2\); \(\cos x = 0.2, -0.5\); \(x = 1.37, 4.91, 2.09, 4.19\) | M1 A1 B1F A1 | or use of formula (attempt); any 2 correct or ft their 2 answers in (a); all 4 correct, no extras; 4 marks total |
### 3(a)
$\sec x = 5$; $\cos x = 0.2$; $x = 1.37, 4.91$ AWRT | M1 A1 A1 | 3 marks total
### 3(b)
$\tan^2 x = 3\sec x + 9$; $\sec^2 x - 1 = 3\sec x + 9$; $\sec^2 x - 3\sec x - 10 = 0$ | M1 A1 | for using $\sec^2 x = 1 + \tan^2 x$ OE; AG; 2 marks total
### 3(c)
$(\sec x - 5)(\sec x + 2) = 0$; $\sec x = 5, -2$; $\cos x = 0.2, -0.5$; $x = 1.37, 4.91, 2.09, 4.19$ | M1 A1 B1F A1 | or use of formula (attempt); any 2 correct or ft their 2 answers in (a); all 4 correct, no extras; 4 marks total3
\begin{enumerate}[label=(\alph*)]
\item Solve the equation $\sec x = 5$, giving all the values of $x$ in the interval $0 \leqslant x \leqslant 2 \pi$ in radians to two decimal places.
\item Show that the equation $\tan ^ { 2 } x = 3 \sec x + 9$ can be written as
$$\sec ^ { 2 } x - 3 \sec x - 10 = 0$$
\item Solve the equation $\tan ^ { 2 } x = 3 \sec x + 9$, giving all the 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 2006 Q3 [9]}}