| Exam Board | AQA |
|---|---|
| Module | C3 (Core Mathematics 3) |
| Year | 2016 |
| Session | June |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Reciprocal Trig & Identities |
| Type | Solve equation with reciprocal functions |
| Difficulty | Standard +0.3 This is a structured multi-part question testing standard reciprocal trig identities. Part (a)(i) uses the identity sec²x - tan²x = 1 to find sec x + tan x (routine algebraic manipulation), part (a)(ii) requires adding equations to find sec x then cos x (straightforward), and part (b) applies the result to a compound angle (simple substitution and solving). The question guides students through each step with clear scaffolding, making it slightly easier than average despite involving reciprocal functions. |
| Spec | 1.05h Reciprocal trig functions: sec, cosec, cot definitions and graphs1.05o Trigonometric equations: solve in given intervals |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \((\sec x + \tan x)(\sec x - \tan x) = \sec^2 x - \tan^2 x\) | M1 | Or correct use of \(\sec^2 x = 1 + \tan^2 x\) in a correct expression |
| \(-5(\sec x + \tan x) = 1\), \(\sec x + \tan x = -0.2\) | A1 | AG: no errors seen |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(2\sec x = -5.2\) or \(2\tan x = 4.8\) | M1 | Correctly reducing to a linear equation in one trig function |
| \(\sec x = -2.6\) | A1 | PI by correct value for \(\cos x\) |
| \(\cos x = -\frac{5}{13}\) | A1 | \(\frac{a}{b}\) where \(a, b\) are correct integers |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(\sec y = -2.6\), \([y =]\ [\pm]112.6°\) | B1 | AWRT \([\pm]112.6°\), PI by correct final answer |
| \(2x - 70 = [\pm]\) their \(y\) | M1 | |
| \(x = -21.3°,\ [-88.7°]\) | A1 | And no other extras in interval (ignore answers outside interval) |
# Question 9:
## Part (ai):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $(\sec x + \tan x)(\sec x - \tan x) = \sec^2 x - \tan^2 x$ | M1 | Or correct use of $\sec^2 x = 1 + \tan^2 x$ in a correct expression |
| $-5(\sec x + \tan x) = 1$, $\sec x + \tan x = -0.2$ | A1 | AG: no errors seen |
**Subtotal: 2 marks**
## Part (ii):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $2\sec x = -5.2$ or $2\tan x = 4.8$ | M1 | Correctly reducing to a linear equation in one trig function |
| $\sec x = -2.6$ | A1 | PI by correct value for $\cos x$ |
| $\cos x = -\frac{5}{13}$ | A1 | $\frac{a}{b}$ where $a, b$ are correct integers |
**Subtotal: 3 marks**
## Part (b):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $\sec y = -2.6$, $[y =]\ [\pm]112.6°$ | B1 | AWRT $[\pm]112.6°$, PI by correct final answer |
| $2x - 70 = [\pm]$ their $y$ | M1 | |
| $x = -21.3°,\ [-88.7°]$ | A1 | And no other extras in interval (ignore answers outside interval) |
**Subtotal: 3 marks**
**Total: 8 marks**9
\begin{enumerate}[label=(\alph*)]
\item It is given that $\sec x - \tan x = - 5$.
\begin{enumerate}[label=(\roman*)]
\item Show that $\sec x + \tan x = - 0.2$.
\item Hence find the exact value of $\cos x$.
\end{enumerate}\item Hence solve the equation
$$\sec \left( 2 x - 70 ^ { \circ } \right) - \tan \left( 2 x - 70 ^ { \circ } \right) = - 5$$
giving all values of $x$, to one decimal place, in the interval $- 90 ^ { \circ } \leqslant x \leqslant 90 ^ { \circ }$.\\[0pt]
[3 marks]
\section*{DO NOT WRITE ON THIS PAGE ANSWER IN THE SPACES PROVIDED}
\end{enumerate}
\hfill \mbox{\textit{AQA C3 2016 Q9 [8]}}