| Exam Board | AQA |
|---|---|
| Module | C3 (Core Mathematics 3) |
| Year | 2010 |
| Session | June |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Reciprocal Trig & Identities |
| Type | Sketch reciprocal function graphs |
| Difficulty | Moderate -0.3 This question tests standard knowledge of reciprocal trig functions with routine transformations. Part (a)(i) is trivial recall (sec 0° = 1), part (a)(ii) requires sketching a transformed graph using known techniques (period halving and absolute value), and parts (b)-(c) involve solving sec equations by converting to cos, which is a standard textbook exercise. While multi-part, each component is straightforward application of learned techniques without requiring problem-solving insight. |
| Spec | 1.02w Graph transformations: simple transformations of f(x)1.05h Reciprocal trig functions: sec, cosec, cot definitions and graphs1.05o Trigonometric equations: solve in given intervals |
| Answer | Marks |
|---|---|
| \(y = 1\) | B1 |
| Answer | Marks | Guidance |
|---|---|---|
| Graph of \( | \sec 2x | \): period \(90°\), correct shape with cusps on \(x\)-axis, amplitude correct, all correct |
| Answer | Marks | Guidance |
|---|---|---|
| \(\sec x = 2 \Rightarrow \cos x = \frac{1}{2} \Rightarrow x = 60°, 300°\) | M1 A1 | M1 for \(\cos x = \frac{1}{2}\) |
| Answer | Marks | Guidance |
|---|---|---|
| \( | \sec(2x-10°) | = 2 \Rightarrow \sec(2x-10°) = \pm 2 \Rightarrow \cos(2x-10°) = \pm\frac{1}{2}\) |
| \(2x - 10° = 60°, 120°, 240°, 300°\) (and negatives) | M1 | |
| \(x = 35°, 65°, 125°, 155°\) | A1 A1 | A1 for each correct pair |
# Question 2:
## Part (a)(i)
| $y = 1$ | B1 | |
## Part (a)(ii)
| Graph of $|\sec 2x|$: period $90°$, correct shape with cusps on $x$-axis, amplitude correct, all correct | B1 B1 B1 | B1 shape/period, B1 cusps touching axis, B1 fully correct |
## Part (b)
| $\sec x = 2 \Rightarrow \cos x = \frac{1}{2} \Rightarrow x = 60°, 300°$ | M1 A1 | M1 for $\cos x = \frac{1}{2}$ |
## Part (c)
| $|\sec(2x-10°)| = 2 \Rightarrow \sec(2x-10°) = \pm 2 \Rightarrow \cos(2x-10°) = \pm\frac{1}{2}$ | M1 | |
| $2x - 10° = 60°, 120°, 240°, 300°$ (and negatives) | M1 | |
| $x = 35°, 65°, 125°, 155°$ | A1 A1 | A1 for each correct pair |
---2
\begin{enumerate}[label=(\alph*)]
\item The diagram shows the graph of $y = \sec x$ for $0 ^ { \circ } \leqslant x \leqslant 360 ^ { \circ }$.\\
\begin{tikzpicture}[>=stealth, scale=1]
% --- Clipping region ---
\clip (-0.8,-4.5) rectangle (10.8,4.8);
% --- Axes ---
\draw[->] (-0.5,0) -- (10.5,0) node[right] {$x$};
\draw[->] (0,-4.3) -- (0,4.5) node[above] {$y$};
\node[below left] at (0,0) {$O$};
% Scaling: 90^\circ = 2.5 units on x-axis
% x-ticks
\draw (2.5,0.1) -- (2.5,-0.1) node[below] {$90^\circ$};
\draw (5,0.1) -- (5,-0.1) node[below] {$180^\circ$};
\draw (7.5,0.1) -- (7.5,-0.1) node[below] {$270^\circ$};
\draw (10,0.1) -- (10,-0.1) node[below] {$360^\circ$};
% y-tick at A = 1
\draw (0.1,1) -- (-0.1,1) node[left] {$A$};
% --- Dashed asymptotes at 90^\circ and 270^\circ ---
\draw[dashed, thick, gray] (2.5,-4.3) -- (2.5,4.5);
\draw[dashed, thick, gray] (7.5,-4.3) -- (7.5,4.5);
% --- y = sec(x) ---
% Branch 1: 0^\circ to ~88^\circ (sec > 0, going to +inf)
\draw[thick, domain=0:88, samples=100]
plot ({2.5*\x/90}, {1/cos(\x)});
% Branch 2: ~92^\circ to ~268^\circ (sec < 0 in the middle, U-shape below)
\draw[thick, domain=92:268, samples=150]
plot ({2.5*\x/90}, {1/cos(\x)});
% Branch 3: ~272^\circ to 360^\circ (sec > 0, coming from +inf)
\draw[thick, domain=272:360, samples=100]
plot ({2.5*\x/90}, {1/cos(\x)});
\end{tikzpicture}
\begin{enumerate}[label=(\roman*)]
\item The point $A$ on the curve is where $x = 0$. State the $y$-coordinate of $A$.
\item Sketch, on the axes given on page 3, the graph of $y = | \sec 2 x |$ for $0 ^ { \circ } \leqslant x \leqslant 360 ^ { \circ }$.
\end{enumerate}\item Solve the equation $\sec x = 2$, giving all values of $x$ in degrees in the interval $0 ^ { \circ } \leqslant x \leqslant 360 ^ { \circ }$.
\item Solve the equation $\left| \sec \left( 2 x - 10 ^ { \circ } \right) \right| = 2$, giving all values of $x$ in degrees in the interval $0 ^ { \circ } \leqslant x \leqslant 180 ^ { \circ }$.
\end{enumerate}
\hfill \mbox{\textit{AQA C3 2010 Q2 [10]}}