| Exam Board | Pre-U |
|---|---|
| Module | Pre-U 9794/2 (Pre-U Mathematics Paper 2) |
| Year | 2011 |
| Session | June |
| Marks | 9 |
| Topic | Standard trigonometric equations |
| Type | Equation with non-equation preliminary part (sketch/proof/identity) |
| Difficulty | Standard +0.3 This question requires sketching sec and cos graphs (straightforward with knowledge of asymptotes and key values) and solving a trigonometric equation that reduces to a quadratic in cos x. The algebraic manipulation is standard (multiply by cos x, rearrange to 3cos²x + cos x - 2 = 0), and factorization or the quadratic formula yields solutions. While it requires multiple techniques, each step is routine for A-level students, making it slightly easier than average. |
| Spec | 1.05f Trigonometric function graphs: symmetries and periodicities1.05h Reciprocal trig functions: sec, cosec, cot definitions and graphs1.05o Trigonometric equations: solve in given intervals |
| Answer | Marks | Guidance |
|---|---|---|
| Shape of each graph (concavity) | B1 B1 | |
| Asymptote at \(\frac{\pi}{2}\) | B1 | |
| Max/Min points clearly indicated at \(x = 0\) and \(\pi\) | B1 | [4] |
| Answer | Marks | Guidance |
|---|---|---|
| Evidence that \(\sec x = \frac{1}{\cos x}\) | B1 | |
| Multiply by \(\cos x\), obtaining a quadratic | M1 | |
| Solve quadratic | M1 | |
| Solutions \(x = \pi\) | A1 | |
| and \(x = 0.841\) | A1 | [5] |
**Part (i)**
Shape of each graph (concavity) | B1 B1
Asymptote at $\frac{\pi}{2}$ | B1
Max/Min points clearly indicated at $x = 0$ and $\pi$ | B1 | [4]
**Part (ii)**
Evidence that $\sec x = \frac{1}{\cos x}$ | B1
Multiply by $\cos x$, obtaining a quadratic | M1
Solve quadratic | M1
Solutions $x = \pi$ | A1
and $x = 0.841$ | A1 | [5]
*SC: For either both in degrees or one in degrees and one in radians – A1A0*\begin{enumerate}[label=(\roman*)]
\item On the same diagram, sketch the graphs of $y = 2 \sec x$ and $y = 1 + 3 \cos x$, for $0 \leqslant x \leqslant \pi$. [4]
\item Solve the equation $2 \sec x = 1 + 3 \cos x$, where $0 \leqslant x \leqslant \pi$. [5]
\end{enumerate}
\hfill \mbox{\textit{Pre-U Pre-U 9794/2 2011 Q4 [9]}}