| Exam Board | Pre-U |
|---|---|
| Module | Pre-U 9794/1 (Pre-U Mathematics Paper 1) |
| Year | 2019 |
| Session | Specimen |
| Marks | 4 |
| Topic | Standard trigonometric equations |
| Type | Prove identity then solve |
| Difficulty | Standard +0.3 Part (a) is a standard trigonometric identity proof requiring algebraic manipulation with common denominators. Part (b) applies the proven identity with a compound angle substitution and solves a linear equation in cosec, requiring knowledge of the cosec graph but following a clear path once the identity is recognized. This is slightly easier than average due to the direct 'hence' connection and straightforward final equation. |
| Spec | 1.05j Trigonometric identities: tan=sin/cos and sin^2+cos^2=11.05o Trigonometric equations: solve in given intervals1.05p Proof involving trig: functions and identities |
(a) Dealing with cot — **B1**
Adding fractions in terms of sin and cos — **M1**
Use of $\cos^2 + \sin^2$ — **M1**
Simplification to given answer — **A1**
**Total: 4**
(b) Substituting $\text{cosec}\!\left(\theta + \dfrac{\pi}{4}\right)$ — **M1**
Converting equation in sin — **M1**
$\theta + \dfrac{\pi}{4} = 0.4115,\ 2.730,\ 6.695$ — **M1**
$\theta = 1.94,\ 5.91$ — **A1**
**Total: 4**10
\begin{enumerate}[label=(\alph*)]
\item Prove that $\cot \theta + \frac { \sin \theta } { 1 + \cos \theta } = \operatorname { cosec } \theta$.
\item Hence solve the equation $\cot \left( \theta + \frac { \neq } { 4 } \right) + \frac { \sin \left( \theta + \frac { \neq } { 4 } \right) } { 1 + \cos \left( \theta + \frac { \neq } { 4 } \right) } = \frac { 5 } { 2 }$ for $0 \leqslant \theta \leqslant 2 \pi$.
\end{enumerate}
\hfill \mbox{\textit{Pre-U Pre-U 9794/1 2019 Q10 [4]}}