| Exam Board | OCR |
|---|---|
| Module | H240/03 (Pure Mathematics and Mechanics) |
| Year | 2022 |
| Session | June |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Topic | Trig Proofs |
| Type | Prove trigonometric identity |
| Difficulty | Standard +0.8 This question requires multiple sophisticated steps: converting a trigonometric equation using sec θ = 1/cos θ, multiplying through by cos θ, then using tan θ = sin θ/cos θ to reach the quadratic form. Part (b) demands understanding that 'only one solution' means the discriminant equals zero, solving for m, then applying the constraint that m is a negative integer. This combines algebraic manipulation, trigonometric identities, discriminant analysis, and constraint satisfaction—significantly above average difficulty but not requiring truly novel insight. |
| Spec | 1.02d Quadratic functions: graphs and discriminant conditions1.05j Trigonometric identities: tan=sin/cos and sin^2+cos^2=11.05k Further identities: sec^2=1+tan^2 and cosec^2=1+cot^21.05o Trigonometric equations: solve in given intervals |
\textbf{In this question you must show detailed reasoning.}
\begin{enumerate}[label=(\alph*)]
\item Show that the equation $m \sec \theta + 3 \cos \theta = 4 \sin \theta$ can be expressed in the form
$$m \tan^2 \theta - 4 \tan \theta + (m + 3) = 0.$$ [3]
\item It is given that there is only one value of $\theta$, for $0 < \theta < \pi$, satisfying the equation $m \sec \theta + 3 \cos \theta = 4 \sin \theta$.
Given also that $m$ is a negative integer, find this value of $\theta$, correct to 3 significant figures. [5]
\end{enumerate}
\hfill \mbox{\textit{OCR H240/03 2022 Q7 [8]}}