Standard +0.2 This is a straightforward C2 question testing standard skills: exact trig values (routine recall), verification by substitution (mechanical), sketching basic trig graphs to find solutions graphically (standard technique), and trapezium rule application (algorithmic). All parts are textbook exercises requiring no problem-solving insight, making it slightly easier than average.
Write down the exact values of \(\cos \frac{1}{6}\pi\) and \(\tan \frac{1}{6}\pi\) (where the angles are in radians). Hence verify that \(x = \frac{1}{6}\pi\) is a solution of the equation
$$2 \cos x = \tan 2x.$$ [3]
Sketch, on a single diagram, the graphs of \(y = 2 \cos x\) and \(y = \tan 2x\), for \(x\) (radians) such that \(0 \leqslant x \leqslant \pi\). Hence state, in terms of \(\pi\), the other values of \(x\) between 0 and \(\pi\) satisfying the equation
$$2 \cos x = \tan 2x.$$ [4]
Use the trapezium rule, with 3 strips, to find an approximate value for the area of the region bounded by the curve \(y = \tan x\), the \(x\)-axis, and the lines \(x = 0.1\) and \(x = 0.4\). (Values of \(x\) are in radians.) [4]
State with a reason whether this approximation is an underestimate or an overestimate. [1]
\begin{enumerate}[label=(\alph*)]
\item \begin{enumerate}[label=(\roman*)]
\item Write down the exact values of $\cos \frac{1}{6}\pi$ and $\tan \frac{1}{6}\pi$ (where the angles are in radians). Hence verify that $x = \frac{1}{6}\pi$ is a solution of the equation
$$2 \cos x = \tan 2x.$$ [3]
\item Sketch, on a single diagram, the graphs of $y = 2 \cos x$ and $y = \tan 2x$, for $x$ (radians) such that $0 \leqslant x \leqslant \pi$. Hence state, in terms of $\pi$, the other values of $x$ between 0 and $\pi$ satisfying the equation
$$2 \cos x = \tan 2x.$$ [4]
\end{enumerate}
\item \begin{enumerate}[label=(\roman*)]
\item Use the trapezium rule, with 3 strips, to find an approximate value for the area of the region bounded by the curve $y = \tan x$, the $x$-axis, and the lines $x = 0.1$ and $x = 0.4$. (Values of $x$ are in radians.) [4]
\item State with a reason whether this approximation is an underestimate or an overestimate. [1]
\end{enumerate}
\end{enumerate}
\hfill \mbox{\textit{OCR C2 Q9 [12]}}