| Exam Board | OCR |
|---|---|
| Module | C2 (Core Mathematics 2) |
| Year | 2005 |
| Session | June |
| Marks | 12 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Trig Graphs & Exact Values |
| Type | State exact trig values at special angles |
| Difficulty | Standard +0.3 This is a straightforward C2 question testing standard skills: recalling exact trig values at special angles (π/6, π/3), verifying a solution by substitution, sketching basic trig graphs, applying the trapezium rule, and commenting on concavity. All parts are routine textbook exercises requiring no problem-solving insight, though slightly above average difficulty due to the multi-part nature and the double angle in tan(2x). |
| Spec | 1.05g Exact trigonometric values: for standard angles1.05o Trigonometric equations: solve in given intervals1.09f Trapezium rule: numerical integration |
| Answer | Marks | Guidance |
|---|---|---|
| (a) (i) \(\cos \frac{\pi}{6} = \pm\frac{\sqrt{3}}{2}\) | B1 | For any correct exact value |
| \(\tan \frac{\pi}{6} = \sqrt{3}\) | B1 | For any correct exact value |
| Hence \(2\cos\frac{\pi}{6} = 2 \times \frac{1}{2}\sqrt{3} = \tan\frac{\pi}{6}\) | B1 | For correct verification (allow via decimals) |
| (ii) | B1, B1 | For correct sketch of either \(y = \tan 2x\) or \(y = 2\cos x\); For second correct sketch, with both graphs in proportion (ie 3 points of intersection) |
| Other roots are \(\pi/2\) and \(5\pi/6\) | B1, B1 | For one of \(\pi/2\) or \(5\pi/6\) (or equiv in degrees); For second correct value, and no others in range |
| Answer | Marks | Guidance |
|---|---|---|
| (i) \(0.05(0.1003 + 2(0.2027 + 0.3093) + 0.4228) = 0.0774\) | M1, M1 | State at least three of \(\tan 0.1, \tan 0.2, \tan 0.3, \tan 0.4\); Substitute numerical values (must be attempt at y-coords, not x-coords) into correct trapezium rule, with \(h\) consistent with number of strips |
| A1 | Obtain \(0.05(\tan 0.1 + 2(\tan 0.2 + \tan 0.3) + \tan 0.4)\) or equiv in decimals (SC – award A1 if values are now decimals from using degrees – gives final answer of 0.00131) | |
| A1 | Obtain 0.077 or better | |
| (ii) Overestimate; tops of trapezia above the curve or equiv | B1 | For correct statement and justification |
**(a) (i)** $\cos \frac{\pi}{6} = \pm\frac{\sqrt{3}}{2}$ | B1 | For any correct exact value
$\tan \frac{\pi}{6} = \sqrt{3}$ | B1 | For any correct exact value
Hence $2\cos\frac{\pi}{6} = 2 \times \frac{1}{2}\sqrt{3} = \tan\frac{\pi}{6}$ | B1 | For correct verification (allow via decimals)
**(ii)** | B1, B1 | For correct sketch of either $y = \tan 2x$ or $y = 2\cos x$; For second correct sketch, with both graphs in proportion (ie 3 points of intersection)
Other roots are $\pi/2$ and $5\pi/6$ | B1, B1 | For one of $\pi/2$ or $5\pi/6$ (or equiv in degrees); For second correct value, and no others in range
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## Question 9(b):
**(i)** $0.05(0.1003 + 2(0.2027 + 0.3093) + 0.4228) = 0.0774$ | M1, M1 | State at least three of $\tan 0.1, \tan 0.2, \tan 0.3, \tan 0.4$; Substitute numerical values (must be attempt at y-coords, not x-coords) into correct trapezium rule, with $h$ consistent with number of strips
| A1 | Obtain $0.05(\tan 0.1 + 2(\tan 0.2 + \tan 0.3) + \tan 0.4)$ or equiv in decimals (SC – award A1 if values are now decimals from using degrees – gives final answer of 0.00131)
| A1 | Obtain 0.077 or better
**(ii)** Overestimate; tops of trapezia above the curve or equiv | B1 | For correct statement and justification
---9
\begin{enumerate}[label=(\alph*)]
\item \begin{enumerate}[label=(\roman*)]
\item Write down the exact values of $\cos \frac { 1 } { 6 } \pi$ and $\tan \frac { 1 } { 3 } \pi$ (where the angles are in radians). Hence verify that $x = \frac { 1 } { 6 } \pi$ is a solution of the equation
$$2 \cos x = \tan 2 x$$
\item Sketch, on a single diagram, the graphs of $y = 2 \cos x$ and $y = \tan 2 x$, 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
$$2 \cos x = \tan 2 x$$
\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.)
\item State with a reason whether this approximation is an underestimate or an overestimate.
\end{enumerate}\end{enumerate}
\hfill \mbox{\textit{OCR C2 2005 Q9 [12]}}