| Exam Board | OCR |
|---|---|
| Module | C3 (Core Mathematics 3) |
| Year | 2008 |
| Session | June |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Reciprocal Trig & Identities |
| Type | Given one function find others |
| Difficulty | Moderate -0.3 Part (a) requires recall of the double angle formula for tan and solving a resulting quadratic, which is standard C3 material. Part (b) involves direct application of reciprocal trig definitions and Pythagorean identity - straightforward bookwork with minimal problem-solving. This is slightly easier than a typical C3 question due to the routine nature of both parts. |
| Spec | 1.05h Reciprocal trig functions: sec, cosec, cot definitions and graphs1.05l Double angle formulae: and compound angle formulae1.05o Trigonometric equations: solve in given intervals |
| Answer | Marks | Guidance |
|---|---|---|
| (a) | Obtain expression of form \(\frac{a\tan\alpha}{b+c\tan^2\alpha}\) | M1 |
| State correct \(\frac{2\tan\alpha}{1-\tan^2\alpha}\) | A1 | or equiv |
| Attempt to produce polynomial equation in \(\tan\alpha\) | M1 | using sound process |
| Obtain at least one correct value of \(\tan\alpha\) | A1 | \(\tan\alpha = \pm\frac{1}{\sqrt{3}}\) |
| Obtain 41.8 | A1 | allow 42 or greater accuracy; allow 0.73 |
| Obtain 138.2 and no other values between 0 and 180 | A1 | allow 138 or greater accuracy; [SC: Answers only 41.8 or …, B1; 138.2 or … and no others B1] |
| (b)(i) | State \(\frac{2}{6}\) | B1 |
| (ii) | Attempt use of identity linking \(\cot^2\beta\) and \(\cosec^2\beta\) | M1 |
| Obtain \(\frac{13}{36}\) | A1 | or exact equiv |
(a) | Obtain expression of form $\frac{a\tan\alpha}{b+c\tan^2\alpha}$ | M1 | any non-zero constants $a, b, c$ |
| State correct $\frac{2\tan\alpha}{1-\tan^2\alpha}$ | A1 | or equiv |
| Attempt to produce polynomial equation in $\tan\alpha$ | M1 | using sound process |
| Obtain at least one correct value of $\tan\alpha$ | A1 | $\tan\alpha = \pm\frac{1}{\sqrt{3}}$ |
| Obtain 41.8 | A1 | allow 42 or greater accuracy; allow 0.73 |
| Obtain 138.2 and no other values between 0 and 180 | A1 | allow 138 or greater accuracy; [SC: Answers only 41.8 or …, B1; 138.2 or … and no others B1] |
(b)(i) | State $\frac{2}{6}$ | B1 | |
(ii) | Attempt use of identity linking $\cot^2\beta$ and $\cosec^2\beta$ | M1 | or equiv retaining exactness; condone sign errors |
| Obtain $\frac{13}{36}$ | A1 | or exact equiv |5
\begin{enumerate}[label=(\alph*)]
\item Express $\tan 2 \alpha$ in terms of $\tan \alpha$ and hence solve, for $0 ^ { \circ } < \alpha < 180 ^ { \circ }$, the equation
$$\tan 2 \alpha \tan \alpha = 8 .$$
\item Given that $\beta$ is the acute angle such that $\sin \beta = \frac { 6 } { 7 }$, find the exact value of
\begin{enumerate}[label=(\roman*)]
\item $\operatorname { cosec } \beta$,
\item $\cot ^ { 2 } \beta$.
\end{enumerate}\end{enumerate}
\hfill \mbox{\textit{OCR C3 2008 Q5 [9]}}