| Exam Board | OCR |
|---|---|
| Module | C3 (Core Mathematics 3) |
| Year | 2005 |
| Session | June |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Reciprocal Trig & Identities |
| Type | Prove identity then solve equation |
| Difficulty | Standard +0.3 This is a slightly above-average C3 question. Part (i) is trivial recall, part (ii) is a standard identity proof using the double angle formula and basic algebraic manipulation, and part (iii) requires solving a trigonometric equation by substitution and factorization. While it involves multiple steps, the techniques are all standard C3 material with no novel insights required. |
| Spec | 1.05l Double angle formulae: and compound angle formulae1.05o Trigonometric equations: solve in given intervals1.05p Proof involving trig: functions and identities |
| Answer | Marks | Guidance |
|---|---|---|
| (i) State \(2\cos^2 x - 1\) | B1 | Total: 1 mark |
| (ii) Attempt to express left hand side in terms of \(\cos x\) | M1 | [using expression of form \(a\cos^2 x + b\)] |
| Identify \(\frac{1}{\cos x}\) as \(\sec x\) | M1 | [maybe implied] |
| Confirm result | A1 | Total: 3 marks [AG; necessary detail required] |
| (iii) Use identity \(\sec^2 x = 1 + \tan^2 x\) | B1 | |
| Attempt solution of quadratic equation in \(\tan x\) | M1 | [or equiv] |
| Obtain \(2\tan^2 x + 3\tan x - 9 = 0\) and hence \(\tan x = -3, \frac{3}{2}\) | A1 | |
| Obtain at least two of 0.983, 4.12, 1.89, 5.03 | A1 | [allow answers with only 2 s.f.; allow greater accuracy; allow \(0.983 + \pi, 1.89 + \pi\) allow degrees: 56, 236, 108, 288] |
| Obtain all four solutions | A1 | Total: 5 marks [now with at least 3 s.f.; must be radians; no other solutions in the range 0 – \(2\pi\); ignore solutions outside range 0 – \(2\pi\)] |
**(i)** State $2\cos^2 x - 1$ | B1 | Total: 1 mark
**(ii)** Attempt to express left hand side in terms of $\cos x$ | M1 | [using expression of form $a\cos^2 x + b$]
Identify $\frac{1}{\cos x}$ as $\sec x$ | M1 | [maybe implied]
Confirm result | A1 | Total: 3 marks [AG; necessary detail required]
**(iii)** Use identity $\sec^2 x = 1 + \tan^2 x$ | B1 |
Attempt solution of quadratic equation in $\tan x$ | M1 | [or equiv]
Obtain $2\tan^2 x + 3\tan x - 9 = 0$ and hence $\tan x = -3, \frac{3}{2}$ | A1 |
Obtain at least two of 0.983, 4.12, 1.89, 5.03 | A1 | [allow answers with only 2 s.f.; allow greater accuracy; allow $0.983 + \pi, 1.89 + \pi$ allow degrees: 56, 236, 108, 288]
Obtain all four solutions | A1 | Total: 5 marks [now with at least 3 s.f.; must be radians; no other solutions in the range 0 – $2\pi$; ignore solutions outside range 0 – $2\pi$]7 (i) Write down the formula for $\cos 2 x$ in terms of $\cos x$.\\
(ii) Prove the identity $\frac { 4 \cos 2 x } { 1 + \cos 2 x } \equiv 4 - 2 \sec ^ { 2 } x$.\\
(iii) Solve, for $0 < x < 2 \pi$, the equation $\frac { 4 \cos 2 x } { 1 + \cos 2 x } = 3 \tan x - 7$.
\hfill \mbox{\textit{OCR C3 2005 Q7 [9]}}