| Exam Board | CAIE |
|---|---|
| Module | P1 (Pure Mathematics 1) |
| Year | 2012 |
| Session | June |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Trig Proofs |
| Type | Solve equation using proven identity |
| Difficulty | Moderate -0.3 Part (i) is a straightforward algebraic manipulation of trigonometric identities requiring students to combine fractions and use basic definitions (tan x = sin x/cos x). Part (ii) applies the proven identity to solve an equation, requiring substitution and solving a quadratic in tan x, but the steps are fairly routine for P1 level with no novel insight needed. The 6-mark total and standard techniques place this slightly below average difficulty. |
| Spec | 1.05j Trigonometric identities: tan=sin/cos and sin^2+cos^2=11.05o Trigonometric equations: solve in given intervals1.05p Proof involving trig: functions and identities |
| Answer | Marks | Guidance |
|---|---|---|
| \(\tan x + \frac{1}{\tan x} = \frac{\sin x \cos x}{\sin x \cos x}\) | ||
| (i) LHS = \(\frac{\sin x}{\cos x} + \frac{\cos x}{\sin x}\) | M1 | Use of tan = sin/cos twice |
| \(= \frac{\sin^2 x + \cos^2 x}{\sin x \cos x} = \frac{1}{\sin x \cos x}\) | M1 | [2] |
| (ii) \(\frac{2}{\sin x \cos x} = 3 \tan x + 1\) | M1 | Uses part (i) to obtain eqn in tanx only |
| Uses (i) \(2(\tan x + \frac{1}{\tan x}) = 3 \tan x + 1\) | M1 | |
| \(\to \tan^2 x + \tan x - 2 = 0\) | DM1 | Correct soln of quadratic eqn |
| \(\to \tan x = 1\) or \(-2\) | B1 A1 | co. Must have correct quadratic co |
| \(\to x = 45°\) or \(116.6°\) | [4] |
$\tan x + \frac{1}{\tan x} = \frac{\sin x \cos x}{\sin x \cos x}$ | |
(i) LHS = $\frac{\sin x}{\cos x} + \frac{\cos x}{\sin x}$ | M1 | Use of tan = sin/cos twice
$= \frac{\sin^2 x + \cos^2 x}{\sin x \cos x} = \frac{1}{\sin x \cos x}$ | M1 | [2] | Use of $s^2 + c^2 = 1$ appropriately – everything correct.
(ii) $\frac{2}{\sin x \cos x} = 3 \tan x + 1$ | M1 | Uses part (i) to obtain eqn in tanx only
Uses (i) $2(\tan x + \frac{1}{\tan x}) = 3 \tan x + 1$ | M1 |
$\to \tan^2 x + \tan x - 2 = 0$ | DM1 | Correct soln of quadratic eqn
$\to \tan x = 1$ or $-2$ | B1 A1 | co. Must have correct quadratic co
$\to x = 45°$ or $116.6°$ | [4] |\begin{enumerate}[label=(\roman*)]
\item Prove the identity $\tan x + \frac{1}{\tan x} = \frac{1}{\sin x \cos x}$. [2]
\item Solve the equation $\frac{2}{\sin x \cos x} = 1 + 3 \tan x$, for $0° \leqslant x \leqslant 180°$. [4]
\end{enumerate}
\hfill \mbox{\textit{CAIE P1 2012 Q5 [6]}}