| Exam Board | CAIE |
|---|---|
| Module | P1 (Pure Mathematics 1) |
| Year | 2014 |
| Session | June |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Reciprocal Trig & Identities |
| Type | Prove identity then solve equation |
| Difficulty | Standard +0.3 This is a straightforward two-part question requiring standard algebraic manipulation of trig identities (converting tan to sin/cos, factoring) followed by a routine equation solve using the R-formula or auxiliary angle method. While it requires multiple steps, the techniques are standard P1 material with no novel insight needed, making it slightly easier than average. |
| 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 |
|---|---|---|
| \(\frac{\tan x + 1}{\sin x \tan x + \cos x} = \sin x + \cos x\) | ||
| (i) LHS \(= \frac{\left[\frac{s}{c}\right]+1}{\frac{s^2}{c}+c} = \frac{s+c}{s^2+c^2}\) | M1 M1 | Use of \(t = s/c\) twice. Correct algebra and use of \(s^2 + c^2 = 1\) |
| \(= \) RHS | A1 | AG all ok |
| [3] | ||
| (ii) \(s + c = 3s - 2c\) | ||
| \(\rightarrow \tan x = \frac{3}{2}\) | M1 | Uses (i) and \(t = \frac{s}{c}\) or 0 is M0 |
| Allow \(\cos^2 x = \frac{4}{13}\), \(\sin^2 x = \frac{9}{13}\) | ||
| \(\rightarrow x = 0.983\) and 4.12 or 4.13 | A1 A1✓ | co. ✓ 1st \(+ \pi\), providing no excess solns in range. Allow \(0.313\pi, 1.31\pi\) |
| [3] |
$\frac{\tan x + 1}{\sin x \tan x + \cos x} = \sin x + \cos x$ | | |
(i) LHS $= \frac{\left[\frac{s}{c}\right]+1}{\frac{s^2}{c}+c} = \frac{s+c}{s^2+c^2}$ | M1 M1 | Use of $t = s/c$ twice. Correct algebra and use of $s^2 + c^2 = 1$ |
$= $ RHS | A1 | AG all ok |
| | [3] |
(ii) $s + c = 3s - 2c$ | | |
$\rightarrow \tan x = \frac{3}{2}$ | M1 | Uses (i) and $t = \frac{s}{c}$ or 0 is M0 |
Allow $\cos^2 x = \frac{4}{13}$, $\sin^2 x = \frac{9}{13}$ | | |
$\rightarrow x = 0.983$ and 4.12 or 4.13 | A1 A1✓ | co. ✓ 1st $+ \pi$, providing no excess solns in range. Allow $0.313\pi, 1.31\pi$ |
| | [3] |4 (i) Prove the identity $\frac { \tan x + 1 } { \sin x \tan x + \cos x } \equiv \sin x + \cos x$.\\
(ii) Hence solve the equation $\frac { \tan x + 1 } { \sin x \tan x + \cos x } = 3 \sin x - 2 \cos x$ for $0 \leqslant x \leqslant 2 \pi$.
\hfill \mbox{\textit{CAIE P1 2014 Q4 [6]}}