| Exam Board | Edexcel |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2020 |
| Session | January |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Reciprocal Trig & Identities |
| Type | Substitution t equals tan |
| Difficulty | Standard +0.8 This question requires multiple non-trivial trigonometric manipulations: expressing tan 2x, cot x, and sec²x in terms of t = tan x, then algebraic simplification to reach the quartic. Part (b) requires solving the quartic and careful consideration of which solutions are valid in the given domain. While systematic, it demands fluency with multiple identities and extended algebraic manipulation beyond routine exercises. |
| Spec | 1.05j Trigonometric identities: tan=sin/cos and sin^2+cos^2=11.05k Further identities: sec^2=1+tan^2 and cosec^2=1+cot^21.05l Double angle formulae: and compound angle formulae1.05o Trigonometric equations: solve in given intervals |
| Answer | Marks | Guidance |
|---|---|---|
| \(12 \times \frac{2t}{1-t^2} + 5 \times \left(1+t^2\right)\left(1-t^2\right) = 0\) | B1 M1 A1 | |
| \(12 \times 2t^2 + 5\left(1+t^2\right)\left(1-t^2\right) = 0 \Rightarrow 5t^4 - 24t^2 - 5 = 0\) * | A1* | (4) |
| Answer | Marks | Guidance |
|---|---|---|
| \((5t^2+1)(t^2-5) = 0\) | M1 | |
| Correct order of operations \(t = (\pm)\sqrt{5} \Rightarrow x = ...\) | dM1 | |
| Two of awrt \(x = 66°, 114°, 246°, 294°\) | A1 | |
| All four of awrt \(x = 65.9°, 114.1°, 245.9°, 294.1°\) | A1 | (4) |
**(a)** $12\tan 2x + 5\cot x\sec^2 x = 0$
$12 \times \frac{2t}{1-t^2} + 5 \times \left(1+t^2\right)\left(1-t^2\right) = 0$ | B1 M1 A1 |
$12 \times 2t^2 + 5\left(1+t^2\right)\left(1-t^2\right) = 0 \Rightarrow 5t^4 - 24t^2 - 5 = 0$ * | A1* | (4)
**(b)** $5t^4 - 24t^2 - 5 = 0$
$(5t^2+1)(t^2-5) = 0$ | M1 |
Correct order of operations $t = (\pm)\sqrt{5} \Rightarrow x = ...$ | dM1 |
Two of awrt $x = 66°, 114°, 246°, 294°$ | A1 |
All four of awrt $x = 65.9°, 114.1°, 245.9°, 294.1°$ | A1 | (4)
**Total: 8 marks**
---5. (a) Use the substitution $t = \tan x$ to show that the equation
$$12 \tan 2 x + 5 \cot x \sec ^ { 2 } x = 0$$
can be written in the form
$$5 t ^ { 4 } - 24 t ^ { 2 } - 5 = 0$$
(b) Hence solve, for $0 \leqslant x < 360 ^ { \circ }$, the equation
$$12 \tan 2 x + 5 \cot x \sec ^ { 2 } x = 0$$
Show each stage of your working and give your answers to one decimal place.\\
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\hfill \mbox{\textit{Edexcel P3 2020 Q5 [8]}}