Standard +0.8 This question requires multiple non-trivial substitutions and algebraic manipulations: expressing tan 2x, cot x, and sec²x in terms of t = tan x, then factorizing a quartic equation and carefully considering all solutions in the given domain. While the individual techniques are A-level standard, the combination and the need to track multiple solutions across 0 ≤ x < 2π makes this moderately challenging, above average difficulty.
7. (a) Use the substitution \(t = \tan x\) to show that the equation
$$4 \tan 2 x - 3 \cot x \sec ^ { 2 } x = 0$$
can be written in the form
$$3 t ^ { 4 } + 8 t ^ { 2 } - 3 = 0$$
(b) Hence solve, for \(0 \leqslant x < 2 \pi\),
$$4 \tan 2 x - 3 \cot x \sec ^ { 2 } x = 0$$
Give each answer in terms of \(\pi\). You must make your method clear.
For obtaining two answers from their \(t\) values in different quadrants
All 4 exact, correct, no extra values in range
A1
If in degrees (30, 210, 150, 330) lose final A; if decimals lose final A
# Question 7:
## Part (a)
| Answer/Working | Mark | Guidance |
|---|---|---|
| Uses $4\tan 2x = 4\times\frac{2t}{1-t^2}$ or $4\times\frac{2\tan x}{1-\tan^2 x}$ | B1 | B0 for $\tan 2x = \frac{\sin 2x}{\cos 2x}$ |
| Uses either $\cot x = \frac{1}{\tan x}$ or $\frac{1}{t}$ **or** $\sec^2 x=1+\tan^2 x$ or $1+t^2$ (quoted correctly) | M1 | |
| Uses **both** $\cot x=\frac{1}{\tan x}$ or $\frac{1}{t}$ **and** $\sec^2 x=1+\tan^2 x$ or $1+t^2$ both quoted correctly | A1 | |
| Reaches $4\times 2t^2 - 3\times(1+t^2)(1-t^2)=0$ and $3t^4+8t^2-3=0$ | A1* | Fully correct intermediate line of working must be seen |
## Part (b)
| Answer/Working | Mark | Guidance |
|---|---|---|
| $3t^4+8t^2-3=0 \Rightarrow (3t^2-1)(t^2+3)=0$ so $t=$ | M1 | Solves quadratic in $t^2$; correct factorisation/formula/completing square; must take square root |
| $\tan x\,(t) = \pm\frac{1}{\sqrt{3}}$ or $\pm 0.5774$ | A1 | Need both plus and minus; ignore further values of $\tan x$ from quadratic |
| $x=\frac{\pi}{6},\frac{7\pi}{6},\frac{5\pi}{6},\frac{11\pi}{6}$ | M1 | For obtaining two answers from their $t$ values in different quadrants |
| All 4 exact, correct, no extra values in range | A1 | If in degrees (30, 210, 150, 330) lose final A; if decimals lose final A |
7. (a) Use the substitution $t = \tan x$ to show that the equation
$$4 \tan 2 x - 3 \cot x \sec ^ { 2 } x = 0$$
can be written in the form
$$3 t ^ { 4 } + 8 t ^ { 2 } - 3 = 0$$
(b) Hence solve, for $0 \leqslant x < 2 \pi$,
$$4 \tan 2 x - 3 \cot x \sec ^ { 2 } x = 0$$
Give each answer in terms of $\pi$. You must make your method clear.
\hfill \mbox{\textit{Edexcel C34 2015 Q7 [8]}}