Equation with 'show that' rewriting preliminary part

Questions where one part requires showing that an equation can be rewritten in a specific form (e.g. as a quadratic), and a subsequent part solves that rewritten equation.

3 questions · Moderate -0.1

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AQA C3 2009 June Q3

8 marks Moderate -0.3

Solve the equation $\tan x = - \frac { 1 } { 3 }$, giving all the values of $x$ in the interval $0 < x < 2 \pi$ in radians to two decimal places.
Show that the equation $$3 \sec ^ { 2 } x = 5 ( \tan x + 1 )$$ can be written in the form $3 \tan ^ { 2 } x - 5 \tan x - 2 = 0$.
Hence, or otherwise, solve the equation $$3 \sec ^ { 2 } x = 5 ( \tan x + 1 )$$ giving all the values of $x$ in the interval $0 < x < 2 \pi$ in radians to two decimal places.
(4 marks)

OCR H240/03 2018 September Q3

7 marks Standard +0.3

Given that $\sqrt { 2 \sin ^ { 2 } \theta + \cos \theta } = 2 \cos \theta$, show that $6 \cos ^ { 2 } \theta - \cos \theta - 2 = 0$.
In this question you must show detailed reasoning. Solve the equation $$6 \cos ^ { 2 } \theta - \cos \theta - 2 = 0$$ giving all values of $\theta$ between $0 ^ { \circ }$ and $360 ^ { \circ }$ correct to 1 decimal place.
Explain why not all the solutions from part (ii) are solutions of the equation $$\sqrt { 2 \sin ^ { 2 } \theta + \cos \theta } = 2 \cos \theta$$

AQA C2 2009 June Q8

9 marks Moderate -0.3

Given that $\frac{\sin \theta - \cos \theta}{\cos \theta} = 4$, prove that $\tan \theta = 5$. [2]
1. Use an appropriate identity to show that the equation $$2 \cos^2 x - \sin x = 1$$ can be written as $$2 \sin^2 x + \sin x - 1 = 0$$ [2]
2. Hence solve the equation $$2 \cos^2 x - \sin x = 1$$ giving all solutions in the interval $0° \leq x \leq 360°$. [5]