# Exam-Style Questions.

## Problems adapted from questions set for previous Mathematics exams.

### 1.

IB Analysis and Approaches

(a) Show that:

$$\cos 2x - \sin 2x + 1 = 2 \cos x ( \cos x - \sin x)$$

(b) Hence or otherwise, solve the following equation for $$\pi \lt x \lt 3\pi$$.

$$\cos 2x - \sin 2x + 1 = \sin x - \cos x$$

### 2.

IB Analysis and Approaches

(a) Show that the equation $$2 \sin^2 x - 5 \cos x = -1$$ may be written in the form $$2 \cos^2 x + 5 \cos x = 3$$

(b) Hence, solve the equation $$2 \sin^2 x - 5 \cos x = -1$$, $$2\pi \lt x \lt 4\pi$$.

### 3.

A-Level

The cosine of acute angle $$\alpha$$ is $$\frac{1}{ \sqrt 5}$$

The angle $$\beta$$ is obtuse and $$\sin \beta = \sqrt \frac{2}{3}$$.

(a) Find exact values of $$\tan \alpha$$ and $$\tan \beta$$.

(b) Hence show that $$\tan( \alpha - \beta )$$ can be written as $$a+b \sqrt 2$$ where $$a$$ and $$b$$ are rational numbers

### 4.

IB Analysis and Approaches

Consider the functions $$f(x)=\sin(x) \text{ and } g(x) = \tan(x+\frac{\pi}{2})$$ in the region where $$\frac{\pi}{2}\le x \le \pi$$

The graphs $$y=f (x)$$ and $$y = g(x)$$ intersect at a point A whose x-coordinate is $$a$$.

(a) Show that $$\sin^2{a}=-\cos{a}$$.

(b) Hence, show that the tangents to the curves at A intersect at right angles.

(c) Find the value of $$\cos(a)$$. Give your answer in the form $$\dfrac{a + \sqrt{b}}{c}$$.

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