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Exam-Style Questions.

Problems adapted from questions set for previous Mathematics exams.


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 \( \pi \lt x \lt 3\pi \).

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


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 \( 2 \sin^2 x - 5 \cos x = -1 \), \( 2\pi \lt x \lt 4\pi \).



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


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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The exam-style questions appearing on this site are based on those set in previous examinations (or sample assessment papers for future examinations) by the major examination boards. The wording, diagrams and figures used in these questions have been changed from the originals so that students can have fresh, relevant problem solving practice even if they have previously worked through the related exam paper.

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