# Exam-Style Questions.

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

### 1.

GCSE Higher

A circular dart board has radius of 30 cm.

(a) Calculate the area of the front face of the dart board in cm2, giving your answer as a multiple of $$\pi$$.

(b) The volume of the dart board is 4500$$\pi$$ cm3. Calculate the thickness of the dart board.

### 2.

GCSE Higher

(a) Express $$\sqrt{5} + \sqrt{20}$$ in the form $$a \sqrt{5}$$ where $$a$$ is an integer.

(a) Express $$( \frac{1}{ \sqrt{5} } ) ^ 9$$ in the form $$\frac{ \sqrt{b}}{c}$$ where $$b$$ and $$c$$ are integers.

### 3.

GCSE Higher

(a) Write this list of numbers in order, smallest first.

$$\sqrt{47}, \frac{22}{3}, 3.5^2, 8.64$$

(b) Write $$(1 + \sqrt5)^2$$ in the form $$a + b\sqrt5$$ .

### 4.

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

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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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