## 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 cm^{2}, giving your answer as a multiple of \(\pi\).

(b) The volume of the dart board is 4500\(\pi\) cm^{3}.
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 numbersIf you would like space on the right of the question to write out the solution try this Thinning Feature. It will collapse the text into the left half of your screen but large diagrams will remain unchanged.

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