Exam-Style Questions.

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.

Worked Solution

---

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.

Worked Solution

---

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 \(a + b\sqrt5\) .

Worked Solution

---

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

Worked Solution

---