Free questions

Non-calculator

A taxi company charges a fixed fee plus a rate per kilometre. A 5 km journey costs $11.50 and a 12 km journey costs $22.00.

1. Find the cost per kilometre. [2 marks]
2. Find the fixed fee and write the cost $C$ dollars in terms of the distance $d$ km. [3 marks]
3. Find the cost of a 20 km journey. [1 mark]
4. Find the greatest distance that can be travelled for $50. [2 marks]

Non-calculator

In a survey of $100$ people, $60$ like coffee, $45$ like tea and $20$ like neither.

1. Find the number who like both coffee and tea. [3 marks]
2. Find the number who like coffee only. [1 mark]
3. A person is chosen at random. Find the probability that they like tea but not coffee. [2 marks]

Non-calculator

A right-angled triangle has perpendicular sides of lengths $2\sqrt{3}$ cm and $6$ cm.

1. Find the length of the hypotenuse, giving your answer as a simplified surd. [3 marks]
2. Find the exact area of the triangle. [2 marks]
3. Find the exact perimeter of the triangle. [2 marks]

Non-calculator

A rectangle has perimeter $34$ cm and area $60$ cm${}^2$.

1. Let one side be $x$ cm. Show that $x^2-17x+60=0$. [2 marks]
2. Solve the equation and state the dimensions of the rectangle. [2 marks]
3. Find the length of a diagonal. [2 marks]

Non-calculator

Solve each equation for $x\in \mathbb{R}$.

1. $x^3=27$ [1 mark]
2. $x^4=81$ [1 mark]
3. $2x^3=-54$ [1 mark]
4. $(x-1{)}^2=16$ [2 marks]
5. $x^{-2}=\frac{1}{9}$ [2 marks]

Non-calculator

A student saves $200 in the first month and increases the amount saved by $15 each month.

1. How much is saved in the 24th month? [2 marks]
2. Find the total saved over the first 24 months. [3 marks]
3. In which month does the monthly saving first exceed $500? [2 marks]

A running track has two straight sections, each $100$ m long, joined by two semicircular ends of radius $36.5$ m.

1. Find the length of one lap. [3 marks]
2. Find the area enclosed by the track. [3 marks]

Non-calculator

A solid paperweight is made by joining a hemisphere of radius $4$ cm to a cylinder of the same radius and height $6$ cm, so that the flat face of the hemisphere covers one circular face of the cylinder.

1. Find the volume of the paperweight, giving your answer in terms of $\pi$. [4 marks]
2. Find the total outer surface area of the paperweight, giving your answer in terms of $\pi$. [4 marks]

A wheelchair ramp rises $0.5$ m over a horizontal distance of $6$ m.

1. Find the angle the ramp makes with the horizontal, correct to $1$ decimal place. [2 marks]
2. Find the length of the ramp surface, correct to $3$ significant figures. [2 marks]
3. A regulation requires the angle to be at most $5^{\circ }$. State, with a reason, whether this ramp meets the regulation. [1 mark]

A vertical tower stands on horizontal ground. From a point $A$ on the ground, the angle of elevation of the top of the tower is $28^{\circ }$. From a point $B$, which is $40$ m closer to the tower on the same straight line as $A$, the angle of elevation is $51^{\circ }$. Let the height of the tower be $h$ m.

1. Write down expressions, in terms of $h$, for the horizontal distance from $A$ to the tower and from $B$ to the tower. [3 marks]
2. Hence find the height of the tower, correct to $3$ significant figures. [4 marks]