Free questions

A car rental company charges a fixed fee plus a rate per kilometre driven. For 100 km the total charge is $85, and for 300 km the total charge is $145.

1. Find the rate per kilometre and interpret its meaning. [2 marks]
2. Find the equation $C=md+f$, where $C$ is the total cost in dollars and $d$ is the distance in km. [2 marks]
3. State the fixed fee and explain what it represents. [1 mark]
4. Find the maximum distance a customer can drive on a budget of $100. [2 marks]
5. A competitor charges $0.40 per km with no fixed fee. Find the distance for which both companies charge the same. [2 marks]

State the gradient and $y$-intercept of the line with equation $y=\frac{2x-3}{6}$. [3 marks]

A data analyst classifies real numbers by the sets they belong to.

1. Copy and complete the table by placing a tick in each set the number belongs to. [6 marks]

   Number	$\mathbb{N}$	$\mathbb{Z}$	$\mathbb{Q}$	${\mathbb{Q}}^{\prime }$	$\mathbb{R}$
   $-4$
   $\frac{5}{8}$
   $\sqrt{49}$
   $\pi -\pi$
   $\sqrt{11}$
   $-2.4\overline{7}$

2. State whether each statement is true or false. Justify your answer. [4 marks]

   (i) ${\mathbb{Z}}^{+}\subset \mathbb{N}$

   (ii) $\mathbb{Q}\subset \mathbb{Z}$

   (iii) $\sqrt{25}\in {\mathbb{Z}}^{+}$

   (iv) $\mathbb{R}=\mathbb{Q}\cup {\mathbb{Q}}^{\prime }$

A rectangular courtyard has length $(3+\sqrt{2})$ m and width $(3-\sqrt{2})$ m.

1. Show that the area of the courtyard is an integer, and state its value. [3 marks]
2. Find the exact perimeter of the courtyard. [2 marks]
3. Paving slabs are squares of side $\sqrt{2}$ m, and cut pieces may be used. Each slab costs $15. Find the smallest number of slabs that must be bought, and the total cost. [3 marks]
4. A second courtyard has length $(\sqrt{5}+1)$ m and width $\frac{1}{\sqrt{5}+1}$ m. Find the exact area of this courtyard. [2 marks]

A spherical water tank has volume $V=\frac{4}{3}\pi r^3$ cm${}^3$, where $r$ is the radius in centimetres.

1. Find the radius of the tank when $V=36\pi$ cm${}^3$. [2 marks]
2. Find the exact radius when $V=4\pi$ cm${}^3$. [2 marks]
3. A cube-shaped storage box has the same volume as the tank in part (a). Find the exact side length of the box. [2 marks]
4. Show that doubling the radius of a sphere multiplies its volume by 8. [2 marks]
5. Two spherical tanks have radii in the ratio $1:2$. Write down the ratio of their volumes. [2 marks]

Ava saves $50 in the first week, $55 in the second week and $60 in the third week. She continues this pattern.

1. Show that Ava's weekly savings form an arithmetic sequence, and state $u_1$ and $d$. [2 marks]
2. Find the amount Ava saves in week 20. [2 marks]
3. Find the total amount Ava saves in the first 26 weeks. [3 marks]
4. In which week do Ava's weekly savings first exceed $200? [3 marks]

A clock face has radius 20 cm. The minute hand is 20 cm long and the hour hand is 14 cm long. Between 12:00 and 12:15 the minute hand turns through a quarter turn.

1. Find the length of the arc traced by the tip of the minute hand. [2 marks]
2. Find the area swept by the minute hand. [2 marks]
3. Find the angle, in degrees, through which the hour hand turns between 12:00 and 12:15, and hence find the length of the arc traced by its tip. [3 marks]
4. Find the difference between the areas swept by the two hands in these 15 minutes. [3 marks]

A yacht leaves buoy $A$ and sails 6 km on a bearing of $050^{\circ }$ to buoy $B$. It then sails 9 km on a bearing of $140^{\circ }$ to buoy $C$.

1. Show that $\angle ABC=90^{\circ }$. [2 marks]
2. Find the distance $AC$. [2 marks]
3. Find the bearing of $C$ from $A$. [3 marks]
4. Write down the bearing of $A$ from $C$. [1 mark]

A right-angled triangle has hypotenuse 17 m. The side opposite the angle $\theta$ is 8 m and the side adjacent to $\theta$ is 15 m.

Find $\sin \theta$, $\cos \theta$ and $\tan \theta$. [3 marks]

An archaeological dig uses a coordinate grid measured in metres. The $z$-axis points vertically upwards and $z=0$ is ground level, so negative $z$ is below ground. Two fossils are found at ${\mathrm{F}}_1(3,7,-2)$ and ${\mathrm{F}}_2(9,2,-5)$.

1. Find the distance between the two fossils. [3 marks]
2. Find the midpoint M of the segment joining the two fossils. [2 marks]
3. Find the angle of depression of ${\mathrm{F}}_2$ from ${\mathrm{F}}_1$. [2 marks]
4. A third fossil is found at ${\mathrm{F}}_3(k,2,-5)$, where ${\mathrm{F}}_1{\mathrm{F}}_3=8$ m. Find the possible values of $k$. [2 marks]