Practice Examination
Level 1 Mathematical Reasoning
Our 2025 prediction paper, free to use as a practice exam. Every question has a worked answer.
Credits: Five
Instructions
- Suggested time: 60 minutes.
- You should attempt ALL the questions in this paper.
- Show ALL relevant working.
- A scientific calculator is permitted.
- Hover on the right of a question for tools.
Main Exam Timer
QUESTION ONE
(a)
Expand and simplify the following expression:
$$3p(p^2 + 4)(2p - 1)$$
(b)
A company makes a spherical chocolate shell completely filled with gooey caramel, as shown in the diagram. The inner radius is 5 cm and the shell is made of 1 cm thick chocolate.
What percentage of the sphere's total volume is the caramel inside?
(c)
After drinking a coffee, the amount of caffeine in a person's bloodstream decreases over time. The table below shows the amount of caffeine, $C$, in milligrams, remaining in the body $t$ hours after drinking the coffee.
| Time after coffee ($t$ in hours) | Caffeine in bloodstream ($C$ in mg) |
|---|---|
| 1 | 120 |
| 2 | 60 |
| 3 | 30 |
| 4 | 15 |
(i) Find the equation that models the amount of caffeine, $C$, in the body after $t$ hours, assuming the pattern continues.
(ii) How much caffeine was in the coffee initially (at $t=0$)?
(iii) If the graph of this equation was drawn for all $t$, state three key features of the graph and relate each feature to this context.
(d)
A dessert chef is creating a dish that consists of six spherical scoops of ice cream, each with a diameter $d$, placed in a rectangular bowl. The scoops are arranged in a 3x2 grid and fit snugly, as shown in the diagram.
Find a simplified expression, in terms of $d$, for the height the ice cream will reach in the bowl if it all melts.
QUESTION TWO
(a)
Jacob is planning a road trip and needs to rent a car for a day. He is comparing two local companies, Kiwi Ride and Go Rentals.
The total daily cost for Kiwi Ride, based on the distance driven, is shown in the graph below.
Kiwi Ride Cost ($) vs. Distance (km)
The pricing for Go Rentals is described as follows: they charge a flat daily fee of \$60 plus an additional \$0.20 per kilometre driven.
Let $C$ be the total cost in dollars and $d$ be the distance driven in kilometres.
(i) Find the equation for the total cost, $C$, for each company in terms of the distance driven, $d$.
(ii) Jacob estimates his trip will be 250 km. Based on this estimate, which company should he choose to get the lower price, and by how much will he save?
(iii) After his trip, Jacob finds that both companies would have charged him the exact same amount. How far did he drive? Explain how this is represented on the graph.
(b)
If $3x + y = 12$ and $x + 3y = 8$, what is the value of $x + y$?
(c)
Sophie is building a custom tool shed for her garden as part of a school technology project. The design is a trapezoidal prism that will sit against a wall. The diagram below shows the exterior dimensions of her planned shed.
(i) For proper rainwater runoff, the local council requires the roof's angle of slope to be calculated. Find the angle the roof makes with the horizontal.
(ii) Sophie needs to paint the four exterior walls of the shed (not the roof or the floor). Calculate the total surface area that needs to be painted.
(iii) Sophie's longest tool is a long-handled rake. What is the absolute maximum length of a single straight tool that could be stored inside the shed?
QUESTION THREE
(a)
The diagram below shows an isosceles triangle with side lengths given in terms of $x$.
The perimeter of the triangle is 20.2 cm. Find the value of $x$.
(b)
The graph of the exponential function $y = 1.1^x$ is shown on the axes below.
A second, linear relationship is described by the table of values.
| $x$ | $y$ |
|---|---|
| 0 | 2 |
| 3 | 3 |
| 6 | 4 |
| 9 | 5 |
(i) Find the equation that describes the linear relationship shown in the table.
(ii) On the axes provided, accurately draw the graph of the linear relationship from part (i). Use your completed graph to find and state an estimate for the solution(s) to the equation $\frac{1}{3}x + 2 = 1.1^x$.
(c)
A cylinder of height h and radius 5k is filled completely with molten metal. It is to be poured into the cast which is a cylindrical shell (hollow cylinder). The cast has an inner radius of 12k and the 'shell' has a width of k. The cast is 5cm taller than the original cylinder. Find the volume of left over space in the cast when the contents is poured inside.
Note: The metal is poured into the outer shell of the cast (not the space in the centre)
(d)
In many parts of tropical Australia, stinger nets are set up at beaches during the summer months. These nets create safe swimming enclosures that protect swimmers from potentially deadly jellyfish, such as the Box Jellyfish and Irukandji.
A 30m long stinger net is to be set up to create a rectangular swimming area against a straight section of a beach. The net will form three sides of the enclosure, with the beach itself forming the fourth side. Two of the sides will extend perpendicularly from the beach into the sea, and the third side will run parallel to the beach. The net must extend at least 3 m onto the beach as shown in the diagram.
Let $x$ be the length of the two sides extending from the top beach as shown in the diagram.
(i) Find an expression for the total area enclosed by the stinger net, $A$, in terms of $x$.
(ii) Use a table and a graph to investigate how the area in the net changes as $x$ changes. Based on your investigation, give a recommendation on the best dimensions for the net and state the maximum possible swimming area.
