L1 Mathematical Reasoning
Question Bank
A water tank has a slow leak. The table below shows the relationship between the volume of water, $V$ in litres, and the time, $t$ in hours.
| Time (t) | Volume (V) |
|---|---|
| 1 | 492 |
| 2 | 484 |
| 3 | 476 |
| 4 | 468 |
Find the equation that describes this relationship.
The table below shows the relationship between the total monthly cost of a phone plan, $C$ in dollars, and the amount of data used, $G$ in gigabytes.
| Data (G) | Cost (C) |
|---|---|
| 1 | $34.50 |
| 2 | $39.00 |
| 3 | $43.50 |
| 4 | $48.00 |
Find the equation that describes this relationship.
(i) The table below shows a relationship between the variables $x$ and $y$. Find the equation that describes this relationship.
| x | y |
|---|---|
| 1 | -5 |
| 2 | 0 |
| 3 | 5 |
| 4 | 10 |
| 5 | 15 |
(ii) Describe at least two key features of the graph of your equation from part (i).
A quadratic sequence begins -2, -3, 2, __, 30, ...
What is the fourth term in this sequence?
(i): The diagram below shows the first three patterns in a sequence.
Find an equation that represents the number of small squares, $y$, for any given pattern number, $x$.
(ii): If the graph of your equation was drawn for all $x$, what would be its key features? Describe at least THREE different features.
The diagram below shows the first three patterns in a sequence made of squares.
(i) Find an equation that relates the number of squares, $S$, to the pattern number, $n$.
(ii) Using your equation, how many squares would be in Pattern 10?
(iii) Which pattern number would be made up of 148 squares? You must use an algebraic method.
The value of a new car is observed to decrease over time. The table below shows the car's value, $V$, in dollars, at the end of each year, $t$.
| Year (t) | Value (V) |
|---|---|
| 1 | $20,000 |
| 2 | $16,000 |
| 3 | $12,800 |
| 4 | $10,240 |
(i) Merit Find the equation that models the car's value over time assuming the pattern continues. You must justify your answer.
(ii) Achieved Use your equation to find the initial purchase price of the car (at $t=0$).
(iii) Merit Find in which year the car's value will first fall below $7,000.
The table shows some points on a graph.
| $x$ | $y$ |
|---|---|
| 1 | 7 |
| 2 | 13 |
| 3 | 25 |
| 4 | 49 |
| 5 | 97 |
(i) Find an equation for $y$ in terms of $x$. Show working to support your answer.
(ii) Describe THREE features of the graph of your equation, drawn for all values of $x$.
(i) Merit The table of values below shows the relationship between two variables, $x$ and $y$. Find the equation that describes this relationship. You must justify your answer by showing your algebraic working.
| x | y |
|---|---|
| 1 | 0 |
| 2 | 3 |
| 3 | 10 |
| 4 | 21 |
| 5 | 36 |
(ii) Merit Find the coordinates of the points where the graph of the equation cuts the x-axis.
(iii) Merit Find the coordinates of the vertex (turning point) of the graph.
(i) Merit The table of values below shows the relationship between two variables, $x$ and $y$. Find the equation that describes this relationship. You must justify your answer by showing your algebraic working.
| x | y |
|---|---|
| 1 | -8 |
| 2 | -15 |
| 3 | -20 |
| 4 | -23 |
| 5 | -24 |
(ii) Merit If the graph of your equation from part (i) was drawn for all values of $x$, what would be its key features? Describe at least THREE different features.
(i) The table below represents points on a particular graph, G₁.
Find the equation of this graph.
| x | y |
|---|---|
| 1 | 14 |
| 2 | 16 |
| 3 | 18 |
| 4 | 20 |
(ii) The table below represents points on another graph G₂.
Find the equation of this graph.
| x | y |
|---|---|
| 1 | -1 |
| 2 | 0 |
| 3 | 7 |
| 4 | 20 |
(iii) Use algebra to find the $x$-values of the two points of intersection of the graphs G₂ and G₁.
The diagram below shows the first four polygons in a sequence, with all possible diagonals drawn.
(i) Complete the following table based on the pattern.
| Number of sides ($n$) | Number of diagonals ($d$) |
|---|---|
| 3 | 0 |
| 4 | 2 |
| 5 | ... |
| 6 | ... |
(ii) Find a formula that connects the number of diagonals, $d$, to the number of sides, $n$.
(iii) A polygon has 54 diagonals. Use your formula to find the number of sides it has.
A shopkeeper is stacking gift boxes for a window display. The first three patterns are shown below.
The number of boxes in each pattern is:
- Pattern 1 has 4 boxes.
- Pattern 2 has 12 boxes.
- Pattern 3 has 24 boxes.
(i) Find an equation that connects the total number of boxes, $T$, to the pattern number, $n$.
(ii) How many boxes would be needed for Pattern 8?
(iii) A different display has a total of 180 boxes. Use an algebraic method to find its pattern number.
Liam invests some money in an account that earns compound interest (this means it increases by the same percentage each year so follows an exponential pattern). The table below shows the total amount, $A$, in his account after $t$ years.
| Year (t) | Amount (A) |
|---|---|
| 1 | $840 |
| 2 | $882 |
| 3 | $926.10 |
| 4 | $972.41 |
(i) Merit Find the equation that models the investment's growth.
(ii) Achieved What was Liam's initial investment?
(iii) Excellence Liam's friend, Chloe, invested the same initial amount in a different account that earns $50 per year. After how many full years will Liam's investment first be worth more than Chloe's?
The quadratic sequence beginning 2, 9, 22, 41, 66... has the $n^{th}$ term $T_1 = 3n^2 - 2n + 1$.
Find the $n^{th}$ term of the quadratic sequence that begins 22, 41, 66...
Two new phone apps were launched on the same day. The tables show the number of users of each app at the end of each of the first few months.
| Month, $t$ | App A users |
|---|---|
| 0 | 500 |
| 1 | 700 |
| 2 | 980 |
| 3 | 1372 |
| Month, $t$ | App B users |
|---|---|
| 0 | 2000 |
| 1 | 2600 |
| 2 | 3200 |
| 3 | 3800 |
Assume each pattern continues.
(i) Find an equation for the number of users of each app after $t$ months.
(ii) At the end of which month will App A first have more users than App B? Justify your answer.
(iii) Explain why App A will stay ahead after that.