L1 Mathematical Reasoning
Question Bank
Find the value of $x$ if $2^{x-3} = 32$.
Find the co-ordinates of the y-intercept for the exponential graph $y = 4(3^{2x-1}) + 7$.
Find the coordinates of the y-intercept for the exponential graph $y = (\frac{1}{2})^{x-3} - 5$.
The diagram below shows the graphs of $y = (1.2)^x$ and $y = \frac{1}{2}x + 1$.
Use the graph to estimate the solutions to the equation $1.2^x = \frac{1}{2}x + 1$.
Simplify the following expression:
$ \frac{\displaystyle 5^{4m-2}}{\displaystyle 5^{m+4}} $
Simplify the following expression:
$ 3^{p+1} \times 9^{2p-5} $
Write down THREE different features of the graph of $y = 2^x + 3$.
Simplify the following expression:
$ 2^{a+2} \times 4^{2a-1} \times 2^{-3a} $
The graph of $y = 2^x + c$ is shown below, where $c$ is a constant. The dashed line is an asymptote.
(i) Write down the equation of the asymptote.
(ii) Find the value of $c$.
(iii) Use algebra to find where the graph crosses the $x$-axis.
Solve the equation $9^{2x-1} = 27^{x+4}$.
Find the value of $x$ by solving the equation $8^{x+3} = 16^{x-1}$.
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.
Solve the inequality $5^{2x-1} > 125$.
What values of $x$ satisfy the inequality $7^{3x} \times 7^{x-1} \ge 49$.
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 $x$.
Solve for $x$ in the equation $25^{2x} = (\frac{1}{125})^{x-5}$.
Make $y$ the subject of the formula $\frac{8^{x+2}}{4^x} = 2^{5y-1}$.
Make $y$ the subject of the formula $25^{x-1} \times 5^{3x+1} = 125^{y+2}$.
Find the values of $x$ by solving the equation $9^{x^2} \times 3^{5x} = \frac{1}{9}$.
Given the equation
$6^{2x} + 6^{2x} + 6^{2x}$$+ 6^{2x} + 6^{2x} + 6^{2x} = 6^y$
find an expression for $y$ in terms of $x$.
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.
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?
Find the values of x for which $(\frac{1}{3})^{x+4} \le (\frac{1}{9})^{x}$.
Solve the inequality $3^{x^2-5x} < 3^{-6}$.
Find the values of x that satisfy the inequality $4^{x^2} \le 8^{x+3}$.