Maths AI HL practice questions

Consider the following function, $f(x)$, defined on the domain of integers from 0 to 4 inclusive.

(a) Solve $f(x) = 4$.[1]

(b) Solve $f(x) = x$.[1]

(c) Complete the following table.[3]

Professor Vinculum investigated the migration season of the Bulbul bird from their natural wetlands to a warmer climate.

He found that during the migration season their population, $P$ could be modelled by $P = 1350 + 400(1.25)^{-t}$, $t \geq 0$, where $t$ is the number of days since the start of the migration season.

(a) Find the population of the Bulbul birds,[3]

(i) at the start of the migration season.

(ii) in the wetlands after 5 days.

(b) Calculate the time taken for the population to decrease below 1400.[2]

(c) According to this model, find the smallest possible population of Bulbul birds during the migration season.[1]

The lengths of the seeds from a particular mango tree are approximated by a normal distribution with a mean of 4 cm and a standard deviation of 0.25 cm.

A seed from this mango tree is chosen at random.

(a) Calculate the probability that the length of the seed is less than 3.7 cm.[2]

It is known that 30% of the seeds have a length greater than $k$ cm.

(b) Find the value of $k$.[2]

For a seed of length $d$ cm, chosen at random, $\text{P}(4 - m < d < 4 + m) = 0.6$.

(c) Find the value of $m$.[2]

Taizo plays a game where he throws one ball at two bottles that are sitting on a table. The probability of knocking over bottles, in any given game, is shown in the following table.

(a) Taizo plays two games that are independent of each other. Find the probability that Taizo knocks over a total of two bottles.[4]

In any given game, Taizo will win $k$ points if he knocks over two bottles, win 4 points if he knocks over one bottle and lose 8 points if no bottles are knocked over.

(b) Find the value of $k$ such that the game is fair.[3]

Sergio is interested in whether an adult's favourite breakfast berry depends on their income level. He obtains the following data for 341 adults and decides to carry out a $\chi^2$ test for independence, at the 10% significance level.

(a) Write down the null hypothesis.[1]

(b) Find the value of the $\chi^2$ statistic.[2]

The critical value of this $\chi^2$ test is 7.78.

(c) Write down Sergio's conclusion to the test in context. Justify your answer.[2]

Professor Wei observed that students have difficulty remembering the information presented in his lectures.

He modelled the percentage of information retained, $R$, by the function $R(t) = 100 e^{-pt}$, $t \geq 0$, where $t$ is the number of days after the lecture.

He found that 1 day after a lecture, students had forgotten 50% of the information presented.

(a) Find the value of $p$.[2]

(b) Use this model to find the percentage of information retained by his students 36 hours after Professor Wei's lecture.[2]

Based on his model, Professor Wei believes that his students will always retain some information from his lecture.

(c) State a mathematical reason why Professor Wei might believe this.[1]

(d) Write down one possible limitation of the domain of the model.[1]

On 1 January 2022, Mina deposited $1000 into a bank account with an annual interest rate of 4%, compounded monthly. At the end of January, and the end of every month after that, she deposits $100 into the same account.

(a) Calculate the amount of money in her account at the start of 2024. Give your answer to two decimal places.[3]

(b) Find how many complete months, counted from 1 January 2022, it will take for Mina to have more than $5000 in her account.[2]

Stars are classified by their brightness. The brightest stars in the sky have a magnitude of 1. The magnitude, $m$, of another star can be modelled as a function of its brightness, $b$, relative to a star of magnitude 1, as shown by the following equation.

$$m = 1 - 2.5 \log_{10}(b)$$

The star called Acubens has a brightness of 0.0525.

(a) Find the magnitude of Acubens.[2]

Ceres has a magnitude of 7 and is the least bright star visible without magnification.

(b) Find the brightness of Ceres.[2]

The star Proxima Centauri has a greater magnitude than the planet Neptune. The difference in their magnitudes is 3.2.

(c) Find how many times brighter Neptune is compared to Proxima Centauri.[3]

A climate scientist is modelling an ice sheet as a rectangle.

She believes that the width $(x \text{ km})$ is increasing at a constant rate of $10$ km per year and the length $(y \text{ km})$ is decreasing at a constant rate of $5$ km per year.

The time, $t$, is measured in years, and the area, $A$, is measured in $\text{km}^2$.

When $t = 0$ then $x = 75$ and $y = 40$.

(a) Find $\frac{dA}{dt}$ when $t = 0$.[3]

(b) State, with justification, whether the area of the ice sheet is increasing or decreasing when $t = 0$.[1]

(c) Find the change in the area of the ice sheet between $t = 0$ and $t = 1$.[3]

Radioactive carbon is a material that decays over time.

The mass, $m(t)$ (in nanograms), of radioactive carbon in a fossil of a plant, after $t$ years, can be modelled by the function

$$m(t) = 120e^{-0.00012t}$$

where $t$ is the time since the plant died.

(a) Write down the initial mass of the radioactive carbon.[1]

(b) Find the mass of the radioactive carbon after 20000 years.[2]

(c) Calculate the smallest number of complete years it takes for more than half the sample to decay.[3]

A meteorologist models the height of a hot air balloon launched from the ground. The model assumes the balloon travels vertically upwards and travels 450 metres in the first minute.

Due to the decrease in temperature as the balloon rises, the balloon will continually slow down. The model suggests that each minute the balloon will travel only 82% of the distance travelled in the previous minute.

(a) Find how high the balloon will travel in the first 10 minutes after it is launched.[3]

(b) The balloon is required to reach a height of at least 2520 metres. Determine whether it will reach this height.[2]

(c) Suggest a limitation of the given model.[1]

On 1 January 2025, the Faber Car Company will release a new car to global markets. The company expects to sell 40 cars in January 2025. The number of cars sold each month can be modelled by a geometric sequence where $r = 1.1$.

(a) Use this model to find the number of cars that will be sold in December 2025.[2]

(b) Use this model to find the total number of cars that will be sold in the year

(i) 2025.[5]

(ii) 2026.

The pH scale is a measure of the acidity of a solution. Its value is given by the formula $\text{pH} = -\log_{10}[\text{H}^+]$, where $[\text{H}^+]$ is the concentration of hydrogen ions in the solution (measured in moles per litre).

(a) Calculate the pH value if the concentration of hydrogen ions is 0.0003.[2]

The pH of milk is 6.6.

(b) Calculate the concentration of hydrogen ions in milk.[2]

The strength of an acid is measured by its concentration of hydrogen ions.

A lemon has a pH value of 2 and a tomato has a pH value of 4.5.

(c) Calculate how many times stronger the acid in a lemon is when compared to the acid in a tomato.[3]

A spherical balloon is being inflated such that its volume is increasing at a rate of $15 \text{ cm}^3 \text{ s}^{-1}$.

(a) Find the radius of the balloon when its volume is $288\pi \text{ cm}^3$.[2]

(b) Hence or otherwise, find the rate of change of the radius at this instant.[3]

A museum has an annual membership fee of $200, which includes 10 free visits. Any additional visits are charged at $30 each. The total cost, $C$, of $n$ visits during the year can be modelled by $$C(n) = \begin{cases} 200, & n < p \\ an+b, & n \geq p \end{cases}$$ where $a, b, p, n \in \mathbb{Z}$.

(a) Write down the value of[2]

(i) $a$

(ii) $p$

(b) Find the value of $b$.[2]