Number and algebra: 37 IB Maths AI HL practice questions

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]

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.

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]

An infinite geometric sequence, with terms $u_n$, is such that $u_1 = 2$ and $\sum_{k=1}^{\infty} u_k = 10$.

(a) Find the common ratio, $r$, for the sequence.[2]

(b) Find the least value of $n$ such that $u_n < \frac{1}{2}$.[3]

Roger buys a new laptop for himself at a cost of $495. At the same time, he buys his daughter Chloe a higher specification laptop at a cost of $2200.

It is anticipated that Roger's laptop will depreciate at a rate of 10% per year, whereas Chloe's laptop will depreciate at a rate of 15% per year.

(a) Estimate the value of Roger's laptop after 5 years.[2]

Roger and Chloe's laptops will have the same value $k$ years after they were purchased.

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

(c) Comment on the validity of your answer to part (b).[1]

The growth of a particular type of seashell is being studied by Manon. At the end of each month Manon records the increase in the width of a seashell since the end of the previous month.

She models the monthly increase in the width of the seashell by a geometric sequence with common ratio $0.8$. In the first month, the width of the seashell increases by $4$ mm.

(a) Find by how much the width of the seashell will increase during the third month, according to her model.[2]

(b) Find the total increase in the width of the seashell, predicted by Manon's model, during the first year.[2]

Manon's seashell had a width of $25$ mm at the beginning of the first month.

(c) Find the maximum possible width of the seashell, predicted by Manon's model.[3]

Two complex numbers are $z = 2 + ai$ and $w = b + 4i$, where $a, b \in \mathbb{R}$.

(a) Find an expression for the real part of $z^2$, in terms of $a$.[2]

(b) Find the value of $a$ and of $b$, given that $z^2 + 2w = 3 + 6i$.[3]

Give answers to this question correct to two decimal places.

Pierre invests 1500 euros (EUR) at the end of each month for 10 years into a savings plan that pays a nominal annual interest rate of 3.6% compounded monthly.

(a) Calculate the value of Pierre's savings plan at the end of the 10 years.[3]

At the end of the 10 years, Pierre withdraws 100 000 EUR from the savings plan to use as a deposit on a house.

Pierre invests the remainder into another account for 15 years at a nominal annual interest rate of 4.5% compounded quarterly.

(b) Calculate the amount in Pierre's account at the end of this time.[4]

Maan deposited $100000 into a savings account with a nominal annual interest rate of $I\%$ compounded monthly. At the end of the eighth year, the amount in the account had increased to $150000.

(a) Find the value of $I$.[3]

Maan withdraws the $150000 and places it in an annuity, earning a nominal annual interest rate of 6.1% compounded monthly. At the end of each month, Maan will receive a payment of $1000.

(b) Find the amount of money remaining in the annuity at the end of 10 years. Express your answer to the nearest dollar.[3]

A manufacturer designs hats that businesses can order.

For an order of 15 hats or fewer, the price per hat is $p$ euros (EUR). For larger orders, the price of **each hat** after the first 15 ordered is reduced by 5 EUR.

$C(n)$ represents the total cost of purchasing $n$ hats from the manufacturer.

(a) Write down an expression in terms of $p$ for the total cost of ordering 15 hats.[1]

(b) Write down an equation for $C(n)$, in terms of $p$ and $n$, for the total cost of ordering $n$ hats when $n > 15$.[3]

A company decides to order 100 hats. The mean price per hat for this order is 25.73 EUR.

(c) (i) Calculate the exact total cost of the order.[1]

(ii) Hence, determine the value of $p$, correct to 2 decimal places.[3]

A bakery sells egg tarts in boxes. Each box contains 6 tarts. On a particular morning, the bakery has 400 freshly baked tarts. The number of tarts remaining, after $n$ boxes have been sold, forms an arithmetic sequence, $u_n = u_1 + (n - 1)d$.

(a) Write down the value of $d$.[1]

(b) Find the value of $u_1$.[1]

(c) Find the greatest number of boxes the bakery can sell that morning.[3]

The formula $F = 1.8C + 32$ is used to convert a temperature in degrees Celsius, $C$, to degrees Fahrenheit, $F$.

(a)(i) Find a formula for converting a temperature in degrees Fahrenheit to degrees Celsius.[2]

(a)(ii) Find the temperature in degrees Celsius that is recorded as 77 degrees Fahrenheit.[3]

Over one year, the mean daily temperature in Mexico City was calculated to be 17 degrees Celsius with a standard deviation of 9 degrees Celsius.

(b)(i) For the same year, find in degrees Fahrenheit the mean daily temperature in Mexico City.[1]

(b)(ii) the standard deviation of the daily temperature in Mexico City.[3]

The ticket prices for a concert are shown in the following table.

• A total of 600 tickets were sold.

• The total amount of money from ticket sales was $7816.

• There were twice as many adult tickets sold as child tickets.

Let the number of adult tickets sold be $x$, the number of child tickets sold be $y$, and the number of student tickets sold be $z$.

(a) Write down three equations that express the information given above.[3]

(b) Find the number of each type of ticket sold.[2]

The following table shows the time, in days, from December 1st and the percentage of Christmas trees in stock at a shop on the beginning of that day.

The following table shows the natural logarithm of both $d$ and $x$ on these days to 2 decimal places.

(a) Use the data in the second table to find the value of $m$ and the value of $b$ for the regression line, $\ln x = m(\ln d) + b$.[2]

(b) Assuming that the model found in part (a) remains valid, estimate the percentage of trees in stock when $d = 25$.[3]

Let $M = \begin{pmatrix} -4 & 2 \\ -3 & 3 \end{pmatrix}$.

(a) Find the eigenvalues of $M$.[3]

$M$ can be written in the form $M = PDP^{-1}$, where $D$ is a diagonal matrix.

(b) (i) Write down $D$.[2]

(ii) Find $P$.[5]