Maths AI SL practice questions

A sphere with diameter 3 474 000 metres can model the shape of the Moon.

(a) Use this model to calculate the circumference of the Moon in kilometres. Give your full calculator display.[3]

(b) Give your answer to part (a) correct to three significant figures.[1]

(c) Write your answer to part (b) in the form $a \times 10^k$, where $1 \le a < 10$, $k \in \mathbb{Z}$.[2]

Consider the function $f(x) = x^3 - 2x^2 + 5x - 7$.

(a) Find $f(1)$.[2]

(b) Find $f'(x)$.[2]

(c) Find the equation of the line that is tangent to the curve $y = f(x)$ at $x = 1$.[2]

Zaha is designing a bridge to cross a river. She believes that the weight of the steel needed for this bridge is approximately 53 632 000 kg.

The exact weight of the steel needed for the bridge is 55 625 000 kg.

(a) Find the percentage error in Zaha's approximation.[2]

Zaha's design is used to build five identical bridges.

(b) (i) Find the weight of the steel needed for these five bridges, to three significant figures. [blank]

(ii) Write down your answer to part (b)(i) in the form $a \times 10^k$, where $1 \le a < 10$, $k \in \mathbb{Z}$.[3]

The IB grades attained by a group of students are listed as follows.

6 4 5 3 7 3 5 4 2 5

(a) Find the median grade.[2]

(b) Calculate the interquartile range.[2]

(c) Find the probability that a student chosen at random from the group scored at least a grade 4.[2]

The following table shows four different sets of numbers: $\mathbb{N}$, $\mathbb{Z}$, $\mathbb{Q}$ and $\mathbb{R}$.

(a) Complete the second column of the table by giving one example of a number from each set.[4]

(b) Write down whether Josh's statement is correct. Justify your answer.[2]

Josh states: "Every integer is a natural number".

In this question, give all answers to two decimal places.

Velina travels from New York to Copenhagen with 1200 US dollars (USD). She exchanges her money to Danish kroner (DKK). The exchange rate is 1 USD = 7.0208 DKK.

(a) Calculate the amount that Velina receives in DKK.[2]

At the end of her trip Velina has 3450 DKK left that she exchanges to USD. The bank charges a 5% commission. The exchange rate is still 1 USD = 7.0208 DKK.

(b) (i) Calculate the amount, in DKK, that will be left to exchange after commission.[2]

(ii) Hence, calculate the amount of USD she receives.[2]

The equation of the straight line $L_1$ is $y = 2x - 3$.

(a) Write down the $y$-intercept of $L_1$.[1]

(b) Write down the gradient of $L_1$.[1]

The line $L_2$ is parallel to $L_1$ and passes through the point $(0, 3)$.

(c) Write down the equation of $L_2$.[1]

The line $L_3$ is perpendicular to $L_1$ and passes through the point $(-2, 6)$.

(d) Write down the gradient of $L_3$.[1]

(e) Find the equation of $L_3$. Give your answer in the form $ax + by + d = 0$, where $a$, $b$ and $d$ are integers.[2]

The following data show the heights, in metres, of six players in a basketball team.

(a) For these six players, find[6]

(a)(i) the mean height.

(a)(ii) the median height.

(a)(iii) the modal height.

(a)(iv) the range of the heights.

A new player, Gheorghe, joins the team. Their height is measured as 1.98 metres to the nearest centimetre.

(b) Write down the shortest possible height of Gheorghe.[1]

A calculator fits into a cuboid case with height 29 mm, width 98 mm and length 186 mm.

(a) Find the volume, in $\text{mm}^3$, of this calculator case. Give your answer to two significant figures.[2]

(b) Write down your answer to part (a) in the form $a \times 10^k$ where $1 \le a < 10$ and $k \in \mathbb{Z}$.[2]

(c) Find the volume, in $\text{cm}^3$, of this calculator case.[2]

Consider the following list of 10 data items.

16 19 21 19 18 12 16 21 21 32

This data has a mean of 19.5.

Find the value of the

(a) mode.[1]

(b) median.[1]

(c) standard deviation.[1]

(d) range.[2]

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]

A class of 13 Mathematics students received the following grades in their final IB examination: 3, 5, 3, 4, 7, 3, 2, 7, 5, 6, 5, 3, 4

For these grades, find:

(a) the mode;[1]

(b) the median;[2]

(c) the upper quartile;[1]

(d) the interquartile range.[2]

Each year the soccer team, Peterson United, plays 25 games at their home stadium. The owner of Peterson United claimed that last year the mean attendance per game at their home stadium was 24 500.

(a) Based on the owner's claim, calculate the total attendance for the games at Peterson United's home stadium last year.[2]

The actual total attendance last year was 617 700.

(b) Calculate the percentage error in the owner's claim.[2]

(c) Write down your answer to part (b) in the form $a \times 10^k$ where $1 \leq a < 10, k \in \mathbb{Z}$.[2]

The prices, in dollars, of 10 different garden chairs are:

79, 139, 255, 99, 50, 209, 229, 193, 69, 49

(a) Find the range of the prices of the 10 chairs.[2]

(b) Use your graphic display calculator to find

(i) the mean price of the chairs.

(ii) the standard deviation of the price of the chairs.[3]

In a sale, the price of each of the 10 garden chairs is reduced by $20.

(c) Write down

(i) the new mean.

(ii) the new standard deviation.[2]

The second term of an arithmetic sequence is 30. The fifth term is 90.

(a) Calculate

(i) the common difference of the sequence;[3]

(ii) the first term of the sequence.[3]

The first, second and fifth terms of this arithmetic sequence are the first three terms of a geometric sequence.

(b) Calculate the seventh term of the geometric sequence.[3]