Number and algebra: 132 IB 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]

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 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]

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]

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 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]

Only one of the following four sequences is arithmetic and only one of them is geometric.

$a_n = 1, 2, 3, 5, \ldots$

$b_n = 1, \frac{3}{2}, \frac{9}{4}, \frac{27}{8}, \ldots$

$c_n = 1, \frac{1}{2}, \frac{1}{3}, \frac{1}{4}, \ldots$

$d_n = 1, 0.95, 0.90, 0.85, \ldots$

(a) State which sequence is[2]

(i) arithmetic;

(ii) geometric.

(b) For another geometric sequence $e_n = -6, -3, -\frac{3}{2}, -\frac{3}{4}, \ldots$

(i) write down the common ratio;[1]

(ii) find the exact value of the tenth term. Give your answer as a fraction.[3]

Two propositions $p$ and $q$ are defined as follows

$p$: Eva is on a diet

$q$: Eva is losing weight

(a) Write down the following statement in words. $q \Rightarrow p$[2]

(b) Write down, in words, the contrapositive statement of $q \Rightarrow p$.[2]

(c) Determine whether your statement in part (a) is logically equivalent to your statement in part (b). Justify your answer.[2]

Consider the three propositions $p$, $q$ and $r$.

$p$: The food is well cooked

$q$: The drinks are chilled

$r$: Dinner is spoilt

(a) Write the following compound proposition in words.

$(p \land q) \Rightarrow \neg r$[3]

(b) Complete the following truth table.[3]

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

Dumisani has received a scholarship of 5000 US dollars (USD) to study in Singapore. He has to travel from South Africa and must change USD for his air fare of 6600 South African rand (ZAR).

The exchange rate is 1 USD = 8.2421 ZAR.

(a) Calculate the number of USD that Dumisani must change to pay for his air fare.[2]

On arrival in Singapore, Dumisani changes 3000 USD to Singapore dollars (SGD) to pay for his school fees. There is a 2.8% commission charged on the exchange.

(b) Calculate the value, in USD, of the commission that Dumisani has to pay.[2]

The exchange rate is 1 USD = 1.29903 SGD.

(c) Calculate the number of SGD Dumisani receives.[2]

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]

The table shows the first five terms of three sequences: $u_n$, $v_n$ and $w_n$.

(a) State which sequence is

(i) arithmetic;[2]

(ii) geometric.[2]

(b) Find the exact value of the 11th term of the geometric sequence.[2]

(c) Find the sum of the first 20 terms of the arithmetic sequence.[2]

When Daniel retires, he invests $400 000 in an annuity fund that earns interest at a nominal rate of 4.5% per year, compounded monthly.

Daniel then withdraws $3600 at the end of every month to pay for his living expenses.

(a) Find how much is in the annuity fund after 5 years.[3]

(b) Calculate how many times Daniel is able to make these withdrawals.[3]