Mathematics AA SL
IB Questions
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Question 1 of 10
The population of a city \(t\) years after 1 January 2018 can be modelled by the function \(P(t)=20000e^{kt}\), where \(k<0\) and \(t\geq0\). It is known that between 1 January 2018 and 1 January 2026 the population decreased by 15%. Use this model to estimate the population of this city on 1 January 2045.
Question 2 of 10
In this question, give all answers correct to two decimal places.
Julia invests \(\$2200\) in a savings account that pays a nominal annual rate of interest of \(3.15\%\), compounded quarterly. Julia makes no further payments to, or withdrawals from, this account.
Calculate the amount that Julia will have in her account after 7 years.
Michael also invests \(\$2200\) in a savings account that pays an annual rate of interest of \(r\%\), compounded annually. Michael makes no further payments or withdrawals from this account.
Find the value of \(r\) required so that the amount in Michael's account after 7 years will be equal to the amount in Julia's account.
Calculate the interest Michael will earn over the 7 years.
Question 3 of 10
Consider the arithmetic sequence \(v_1, v_2, v_3, \ldots\) where the sum of the first \(n\) terms is given by \(S_n = 2n^2 + 3n\).
Calculate the sum of the first four terms.
Given that \(S_5 = 65\), find \(v_5\).
Find \(v_1\).
Find an expression for \(v_n\) in terms of \(n\).
Question 4 of 10
All answers in this question should be given to four significant figures.
In a local weekly raffle, tickets cost \($3\) each.
In the first week of the raffle, a player will receive \($P\) for each ticket, with the probability distribution shown in the following table. For example, the probability of a player receiving \($15\) is 0.04. The grand prize in the first week of the raffle is \($1500\).
| \(p\) | 0 | 3 | 15 | 75 | Grand Prize |
|---|---|---|---|---|---|
| \(P(P=p)\) | 0.80 | \(k\) | 0.04 | 0.003 | 0.0002 |
A table showing the probability distribution for the raffle.
Find the value of \(k\).
Determine whether this raffle is a fair game in the first week. Justify your answer.
If nobody wins the grand prize in the first week, the probabilities will remain the same, but the value of the grand prize will be \($3000\) in the second week, and the value of the grand prize will continue to double each week until it is won. All other prize amounts will remain the same.
Write an expression in terms of \(n\) for the value of the grand prize in the \(n\)th week of the raffle.
Question 5 of 10
Consider any three consecutive integers, \(k, k+1\) and \(k+2\).
Prove that the sum of these three integers is always divisible by 3.
Prove that the sum of the squares of these three integers is never divisible by 3.
Question 6 of 10
Alex and Maya started working for different companies on January 1st 2015. Alex's starting annual salary was \(\$52000\), and his annual salary increases \(2.5\%\) on January 1st each year after 2015.
Calculate Alex's annual salary for the year 2025, to the nearest dollar.
| year (x) | 2015 | 2017 | 2018 | 2022 | 2026 |
|---|---|---|---|---|---|
| annual salary $ S | 52000 | 54500 | 56000 | 61000 | 65500 |
A table showing Maya's annual salary for different years.
Assuming Maya's annual salary can be approximately modelled by the equation \(S = ax + b\), show that Maya had a lower salary than Alex in the year 2025, according to the model.
Question 7 of 10
The expansion of \((x+k)^6\), where \(k > 0\), can be written as \(x^6 + px^5 + qx^4 + rx^3 + sx^2 + ... + k^6\), where \(p, q, r, s, ... \in \mathbb{R}\).
Find an expression, in terms of \(k\), for:
\(p\);
\(q\);
\(s\).
Given that \(p\), \(q\), and \(s\) are the first three terms of a geometric sequence, find the value of \(k\).