Mathematics AI HL
IB Questions
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Question 1 of 10
Sarah purchases a new car for \(\$35,000\). On the same day, her brother Tom buys a luxury vehicle for \(\$85,000\).
Sarah's car is expected to depreciate at a rate of 12% per year, while Tom's car will depreciate at a rate of 18% per year.
Calculate the value of Sarah's car after 4 years.
Find after how many years (\(k\)) both cars will have the same value.
Comment on the validity of your answer to part (b).
Question 2 of 10
Maya has \(\$750\) in her savings account. She considers investing the money for 4 years with a bank. The bank offers an annual interest rate of \(1.8\%\) compounded monthly.
Calculate the amount of money Maya would have at the end of 4 years with the bank. Give your answer correct to two decimal places.
Instead of investing the money, Maya decides to buy a laptop that costs \(\$750\). At the end of 4 years, the laptop will have a value of \(\$120\). It may be assumed that the depreciation rate per year is constant.
Calculate the annual depreciation rate of the laptop.
Question 3 of 10
The transformation \(T\) is represented by the matrix \({M}=\left(\begin{array}{cc}3 & -2 \\ 1 & 4\end{array}\right)\).
A triangle with an area of \(15\text{ cm}^2\) is transformed by \(T\).
Find the area of the image of the triangle.
Under the transformation \(T\), the image of point P has coordinates \((3t+2, 4-2t)\), where \(t \in \mathbb{R}\).
Find, in terms of \(t\), the coordinates of P.
Question 4 of 10
Let \(w_1 = 4\operatorname{cis}(\frac{2\pi}{3})\) and \(w_2 = 3\operatorname{cis}(\frac{m\pi}{12})\), where \(m \in \mathbb{Z}^+\).
In parts (a)(i) and (a)(ii), give your answers in the form \(re^{i\theta}\), where \(r \geq 0\) and \(-\pi < \theta \leq \pi\).
Find the value of \(w_1^2\).
Find the value of \((\frac{w_1}{w_2})^3\) for \(m = 3\).
Find the least value of \(m\) such that \(w_1w_2 \in \mathbb{R}^+\).
Question 5 of 10
Alex and Beth start training for a marathon. On day one, they both swim 300 meters. On each subsequent day, Alex swims 50 meters more than the previous day, whereas Beth increases her distance by \(3\%\) of the distance swum on the previous day.
Calculate how far:
Alex swam on day 15 of the training programme.
Beth swam on day 15 of the training programme.
Question 6 of 10
The equation of the line \(y = kx + d\) can be expressed in vector form \(\mathbf{r} = \mathbf{p} + \lambda\mathbf{q}\).
Find the vectors \(\mathbf{p}\) and \(\mathbf{q}\) in terms of \(k\) and/or \(d\).
The matrix \(\mathbf{N}\) is defined by \(\begin{pmatrix} 4 & 2 \\ 8 & 4 \end{pmatrix}\).
Calculate the value of \(\det \mathbf{N}\).
The line \(y = kx + d\) (where \(k \neq -4\)) is transformed using the matrix \(\mathbf{N}\).
Show that the equation of the resulting line does not depend on \(k\) or \(d\).