Mathematics AA HL
IB Questions
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Question 1 of 10
Let \(w_1 = 2 + a\mathrm{i}\) and \(w_2 = (2-a^2) - 3a\mathrm{i}\), where \(a \in \mathbb{R}, a \neq 0\).
Find an expression for \(w_1w_2\) in terms of \(a\).
Given that \(\arg(w_1w_2) = \frac{\pi}{3}\), find the value of \(a\).
Question 2 of 10
Let \(w_n\) be the complex number defined as \(w_n = (n^2-2n+2) + i\) for \(n \in \mathbb{N}\).
Find \(\arg(w_0)\).
Write down an expression for \(\arg(w_n)\) in terms of \(n\).
Let \(v_n = w_0w_1w_2w_3...w_{n-1}w_n\) for \(n \in \mathbb{N}\).
Show that \(\arctan(a) + \arctan(b) = \arctan(\frac{a+b}{1-ab})\) for \(a,b \in \mathbb{R}^+, ab < 1\).
Hence or otherwise, show that \(\arg(v_1) = \arctan(\frac{3}{2})\).
Prove by mathematical induction that \(\arg(v_n) = \arctan(n+2)\) for \(n \in \mathbb{N}\).
Question 3 of 10
Consider the series \(\ln y + q\ln y + \frac{1}{4}\ln y + \ldots\), where \(y \in \mathbb{R}, y > 1\) and \(q \in \mathbb{R}, q \neq 0\).
Consider the case where the series is geometric.
Show that \(q = \pm \frac{1}{2}\).
Hence or otherwise, show that the series is convergent.
Given that \(q > 0\) and \(S_{\infty} = 4\), find the value of \(y\).
Now consider the case where the series is arithmetic with common difference \(d\).
Show that \(q = \frac{5}{8}\).
Write down \(d\) in the form \(k\ln y\), where \(k \in \mathbb{Q}\).
The sum of the first \(n\) terms of the series is \(\ln(\frac{1}{y^4})\).
Find the value of \(n\).
Question 4 of 10
Consider the equation \(z^4 + qz^3 + 36z^2 - 72z + 48 = 0\) where \(z \in \mathbb{C}\) and \(q \in \mathbb{R}\). Three of the roots of the equation are \(2 + i\), \(\beta\) and \(\beta^2\), where \(\beta \in \mathbb{R}\).
Find the value of \(\beta\) by considering the product of all the roots of the equation.
Find the value of \(q\).
Question 5 of 10
Consider the arithmetic sequence \((a_1, a_2, a_3, ...)\). The sum of the first \(n\) terms of this sequence is given by \(S_n = n^2 - 2n\).
Find the sum of the first four terms.
Given that \(S_5 = 15\), find \(a_5\).
Find \(a_1\).
Find an expression for \(a_n\) in terms of \(n\).
Consider a geometric sequence \((b_n)\), where \(b_2 = a_1\) and \(b_4 = a_5\).
Find the possible values of the common ratio, \(r\).
Question 6 of 10
An arithmetic sequence has first term \(45\) and common difference \(-3.5\).
Given that the \(p\)th term of the sequence is zero, find the value of \(p\).
Let \(S_n\) denote the sum of the first \(n\) terms of the sequence.
Find the maximum value of \(S_n\).