B

2010013106

Level: 
B
Let \(z_1 = 1 + \mathrm{i}\sqrt{3}\), \(z_2=\sqrt3 + \mathrm{i}\). Identify a complex number that is not equal to \(\frac{z_1}{z_2}\).
\(\cos \frac{7\pi}{6} + \mathrm{i} \sin \frac{7\pi}{6}\)
\(\cos \frac{\pi}{6} +\mathrm{i} \sin \frac{\pi}{6}\)
\( \frac{\sqrt{3}}{2} + \frac{\mathrm{i}}{2}\)
\(\cos \left(-\frac{\pi}{6}\right) - \mathrm{i} \sin \left(-\frac{\pi}{6}\right)\)

2010013105

Level: 
B
Let \(z_1 = \sqrt3 + \mathrm{i}\), \(z_2=1 + \mathrm{i}\sqrt3\). Identify a complex number that is not equal to \(\frac{z_1}{z_2}\).
\(\cos \frac{5\pi}{6} + \mathrm{i} \sin \frac{5\pi}{6}\)
\(\cos \left(-\frac{\pi}{6}\right) + \mathrm{i} \sin \left(-\frac{\pi}{6}\right)\)
\( \frac{\sqrt{3}}{2} - \frac{\mathrm{i}}{2}\)
\(\cos \frac{\pi}{6} - \mathrm{i} \sin \frac{\pi}{6}\)

2010013104

Level: 
B
Let \( [x;y]\in\mathbb{R}\times\mathbb{R} \), \( z_1 = -2 + xy\,\mathrm{i} \) and \( z_2 = x + y + 8\,\mathrm{i}\). Find all \( [x;y] \) such that \( z_1 \) and \( z_2 \) are the opposite numbers.
\( [x;y] \in\left\{[4;-2],[-2;4]\right\} \)
\( [x;y] \in \left\{[4;-2]\right\} \)
\( [x;y] \in\left\{[-4;2],[2;-4]\right\} \)
\( [x;y] \in \left\{[-2;4]\right\} \)

2010013103

Level: 
B
Given the complex numbers \( a=3\sqrt{2}\left(\cos120^{\circ}+\mathrm{i}\sin120^{\circ}\right) \), \( b=2\left(\cos\frac{4\pi}3+\mathrm{i}\sin\frac{4\pi}{3}\right) \) and \( c=\frac{\sqrt{2}}{2}\left(\cos\frac{\pi}4+\mathrm{i}\sin\frac{\pi}{4}\right) \), evaluate \( \frac{a\cdot b}{c} \).
\( 12\left(\cos\frac{7\pi}4+\mathrm{i}\sin\frac{7\pi}4\right) \)
\( 12\left(\sin\frac{7\pi}4+\mathrm{i}\cos\frac{7\pi}4\right) \)
\( 6\left(\cos\frac{7\pi}4+\mathrm{i}\sin\frac{7\pi}4\right) \)
\( 12\left(\cos\frac{9\pi}4+\mathrm{i}\sin\frac{9\pi}4\right) \)

2010013102

Level: 
B
Given the complex numbers \( a=2\left(\cos⁡ \frac{\pi}{3}+\mathrm{i}\sin⁡ \frac{\pi}{3}\right) \), \( b=\sqrt{2}\left(\cos⁡ \frac{5\pi}{4}+\mathrm{i}\sin \frac{5\pi}{4}\right) \) and \( c=2\sqrt{2}\left(\cos \left(-\frac{\pi}{6}\right)+\mathrm{i}\sin \left(-\frac{\pi}{6}\right)\right) \), evaluate \( a\cdot b\cdot c \).
\(8\left(\cos \frac{17\pi}{12}+\mathrm{i}\sin \frac{17\pi}{12} \right) \)
\(8\left(\cos \frac{17\pi}{12}-\mathrm{i}\sin \frac{17\pi}{12} \right) \)
\(8\left(\cos \frac{7\pi}{4}+\mathrm{i}\sin \frac{7\pi}{4} \right) \)
\(4\sqrt{2}\left(\cos \frac{17\pi}{12}+\mathrm{i}\sin \frac{17\pi}{12} \right) \)

2010013017

Level: 
B
Let \(f\) be a function defined by \(f(x)=\left(\frac12\right)^{x-m}+m\), where \(m\) is a parameter. Which of the following statements about the function \(f\) and the line \(y=2\) is false?
The graph of \(f\) and the line do not have a common point for any \(m\in\left(-\infty;2\right)\).
The graph of \(f\) and the line do not have a common point for any \(m\in\left. [ 2;\infty\right)\).
The graph of \(f\) and the line do not have a common point for \(m=2\).
The graph of \(f\) and the line do not have a common point for any \(m\in\left(2;\infty\right)\).

2010013016

Level: 
B
Let \(f\) be a function defined by \(f(x)=2^{x+m}-m\), where \(m\) is a parameter. Which of the following statements about the function \(f\) and the line \(y=-2\) is false?
The graph of \(f\) and the line do not have a common point for any \(m\in\left(2;\infty\right)\).
The graph of \(f\) and the line do not have a common point for any \(m\in\left(-\infty;2 \right. ] \).
The graph of \(f\) and the line do not have a common point for \(m=2\).
The graph of \(f\) and the line do not have a common point for any \(m\in\left(-\infty;2\right)\).