A

2010004903

Level: 
A
The seventh term of a geometric sequence is \( 32 \) and the tenth term is \( 4 \). Choose the correct formula to find the eighth term of this sequence.
\( a_8=32\cdot\sqrt[3]{\frac4{32}} \)
\( a_8=32\cdot\sqrt[3]{\frac{32}4} \)
\( a_8=4\cdot\sqrt[3]{\frac4{32}} \)
\( a_8=4\cdot\sqrt[3]{\frac{32}4} \)
\( a_8=8\cdot\sqrt[3]{\frac3{24}} \)

2010004617

Level: 
A
Let \( z \in \mathbb{C}\). The value of the argument of \(z^5\) is \(300^{\circ}\) and \(|z|^5=\frac1{32}\). Find \(z\).
\( z=\frac{1}{4}(1+\mathrm{i}\sqrt{3})\)
\( z=\frac{1}{4}(1-\mathrm{i}\sqrt{3})\)
\( z=-\frac{1}{2}\mathrm{i}\)
\( z=\frac{1}{2}(\cos 60^{\circ} - \mathrm{i} \sin 60^{\circ})\)

2010004616

Level: 
A
Let \( z \in \mathbb{C}\). The value of the argument of \(z^6\) is \(270^{\circ}\) and \(|z|^6=27\). Find \(z\).
\( z=\frac{\sqrt{6}}{2}(1+\mathrm{i})\)
\( z=\frac{\sqrt{6}}{2}(1-\mathrm{i})\)
\( z=\sqrt{3}\mathrm{i}\)
\( z=3(\cos 45^{\circ} + \mathrm{i} \sin 45^{\circ})\)