1003067701
Level:
A
Evaluate \( \mathrm{i}^{100}\cdot\mathrm{i}^{99}\cdot\mathrm{i}^{98}\cdot\mathrm{i}^{97}\cdot\dots\cdot\mathrm{i}^4\cdot\mathrm{i}^3\cdot\mathrm{i}^2\cdot\mathrm{i}\).
\( -1 \)
\( 1 \)
\( \mathrm{i} \)
\( -\mathrm{i} \)
Complex numbers in algebraic and polar form
1003067702
Level:
A
Evaluate \( \mathrm{i}^{100}+\mathrm{i}^{99}+\mathrm{i}^{98}+\mathrm{i}^{97}+\dots+\mathrm{i}^4+\mathrm{i}^3+\mathrm{i}^2+\mathrm{i}\).
\( 0 \)
\( 1-\mathrm{i} \)
\( -1 \)
\( -1+\mathrm{i} \)
1003067703
Level:
A
Evaluate \( \mathrm{i}^{10} - \mathrm{i}^{20} + \mathrm{i}^{30} - \mathrm{i}^{40} + \mathrm{i}^{50} - \mathrm{i}^{60} \).
\( -6 \)
\( 0 \)
\( -1 \)
\( 6 \)
1003067704
Level:
A
Evaluate \( \mathrm{i}^5 - \mathrm{i}^{10} + \mathrm{i}^{15} - \mathrm{i}^{20} + \mathrm{i}^{25} - \mathrm{i}^{30} + \mathrm{i}^{35} - \mathrm{i}^{40} \).
\( 0 \)
\( 1 \)
\( -1 \)
\( -\mathrm{i} \)
1003067705
Level:
A
Given complex numbers \(z_1 = -2 + \mathrm{i} \), \( z_2 = 1 - 4\mathrm{i} \) and \( z_3 = -3\mathrm{i} \), find \( 2\cdot z_1 + 3\cdot z_2- 4\cdot z_3 \).
\( -1 + 2\mathrm{i} \)
\( 7 + 2\mathrm{i} \)
\( 11 + 26\mathrm{i} \)
\( -1 + 26\mathrm{i} \)
1003067706
Level:
A
Given complex numbers \( z_1 = -2 + \mathrm{i} \), \( z_2 = 1 - 4\mathrm{i} \) and \( z_3 = -3\mathrm{i} \), find \(\frac{z_1\cdot z_2}{3\cdot z_3}\).
\( -1 + \frac29\mathrm{i} \)
\( -1 - \frac29\mathrm{i} \)
\( 1 + \frac29\mathrm{i} \)
\( 1 - \frac29\mathrm{i} \)
1003067707
Level:
A
Find the complex conjugate of the following complex number.
\[ \frac{2-\mathrm{i}}{2+\mathrm{i}}-\frac{2+\mathrm{i}}{2-\mathrm{i}} \]
\( \frac85\mathrm{i} \)
\( -\frac85\mathrm{i} \)
\( \frac83\mathrm{i} \)
\( -\frac83\mathrm{i} \)
1003067708
Level:
A
Let \( z=\frac{1-\mathrm{i}}{1+\mathrm{i}}+\frac{1+\mathrm{i}}{1-\mathrm{i}} \). If \( z \) is a complex number, find its absolute value.
\( 0 \)
\( 2 \)
\( 4 \)
Number \( z \) is not a complex number.
1103067709
Level:
A
Choose a diagram, which shows in red all the complex numbers whose absolute value is three.
1103067710
Level:
A
Assuming \( |z - 1| \leq 2 \), choose a diagram, which shows in red all the complex numbers \( z \).
