9000034805
Level:
A
Find the complex number \(z\)
which satisfies \(2z = 2 - 3\mathrm{i}\).
\(1 -\frac{3}
{2}\mathrm{i}\)
\(- 3\mathrm{i}\)
\(4 - 6\mathrm{i}\)
\(- 1 + \frac{3}
{2}\mathrm{i}\)
Complex numbers in algebraic and polar form
9000034807
Level:
B
Find the polar form of the complex number
\(z = 2\mathrm{i}\).
\(2\left (\cos \frac{\pi }{2} + \mathrm{i}\sin \frac{\pi }{2}\right )\)
\(\sqrt{2}\left (\cos \frac{\pi }{2} + \mathrm{i}\sin \frac{\pi }{2}\right )\)
\(\cos \frac{\pi }{2} + \mathrm{i}\sin \frac{\pi }{2}\)
\(2\left (\cos 0 + \mathrm{i}\sin 0\right )\)
9000034808
Level:
B
Find the algebraic form of the complex number
\(z = 2\left (\cos \pi + \mathrm{i}\sin \pi \right )\).
\(- 2\)
\(2\)
\(- 2\mathrm{i}\)
\(2\mathrm{i}\)
9000034804
Level:
A
Find the absolute value of the complex number
\(z = 3 -\mathrm{i}\).
\(\sqrt{10}\)
\(2\)
\(2\sqrt{2}\)
\(\sqrt{2}\)
9000034809
Level:
B
Given complex numbers \(z_{1} = 2\left (\cos \frac{\pi }{6} + \mathrm{i}\sin \frac{\pi }{6}\right )\)
and \(z_{2} = \sqrt{3}\left (\cos \frac{4\pi }
{3} + \mathrm{i}\sin \frac{4\pi }
{3}\right )\), find the angle in the
polar form of the product \(z_{1}z_{2}\).
\(\frac{3\pi }
{2}\)
\(\frac{2}
{9}\pi \)
\(\frac{5}
{9}\pi \)
\(3\pi \)
9000034810
Level:
B
Given complex numbers \(z_{1} = 2\left (\cos \frac{\pi }{4} + \mathrm{i}\sin \frac{\pi }{4}\right )\)
and \(z_{2} = \sqrt{2}\left (\cos \frac{7\pi }
{4} + \mathrm{i}\sin \frac{7\pi }
{4}\right )\),
find the angle in the polar form of the quotient
\(\frac{z_{1}}
{z_{2}} \).
\(\frac{\pi }
{2}\)
\(- \frac{\pi } {2}\)
\(-\frac{3}
{2}\pi \)
\(\frac{3}
{2}\pi \)
9000034801
Level:
A
Given complex numbers \(z_{1} = 4 -\mathrm{i}\)
and \(z_{2} = 1 - 2\mathrm{i}\),
find \(z_{1} - z_{2}\).
\(3 + \mathrm{i}\)
\(3 - 3\mathrm{i}\)
\(5 - 3\mathrm{i}\)
\(3 -\mathrm{i}\)
9000031210
Level:
B
Given complex numbers \(z_{1} = 2\sqrt{3}\left (\cos \frac{\pi }{6} + \mathrm{i}\sin \frac{\pi }{6}\right )\)
and \(z_{2} = \sqrt{3}\left (\cos \frac{4\pi }
{3} + \mathrm{i}\sin \frac{4\pi }
{3}\right )\), find the
quotient \(\frac{z_{1}}
{z_{2}} \).
\(-\sqrt{3} + \mathrm{i}\)
\(\sqrt{3} -\mathrm{i}\)
\(\sqrt{3} + \mathrm{i}\)
\(-\sqrt{3} -\mathrm{i}\)
9000031209
Level:
B
Given complex numbers \(z_{1} = 2\sqrt{2}\left (\cos \frac{\pi }{4} + \mathrm{i}\sin \frac{\pi }{4}\right )\)
and \(z_{2} = \sqrt{2}\left (\cos \frac{7\pi }
{4} + \mathrm{i}\sin \frac{7\pi }
{4}\right )\), find
the product \(z_{1}z_{2}\).
\(4\)
\(4\mathrm{i}\)
\(- 4\mathrm{i}\)
\(- 4\)
9000031205
Level:
A
Find the complex conjugate of \(z = \mathrm{i}^{5} - 3\mathrm{i}^{10}\).
\(3 -\mathrm{i}\)
\(- 3 -\mathrm{i}\)
\(- 3 + \mathrm{i}\)
\(3 + \mathrm{i}\)