2000002201
Level:
A
Identify the solution set of the following inequality.
\[
|2x+5|>0
\]
\( \mathbb{R} \setminus \{-2.5\} \)
\( \emptyset \)
\( (-2.5; \infty) \)
\( (-\infty;-2.5 ) \)
A
2000002110
Level:
A
Find the algebraic form of the complex number \( \left(\cos{ \frac{\pi}{6}} + i\sin{ \frac{\pi}{6}}\right)^{-6} \).
\( -1\)
\( i\)
\( -i \)
\( 1\)
2000002109
Level:
A
Given \( z= \sqrt[3]{3} \left(\cos{\frac{3\pi}{4}}+i\sin{\frac{3\pi}{4}}\right) \), determine which of the numbers below does not represent \( z^6\).
\( 9 \)
\( 9i \)
\( 9\left(\cos{\frac{9\pi}{2}}+i\sin{\frac{9\pi}{2}}\right) \)
\( 9\left(\cos{\frac{\pi}{2}}+i\sin{\frac{\pi}{2}}\right) \)
2000002108
Level:
A
Find the value of the principal argument \( \varphi\) of \( \left(3\left(\cos{\frac{3\pi}{2} }+ i\sin{\frac{3\pi}{2} }\right)\right)^{13} \). The principal value is the corresponding angle \(\varphi \in (-\pi; \pi] \).
\( -\frac{\pi}{2} \)
\( \frac{\pi}{2} \)
\( 0 \)
\( \frac{3}{26}\pi \)
2000002107
Level:
A
Find the algebraic form of \( (\cos{45^{\circ}} +i\sin{45^{\circ}})^9 \).
\( \frac{\sqrt{2}}{2} + i\frac{\sqrt{2}}{2} \)
\( -\frac{\sqrt{2}}{2} - i\frac{\sqrt{2}}{2} \)
\( \frac{\sqrt{3}}{2} - \frac{i}{2} \)
\( i +1 \)
2000002106
Level:
A
Which of the following numbers is not the value of the argument of \( \left(\cos{\frac{\pi}{4}} + i\sin{\frac{\pi}{4}}\right)^5 \)?
\( \frac{\pi}{4} \)
\( \frac{5\pi}{4} \)
\( -\frac{3\pi}{4} \)
\( \frac{13\pi}{4} \)
2000002105
Level:
A
Find the value of the principal argument \( \varphi\) of \( \left(\cos{\frac{2\pi}{3} }+ i\sin{\frac{2\pi}{3} }\right)^{18} \). The principal value is the corresponding angle \(\varphi \in (-\pi; \pi] \).
\( 0 \)
\( \pi \)
\( -\pi \)
\( \frac{\pi}{3} \)
2000002104
Level:
A
Find the algebraic form of the complex number \(\left(\cos{\left(-\frac{\pi}{3}\right)} + i\sin{\left(-\frac{\pi}{3}\right)} \right)^{99} \).
\( -1 \)
\( -i\)
\(1\)
\(i\)
2000002103
Level:
A
Find the algebraic form of the complex number \( \left(\sqrt{2}\left(\cos{\frac{\pi}{2}}+i\sin{\frac{\pi}{2}}\right)\right)^{18} \).
\( -512 \)
\( 512i \)
\( 512 \)
\( -512i \)
2000002102
Level:
A
Consider \( z= \cos{\frac{\pi}{4}} + i\sin{\frac{\pi}{4}} \) and find \(z^9\).
\( \cos{\frac{\pi}{4}} + i\sin{\frac{\pi}{4}} \)
\( 9 \left(\cos{\frac{\pi}{4}} + i\sin{\frac{\pi}{4}}\right) \)
\( \cos{\frac{9\pi}{4}} - i\sin{\frac{9\pi}{4}} \)
\( 9\left(\frac{\sqrt{2}}{2} + i\frac{\sqrt{2}}{2} \right) \)