2000002610
Level:
C
One solution of the equation \( x^2-72i=0\) is \(x_1 = 6(1+i)\). Find the missing solution.
\( x_2 = -6(1+i) \)
\( x_2 = 6(1-i) \)
\( x_2 = 6(-1+i) \)
\( x_2 = -6(1-i) \)
Powers and roots of complex numbers
2000002608
Level:
B
Find the right formula for solving the equation \(x^5 +32=0\).
\( x_k = \sqrt[5]{|-32|}( \cos\frac{\pi +2k\pi}{5}+ i\sin \frac{\pi +2k\pi}{5})\), \(k=0,1,2,3,4\)
\( x_k = \sqrt[5]{-32}( \cos\frac{\pi +2k\pi}{5}+ i\sin \frac{\pi +2k\pi}{5})\), \(k=0,1,2,3,4\)
\( x_k = \sqrt[5]{|-32|}( \cos \frac{\pi +k\pi}{5}+ i\sin \frac{\pi +k\pi}{5})\), \(k=0,1,2,3,4\)
\( x_k = \sqrt[5]{|-32|}( \cos \frac{\pi +2k\pi}{5}+ \sin \frac{\pi +2k\pi}{5})\), \(k=0,1,2,3,4\)
2000002607
Level:
C
Which of the following equations has not the solution \(x=i\)?
\( x^6 -1 =0\)
\( x^6 +1 =0\)
\( x^3 +i =0\)
\( x^5 -i=0\)
2000002606
Level:
B
Imagine all the solutions of the equation \(x^6 -64 =0\) shown as points in the complex plane. Find the false statement.
Two points lie on the imaginary axis.
The values of the arguments of any two solutions differ by an integer multiple of \(\frac{\pi}{3}\).
All solutions of the equation lie on a circle centered at the origin with a radius of \(2\).
Two points lie on the real axis.
2000002605
Level:
B
How many solutions does the equation \(2x^4=32\) have in the set of complex numbers?
four
one
two
eight
2000002604
Level:
B
Find the solution set of the equation \(x^4+81=0\) if you know that one of its roots is \(\frac{3}{\sqrt{2}}(1+i)\).
\( \left\{ \frac{3}{\sqrt{2}}(1+i); -\frac{3}{\sqrt{2}}(1+i); \frac{3}{\sqrt{2}}(1-i);-\frac{3}{\sqrt{2}}(1-i) \right\} \)
\( \left\{ \frac{3}{\sqrt{2}}(1+i); -\frac{3}{\sqrt{2}}(1+i);3;-3 \right\} \)
\( \left\{ \frac{3}{\sqrt{2}}(1+i); \frac{3}{\sqrt{2}}(1-i);3i;-3i \right\} \)
\( \left\{\frac{3}{\sqrt{2}}(1+i);\frac{3}{\sqrt{2}}(1-i) \right\}\)
2000002603
Level:
B
One of the roots of the equation \(x^3-8=0\) is \(x_1 = -1-i\sqrt{3}\). Find the sum of all its roots.
\( 0\)
\( -8\)
\( -2i\sqrt{3} \)
\(-4\)
2000002602
Level:
B
Consider the equation \(x^4 =1\), where \(x\) is a complex variable. Which of the following statements is true?
The equation has four different complex roots.
The equation has no real root.
The equation has two double roots: \(x_{1,2}=1\) and \(x_{3,4}=-1\).
The equation has the root \(x=1+i\).
2000002601
Level:
B
Consider the equation \(x^4+16=0\). Which of the given numbers is the solution of this equation?
\( 2(\cos{\frac{\pi}{4}}+ i \sin{\frac{\pi}{4}}) \)
\( 2i\)
\( -2i\)
\( 2(\cos{{\pi}}+ i \sin{{\pi}}) \)
2000002110
Level:
A
Find the algebraic form of the complex number \( \left(\cos{ \frac{\pi}{6}} + i\sin{ \frac{\pi}{6}}\right)^{-6} \).
\( -1\)
\( i\)
\( -i \)
\( 1\)