2000002704
Level:
B
Given \(n \in \mathbb{N} \), find the sum: \(\left({n+1\above 0.0pt n} \right) + \left({n\above 0.0pt 0} \right)\)
\( n+2\)
\(n\)
\(n+1\)
\(2\)
B
2000002703
Level:
B
Specify the domain of the expression: \(\left({5\above 0.0pt
n} \right) +\left({n\above 0.0pt
3} \right)\)
\( n \in \{3;4;5\} \)
\( n \in \mathbb{N},~n\leq 5\)
\( n \in \mathbb{N} \)
\( n \in \mathbb{N},~n\geq 2\)
2000002702
Level:
B
Calculate the difference: \(\left({17\above 0.0pt
16} \right) - \left({17\above 0.0pt
17} \right)\)
\( 16\)
\(17\)
\(1\)
\(0\)
2000002701
Level:
B
Simplify for \(n \in \mathbb{N}\): \(\left({n+5\above 0.0pt
n+4} \right)\)
\( n+5\)
\(n+4\)
\(1\)
\(5\)
2000002608
Level:
B
Find the right formula for solving the equation \(x^5 +32=0\).
\( x_k = \sqrt[5]{|-32|}( \cos\frac{\pi +2k\pi}{5}+ i\sin \frac{\pi +2k\pi}{5})\), \(k=0,1,2,3,4\)
\( x_k = \sqrt[5]{-32}( \cos\frac{\pi +2k\pi}{5}+ i\sin \frac{\pi +2k\pi}{5})\), \(k=0,1,2,3,4\)
\( x_k = \sqrt[5]{|-32|}( \cos \frac{\pi +k\pi}{5}+ i\sin \frac{\pi +k\pi}{5})\), \(k=0,1,2,3,4\)
\( x_k = \sqrt[5]{|-32|}( \cos \frac{\pi +2k\pi}{5}+ \sin \frac{\pi +2k\pi}{5})\), \(k=0,1,2,3,4\)
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 \(x=1+i\).