9000070109
Level:
A
Evaluate the following complex number.
\[
\left (\sqrt{3} -\mathrm{i}\right )^{3}
\]
\(- 8\mathrm{i}\)
\(8\)
\(- 8\)
\(8\mathrm{i}\)
A
9000065910
Level:
A
Given function
\[
F(x) = x + 2\ln |x|-\frac{1}
{x},
\]
find the function \(f\)
such that \(F\) is
primitive to \(f\)
on \((0;+\infty )\).
\(f(x) = \frac{x^{2}+2x+1}
{x^{2}} \)
\(f(x) = \frac{x^{2}}
{(x+1)^{2}} \)
\(f(x) = \frac{x^{2}-1}
{x^{2}} \)
\(f(x) = \frac{x^{2}}
{(x-1)^{2}} \)
9000070102
Level:
A
Evaluate the following complex number.
\[
\left (\cos \frac{\pi }
{3} + \mathrm{i}\sin \frac{\pi }
{3}\right )^{10}
\]
\(-\frac{1}
{2} -\mathrm{i}\frac{\sqrt{3}}
{2} \)
\(-\frac{\sqrt{3}}
{2} -\frac{1}
{2}\mathrm{i}\)
\(-\frac{\sqrt{3}}
{2} + \frac{1}
{2}\mathrm{i}\)
\(-\frac{1}
{2} + \mathrm{i}\frac{\sqrt{3}}
{2} \)
9000070103
Level:
A
Evaluate the following complex number.
\[
\left (\cos \pi + \mathrm{i}\sin \pi \right )^{9}
\]
\(- 1\)
\(1\)
\(\mathrm{i}\)
\(-\mathrm{i}\)
9000070104
Level:
A
Evaluate the following complex number.
\[
\left (\sin 2\pi + \mathrm{i}\cos 2\pi \right )^{11}
\]
\(-\mathrm{i}\)
\(- 1\)
\(1\)
\(\mathrm{i}\)
9000070105
Level:
A
Evaluate the following complex number.
\[
\mathrm{i}^{13}
\]
\(\cos \frac{\pi }
{2} + \mathrm{i}\sin \frac{\pi }
{2}\)
\(\cos \frac{\pi }
{2} -\mathrm{i}\sin \frac{\pi }
{2}\)
\(\sin \frac{\pi }
{2} + \mathrm{i}\cos \frac{\pi }
{2}\)
\(\cos \frac{3\pi }
{2} + \mathrm{i}\sin \frac{3\pi }
{2}\)
9000069901
Level:
A
Solve the following quadratic equation in the complex plane.
\[
x^{2} + 4x + 5 = 0
\]
\(x_{1} = -2 + \mathrm{i}\),
\(x_{2} = -2 -\mathrm{i}\)
\(x = -2\)
\(x_{1} = 2 + \mathrm{i}\),
\(x_{2} = 2 -\mathrm{i}\)
\(x_{1} = -3\),
\(x_{2} = -1\)
9000069902
Level:
A
Solve the following quadratic equation in the complex plane.
\[
3x^{2} + 2x + 2 = 0
\]
\(x_{1} = -\frac{1}
{3} + \frac{\sqrt{5}}
{3} \mathrm{i}\),
\(x_{2} = -\frac{1}
{3} -\frac{\sqrt{5}}
{3} \mathrm{i}\)
\(x_{1} = -\frac{1}
{3}\)
\(x_{1} = \frac{1}
{3} + \frac{\sqrt{5}}
{3} \),
\(x_{2} = \frac{1}
{3} + \frac{\sqrt{5}}
{3} \)
\(x_{1} = \frac{1}
{3} + \frac{\sqrt{5}}
{3} \mathrm{i}\),
\(x_{2} = \frac{1}
{3} -\frac{\sqrt{5}}
{3} \mathrm{i}\)
9000069903
Level:
A
Find the factorization of the quadratic polynomial
\[
x^{2} + 2x + 2
\]
in the set of polynomials with complex valued coefficients.
\((x + 1 + \mathrm{i})(x + 1 -\mathrm{i})\)
\((x - 1 + \mathrm{i})(x - 1 -\mathrm{i})\)
\((x -\mathrm{i})(x + \mathrm{i})\)
\((x - 1 + \mathrm{i})(x + 1 -\mathrm{i})\)
9000069904
Level:
A
Find the factorization of the quadratic polynomial
\[
x^{2} + 2x + 5
\]
in the set of polynomials with complex valued coefficients.
\((x + 1 - 2\mathrm{i})(x + 1 + 2\mathrm{i})\)
\((x - 1 - 2\mathrm{i})(x - 1 + 2\mathrm{i})\)
\((x + 1 - 2\mathrm{i})(x - 1 + 2\mathrm{i})\)
\((x - 1 - 2\mathrm{i})(x + 1 + 2\mathrm{i})\)