9000069906
Level:
A
Find the sum of all the complex solutions of the following quadratic equation.
\[
x^{2} - 8x + 17 = 0
\]
\(8\)
\(4\)
\(4\mathrm{i}\)
\(0\)
A
9000070107
Level:
A
Find the algebraic form of the complex number:
\[
\left (\frac{1}
{2} +\cos \frac{\pi }
{3} + \mathrm{i}\cos 2\pi \right )^{5}
\]
\(- 4 - 4\mathrm{i}\)
\(- 4 + 4\mathrm{i}\)
\(4 - 4\mathrm{i}\)
\(4 + 4\mathrm{i}\)
9000070101
Level:
A
Find the algebraic form of the complex number:
\[
\left (\cos \frac{\pi }
{4} + \mathrm{i}\sin \frac{\pi }
{4}\right )^{3}
\]
\(-\frac{\sqrt{2}}
{2} + \mathrm{i}\frac{\sqrt{2}}
{2} \)
\(-\frac{\sqrt{2}}
{2} -\mathrm{i}\frac{\sqrt{2}}
{2} \)
\(\frac{\sqrt{2}}
{2} -\mathrm{i}\frac{\sqrt{2}}
{2} \)
\(\frac{\sqrt{2}}
{2} + \mathrm{i}\frac{\sqrt{2}}
{2} \)
9000065902
Level:
A
Evaluate the following integral on the interval \((0;+\infty)\).
\[
\int \left (2 + \frac{1}
{x}\right )\, \text{d}x
\]
\(2x +\ln |x| + c,\ c\in \mathbb{R}\)
\(\ln |x| + c,\ c\in \mathbb{R}\)
\(2 +\ln |x| + c,\ c\in \mathbb{R}\)
\(2x^{2} +\ln |x| + c,\ c\in \mathbb{R}\)
9000070106
Level:
A
Evaluate the following complex number.
\[
(1 -\mathrm{i})^{8}
\]
\(16\)
\(- 16\mathrm{i}\)
\(16\mathrm{i}\)
\(- 16\)
9000065908
Level:
A
Given the function
\[
F(x) = \frac{1}
{2}x^{2} - x,
\]
find the function \(f\)
such that \(F\) is
primitive to \(f\)
on \((1;+\infty )\).
\(f(x) = \frac{x^{2}-1}
{x+1} \)
\(f(x) = \frac{x^{2}-1}
{x-1} \)
\(f(x) = \frac{x+1}
{x^{2}-1}\)
\(f(x) = \frac{x-1}
{x^{2}-1}\)
9000070108
Level:
A
Evaluate the following complex number.
\[
\left (\frac{1}
{2} + \frac{\sqrt{3}}
{2} \mathrm{i}\right )^{6}
\]
\(1\)
\(- 1\)
\(\mathrm{i}\)
\(-\mathrm{i}\)
9000065909
Level:
A
Given function
\[
F(x) = 2\ln |x + 1|,
\]
find the function \(f\)
such that \(F\) is
primitive to \(f\)
on \((-1;+\infty )\).
\(f(x) = \frac{2}
{x+1}\)
\(f(x) = 2\mathrm{e}^{x+1}\)
\(f(x) = \frac{1}
{2(x+1)}\)
\(f(x) = \frac{2}
{2x+2}\)
9000070109
Level:
A
Evaluate the following complex number.
\[
\left (\sqrt{3} -\mathrm{i}\right )^{3}
\]
\(- 8\mathrm{i}\)
\(8\)
\(- 8\)
\(8\mathrm{i}\)
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 \((0;+\infty )\).
\(f(x) = \frac{x^{2}+2x+1}
{x^{2}} \)
\(f(x) = \frac{x^{2}}
{(x+1)^{2}} \)
\(f(x) = \frac{x^{2}-1}
{x^{2}} \)
\(f(x) = \frac{x^{2}}
{(x-1)^{2}} \)