9000070103
Level:
A
Evaluate the following complex number.
\[
\left (\cos \pi + \mathrm{i}\sin \pi \right )^{9}
\]
\(- 1\)
\(1\)
\(\mathrm{i}\)
\(-\mathrm{i}\)
A
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.
\[
x^{2} + 4x + 5 = 0
\]
\(x_{1} = -2 + \mathrm{i}\),
\(x_{2} = -2 -\mathrm{i}\)
\(x = -2\)
\(x_{1} = 2 + \mathrm{i}\),
\(x_{2} = 2 -\mathrm{i}\)
\(x_{1} = -3\),
\(x_{2} = -1\)
9000065502
Level:
A
Evaluate the following integral on \(\mathbb{R}\).
\[
\int (4x + 7)\, \mathrm{d}x
\]
\(2x^{2} + 7x + c,\ c\in \mathbb{R}\)
\(2x^{2} - 7x + c,\ c\in \mathbb{R}\)
\(4 + c,\ c\in \mathbb{R}\)
\(4x^{2} + 7x + c,\ c\in \mathbb{R}\)
9000065601
Level:
A
Find the area of the region bounded by
\(x\)-axis,
graph of \(f(x) = x + 3\)
and lines \(x = -1\)
and \(x = 1\).
\(6\)
\(2\)
\(4\)
\(8\)
9000065503
Level:
A
Evaluate the following integral on the interval \((0;+\infty)\).
\[
\int (4x^{-3} - x^{-4})\, \mathrm{d}x
\]
\(- 2x^{-2} + \frac{1}
{3}x^{-3} + c,\ c\in \mathbb{R}\)
\(-\frac{4}
{3}x^{-2} -\frac{1}
{3}x^{-3} + c,\ c\in \mathbb{R}\)
\(-\frac{3}
{4}x^{-4} -\frac{1}
{5}x^{-5} + c,\ c\in \mathbb{R}\)
\(- 12x^{2} + 4x^{-3} + c,\ c\in \mathbb{R}\)
9000065610
Level:
A
Using definite integral find the area of the triangle defined by the following three
inequalities
\[
\begin{aligned}y& > 0, &
\\y& < x + 3,
\\y& < 3 - x.
\\ \end{aligned}
\]
\(\int _{-3}^{0}(x + 3)\, \mathrm{d}x +\int _{ 0}^{3}(3 - x)\, \mathrm{d}x\)
\(\int _{0}^{3}(x + 3)\, \mathrm{d}x\)
\(\int _{-3}^{3}(3 - x)\, \mathrm{d}x\)
\(\int _{-3}^{0}(3 - x)\, \mathrm{d}x +\int _{ 0}^{3}(x + 3)\, \mathrm{d}x\)
9000065507
Level:
A
Given the function
\[
F(x) = \frac{1}
{4}x^{4} -\frac{2}
{3}x^{3},
\]
find the function \(f\)
such that \(F\) is
primitive to \(f\)
on \(\mathbb{R}\).
\(f(x) = x^{3} - 2x^{2}\)
\(f(x) = x^{5} - 2x^{4}\)
\(f(x) = x^{5} - 3x^{2}\)
\(f(x) = -4x^{-4} - 3x^{2}\)
9000065602
Level:
A
Find the area of the region bounded by
\(x\)-axis,
graph of \(f(x)= x^{2} + 3\)
and lines \(x = -2\)
and \(x = 1\).
\(12\)
\(6\)
\(8\)
\(10\)