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9000034307

Level:

Let

x

be any of the complex solutions of the following equation.

x^{5} + \sqrt{3} - i = 0

Find the absolute value of

x

\sqrt[5]{2}

2

\sqrt[5]{4}

\sqrt[10]{2}

9000034308

Level:

Two solutions of the equation

x^{3} + 1 + i = 0

are

\begin{aligned} x_{1} & = \sqrt[6]{2} (\cos \frac{5}{12} π + i \sin \frac{5}{12} π), \\ x_{2} & = \sqrt[6]{2} (\cos \frac{13}{12} π + i \sin \frac{13}{12} π) . \end{aligned}

Find the third solution.

x_{3} = \sqrt[6]{2} (\cos \frac{21}{12} π + i \sin \frac{21}{12} π)

x_{3} = \sqrt[6]{2} (\cos \frac{9}{12} π + i \sin \frac{9}{12} π)

x_{3} = \sqrt[6]{2} (\cos \frac{17}{12} π + i \sin \frac{17}{12} π)

x_{3} = \sqrt[6]{2} (\cos \frac{19}{12} π + i \sin \frac{19}{12} π)

9000034309

Level:

Find the angle

φ

such that the angles in the polar form of any two solutions of the equation

x^{5} - 1 + i \sqrt{3} = 0

differ by an integer multiple of

φ

φ = \frac{2}{5} π

φ = \frac{3}{5} π

φ = \frac{4}{5} π

φ = π

9000033308

Level:

Find the solution set of the following inequality.

\frac{x^{2} + x + 2}{x^{2} + 4 x + 3} \geq 0

(- \infty; - 3) \cup (- 1; \infty)

(1; 3)

(- \infty; 1) \cup (3; \infty)

(- 3; - 1)

9000033707

Level:

Identify the inequality with a solution graphed in the picture.

| x (3 - x) | > 3 - x

| x (x - 3) | < x - 3

| 3 x - x^{2} | > x - 3

| x^{2} - 3 x | < 3 - x

x^{2} - 3 | x | > 3 - x

x^{2} - 3 | x | < x - 3

9000033705

Level:

Find the domain of the following function.

f (x) = \sqrt{\log (x^{2} + 2 x + 1)}

(- \infty; - 2] \cup [0; \infty)

R ∖ {- 1}

(- 1; \infty)

(- \infty; - 1) \cup (1; \infty)

(- \infty; 0) \cup (2; \infty)

9000033708

Level:

A stone has been thrown vertically up at the velocity

15 m s^{- 1}

from the initial height

10 m

. How long (in seconds) has been the height of the stone at least

20 m

? Hint: The height

h

is given by the expression

h = s_{0} + v_{0} t - \frac{1}{2} g t^{2}

, the standard acceleration is

g ≐ 10 m s^{- 2}

exactly

1 s

less than

1 s

more than

1 s

The information is not sufficient to give a definite answer.

9000033709

Level:

A square shaped garden with the side

a

should be reduced by a length

x

to another square garden. The difference between the areas of the gardens should not be bigger than

25 %

of the original area. Find the possible values of

x

x \leq a - \frac{\sqrt{3}}{2} a

x \leq \sqrt{3} a

x \leq \frac{3}{4} a

x \leq a + \frac{\sqrt{3}}{2} a

9000031007

Level:

Find the sum of the solutions of the following equation.

x^{3} + 2 x^{2} - x - 2 = 0

- 2

3

- 3

- 1

9000028410

Level:

Find the condition which is equivalent to the fact that the equation

a x^{2} + b x + c = 0

with

x \in R

and real coefficients

a

b

c

has two solutions and one of the solutions is a reciprocal value of the second solution.

b^{2} - 4 a c > 0 and \frac{c}{a} = 1

b^{2} - 4 a c > 0 and a = c

b^{2} - 4 a c > 0 and \frac{c}{a} = - 1

b^{2} - 4 a c > 0 and a = - c

C

9000034307

9000034308

9000034309

9000033308

9000033707

9000033705

9000033708

9000033709

9000031007

9000028410

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