2000001508
Level:
C
Which of the given numbers is not the solution of the equation \((x-i)(2x+i)=0\)?
\(x=1\)
\(x=i\)
\(x=-\frac{1}{2}i\)
\(x=\cos{\frac{\pi}{2} }+ i\sin{\frac{\pi}{2}}\)
Quadratic equations with complex roots
2010013202
Level:
C
One of the roots of the equation \( x^{2} + px - 8 = 0\) with the parameter \(p\in \mathbb{C}\) is \(x_{1} = \sqrt{7} +\mathrm{i}\). Find the second root \(x_{2}\) and the corresponding value of the parameter \(p\).
\(x_{2} = \mathrm{i}-\sqrt{7},\ p = -2\mathrm{i}\)
\(x_{2} = -\mathrm{i}-\sqrt{7},\ p = 2\mathrm{i}\)
\(x_{2} = -\mathrm{i}+\sqrt{7},\ p = 2\mathrm{i}\)
\(x_{2} = -\mathrm{i}-\sqrt{7},\ p = 4\mathrm{i}\)
2010013203
Level:
C
Find the complex roots of the following quadratic equation.
\[ 5\mathrm{i}\,x^2 - 3\mathrm{i}\,x = 0 \]
\( x_1=0,\ \ x_2 = \frac35 \)
\( x_1=0,\ \ x_2 = \frac35\mathrm{i} \)
\( x_1=0,\ \ x_2 = -\frac35\mathrm{i} \)
\( x_1=0,\ \ x_2 = -\frac35 \)
2010013204
Level:
C
Find the solution set of the following quadratic equation in the complex plane.
\[ 2x^2-(4-2\mathrm{i})x = 0 \]
\( \{2-\mathrm{i}; 0 \} \)
\( \{-2-\mathrm{i}; 0 \} \)
\( \{2+\mathrm{i}; 0 \} \)
\( \{-2+\mathrm{i}; 0 \} \)
2010013205
Level:
C
Find the complex roots of the following quadratic equation.
\[ (2-\mathrm{i})x^2-(3-2\mathrm{i})x = 0 \]
\( x_1=\frac85-\frac15\mathrm{i},\ \ x_2=0 \)
\( x_1=\frac85+\frac15\mathrm{i},\ \ x_2=0 \)
\( x_1=-\frac85-\frac15\mathrm{i},\ \ x_2=0 \)
\( x_1=-\frac85+\frac15\mathrm{i},\ \ x_2=0 \)
2010013206
Level:
C
Find the complex roots of the following quadratic equation.
\[ 3\mathrm{i}x^2 + 2 = 0 \]
\( x_1=\frac{\sqrt3}3+\frac{\sqrt3}3\mathrm{i},\ \ x_2=-\frac{\sqrt3}3-\frac{\sqrt3}3\mathrm{i} \)
\( x_1=-\frac{\sqrt3}3+\frac{\sqrt3}3\mathrm{i},\ \ x_2=\frac{\sqrt3}3-\frac{\sqrt3}3\mathrm{i} \)
\( x_1=-\frac{\sqrt3}6+\frac{\sqrt3}6\mathrm{i},\ \ x_2=-\frac{\sqrt3}6-\frac{\sqrt3}6\mathrm{i} \)
\( x_1=\frac{\sqrt3}6+\frac{\sqrt3}6\mathrm{i},\ \ x_2=-\frac{\sqrt3}6-\frac{\sqrt3}6\mathrm{i} \)
2010013207
Level:
C
Find the complex roots of the following quadratic equation.
\[ 2x^2 + 5\mathrm{i} = 0 \]
\( x_1=-\frac{\sqrt5}2+\frac{\sqrt5}2\mathrm{i},\ \ x_2=\frac{\sqrt5}2-\frac{\sqrt5}2\mathrm{i} \)
\( x_1=\frac{\sqrt5}2+\frac{\sqrt5}2\mathrm{i},\ \ x_2=-\frac{\sqrt5}2-\frac{\sqrt5}2\mathrm{i} \)
\( x_1=\frac{\sqrt5}2\mathrm{i},\ \ x_2=-\frac{\sqrt5}2\mathrm{i} \)
\( x_1=-\frac{\sqrt5}2,\ \ x_2=\frac{\sqrt5}2\)
2010013208
Level:
C
Which of the given quadratic equations has the roots \( x_1 = -1 - 3\mathrm{i} \) and \( x_2 = 1 +3\mathrm{i} \)?
\( x^2 + (3 - 3\mathrm{i}) + 5 - 3\mathrm{i} = 0 \)
\( x^2 - (3 - 3\mathrm{i}) + 5 + 3\mathrm{i} = 0 \)
\( x^2 + (3 - 3\mathrm{i}) + 5 + 3\mathrm{i} = 0 \)
\( x^2 - (3 - 3\mathrm{i}) + 5 - 3\mathrm{i} = 0 \)
2010013209
Level:
C
Solve the following equation.
\[
x^{2} + 3\mathrm{i}x + 4 = 0
\]
\(\{ -4\mathrm{i};\mathrm{i}\}\)
\(\{ 4\mathrm{i};-\mathrm{i}\}\)
\(\{1 - 3\mathrm{i};1 + 3\mathrm{i}\}\)
\(\{- 2\mathrm{i};-\mathrm{i}\}\)
2010013308
Level:
C
Find the solution set of the following quadratic equation in the complex plane.
\[ 2x^2-(2-4\mathrm{i})x + 3 - 2\mathrm{i}= 0 \]
\( \left\{ \frac12+\frac12\mathrm{i}; \frac12-\frac52\mathrm{i} \right\} \)
\( \left\{ \frac12+\frac12\mathrm{i}; \frac12-\frac12\mathrm{i} \right\} \)
\( \emptyset \)
\( \left\{ -\frac12-\frac12\mathrm{i}; -\frac12+\frac52\mathrm{i} \right\} \)
\( \left\{ 1+\mathrm{i}; 1-5\mathrm{i} \right\} \)