9000035604
Level:
C
Solve the following equation.
\[
x^{2} - 2\mathrm{i}x + 3 = 0
\]
\(\{ -\mathrm{i};3\mathrm{i}\}\)
\(\{ - 4\mathrm{i};4\mathrm{i}\}\)
\(\{1 - 2\mathrm{i};1 + 2\mathrm{i}\}\)
\(\{ - 3\mathrm{i};\mathrm{i}\}\)
Quadratic equations with complex roots
9000035606
Level:
B
Find the quadratic equation with real valued coefficients and one of the solutions
\(x_{1} = -1 + \mathrm{i}\sqrt{3}\).
\(x^{2} + 2x + 4 = 0\)
\(x^{2} - 2x + 4 = 0\)
\(x^{2} + 2x - 4 = 0\)
\(x^{2} - 2x - 2 = 0\)
\(x^{2} + 2x + 2 = 0\)
9000035605
Level:
B
The number \(\cos \frac{7}
{6}\pi + \mathrm{i}\sin \frac{7}
{6}\pi \)
is a solution of a quadratic equation with real valued coefficients. Find the second
solution.
\(\cos \frac{5}
{6}\pi + \mathrm{i}\sin \frac{5}
{6}\pi \)
\(\cos \frac{1}
{6}\pi + \mathrm{i}\sin \frac{1}
{6}\pi \)
\(\cos \frac{7}
{6}\pi + \mathrm{i}\sin \frac{7}
{6}\pi \)
\(\cos \frac{11}
{6} \pi + \mathrm{i}\sin \frac{11}
{6} \pi \)
9000035609
Level:
C
One of the roots of the equation \( x^{2} + px - 11 = 0\) with the parameter \(p\in \mathbb{C}\) is \(x_{1} = 3 -\mathrm{i}\sqrt{2}\). Find the second root \(x_{2}\) and the corresponding value of the parameter \(p\).
\(x_{2} = -3 -\mathrm{i}\sqrt{2},\ p = 2\mathrm{i}\sqrt{2}\)
\(x_{2} = 3 + \mathrm{i}\sqrt{2},\ p = 6\)
\(x_{2} = -3 -\mathrm{i}\sqrt{2},\ p = 6\)
\(x_{2} = 3 + \mathrm{i}\sqrt{2},\ p = -2\mathrm{i}\)
\(x_{2} = -3 -\mathrm{i}\sqrt{2},\ p = -2\mathrm{i}\sqrt{2}\)
9000035601
Level:
B
Find the values of the parameter \(p\in \mathbb{R}\)
which guarantee that the following quadratic equation has solutions with nonzero
imaginary part.
\[
px^{2} - 3x + 4p = 0
\]
\(p\in \left (-\infty ;-\frac{3}
{4}\right )\cup \left (\frac{3}
{4};\infty \right )\)
\(p\in\left (-\frac{3}
{4}; \frac{3}
{4}\right )\)
\(p\in\left (\frac{3}
{4};\infty \right )\)
\(p\in\left \{-\frac{3}
{4}; \frac{3}
{4}\right \}\)
\(p\in\mathbb{R}\setminus \left \{-\frac{3}
{4}; \frac{3}
{4}\right \}\)
9000022803
Level:
B
Establish the values of the parameter \(t\)
which ensure that the equation
\[
x^{2} + tx + t + 8 = 0
\]
with an unknown \(x\)
has complex solutions with a nonzero imaginary part.
\(\left (-4;8\right )\)
\(\left [ -4;8\right ] \)
\(\left (-\infty ;-4\right )\cup \left (8;\infty \right )\)
\(\left (-\infty ;-4\right ] \cup \left [ 8;\infty \right )\)
9000019808
Level:
B
Assuming \(x\in \mathbb{C}\),
find the solution set of the following equation.
\[
x\left (x + 1\right )\left (x^{2} + 1\right ) = 0
\]
\(\left \{-1;0;-\mathrm{i};\mathrm{i}\right \}\)
\(\left \{-1;0;1;-\mathrm{i};\mathrm{i}\right \}\)
\(\left \{-1;1;-\mathrm{i};\mathrm{i}\right \}\)
\(\left \{-1;0;-\mathrm{i}\right \}\)