9000064502
Level:
B
Find the quadratic equation with the solution
\(x_{1, 2} = 2\pm \mathrm{i}\sqrt{2}\).
\(x^{2} - 4x + 6 = 0\)
\(3x^{2} + 4x + 2 = 0\)
\(3x^{2} - 4x + 2 = 0\)
\(x^{2} + 4x + 6 = 0\)
Quadratic equations with complex roots
9000064503
Level:
B
Find the values of the real coefficients
\(a\),
\(b\) and
\(c\) such
that the quadratic equation
\[
ax^{2} + bx + c = 0
\]
has solution \(x_{1, 2} =\pm \mathrm{i}\frac{\sqrt{5}}
{3} \).
\(a = 9\text{, }b = 0\text{, }c = 5\)
\(a = 5\text{, }b = 0\text{, }c = 9\)
\(a = 9\text{, }b = 0\text{, }c = -5\)
\(a = 5\text{, }b = 0\text{, }c = -9\)
9000064504
Level:
B
Find the values of the real coefficients
\(a\),
\(b\) and
\(c\) such
that the quadratic equation
\[
ax^{2} + bx + c = 0
\]
has solutions \(x_{1, 2} = 1\pm \frac{\mathrm{i}}
{2}\).
\(a = 4\text{, }b = -8\text{, }c = 5\)
\(a = 1\text{, }b = -4\text{, }c = 5\)
\(a = 4\text{, }b = 8\text{, }c = 5\)
\(a = 1\text{, }b = 4\text{, }c = 5\)
9000039107
Level:
A
Find the sum of the reciprocal values of the solutions of
\(5x^{2} - 2x + 1 = 0\).
\(2\)
\(\frac{2}
{5}\)
\(\frac{2}
{5} + \frac{4}
{5}\mathrm{i}\)
\(-\frac{2}
{5}\)
9000039106
Level:
B
Find the value of the parameter \(a\)
which guarantees that the quadratic equation
\[
x^{2} + 2ax + a = 0
\]
has a pair of complex conjugate solutions with a nonzero imaginary part.
\(a\in (0;1)\)
\(a\in [ 0;1] \)
\(a\in (-\infty ;0)\cup (1;\infty )\)
Such an \(a\)
does not exist
9000039105
Level:
B
Find the quadratic equation with real coefficients such that one of the solutions is the complex
number \(x_{1} = 1 + 2\mathrm{i}\).
\(x^{2} - 2x + 5 = 0\)
\(x^{2} - 2x + 3 = 0\)
\(x^{2} + 2x + 5 = 0\)
\(x^{2} + 2x - 3 = 0\)
9000035604
Level:
C
Solve the following equation.
\[
x^{2} - 2\mathrm{i}x + 3 = 0
\]
\(\{ -\mathrm{i};3\mathrm{i}\}\)
\(\{ - 4\mathrm{i};4\mathrm{i}\}\)
\(\{1 - 2\mathrm{i};1 + 2\mathrm{i}\}\)
\(\{ - 3\mathrm{i};\mathrm{i}\}\)
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 \(p\).
\(x_{2} = -3 -\mathrm{i}\sqrt{2},\ p = 2\mathrm{i}\sqrt{2}\)
\(x_{2} = 3 + \mathrm{i}\sqrt{2},\ p = 6\)
\(x_{2} = -3 -\mathrm{i}\sqrt{2},\ p = 6\)
\(x_{2} = 3 + \mathrm{i}\sqrt{2},\ p = -2\mathrm{i}\)
\(x_{2} = -3 -\mathrm{i}\sqrt{2},\ p = -2\mathrm{i}\sqrt{2}\)