2010013210
Level:
B
Find the quadratic equation with real valued coefficients and one of the solutions
\(x_{1} = -2 + \mathrm{i}\sqrt{2}\).
\(x^{2} + 4x + 6 = 0\)
\(x^{2} - 4x + 6 = 0\)
\(x^{2} + 4x - 6 = 0\)
\(x^{2} - 4x - 6 = 0\)
Quadratic equations with complex roots
2010013301
Level:
B
Find the set of all complex roots of the following quadratic equation.
\[ x^2 - 2x + 2 = 0 \]
\( \left\{ \sqrt2\left(\cos\frac{\pi}4+\mathrm{i}\cdot\sin\frac{\pi}4\right); \sqrt2\left(\cos\frac{7\pi}4+\mathrm{i}\cdot\sin\frac{7\pi}4\right) \right\} \)
\( \left\{ 2\left(\cos\frac{\pi}4+\mathrm{i}\cdot\sin\frac{\pi}4\right); 2\left(\cos\frac{7\pi}4+\mathrm{i}\cdot\sin\frac{7\pi}4\right) \right\} \)
\( \left\{ \sqrt2\left(\cos\frac{\pi}4+\mathrm{i}\cdot\sin\frac{\pi}4\right); \sqrt2\left(\cos\frac{3\pi}4+\mathrm{i}\cdot\sin\frac{3\pi}4\right) \right\} \)
\( \left\{ \sqrt2\left(\cos\frac{\pi}4+\mathrm{i}\cdot\sin\frac{\pi}4\right); \sqrt2\left(\cos\frac{5\pi}4+\mathrm{i}\cdot\sin\frac{5\pi}4\right) \right\} \)
2010013302
Level:
B
Find the quadratic equation with real coefficients such that one of the solutions is the complex
number \(x_{1} = 1 - 2\sqrt{2}\mathrm{i}\).
\(x^{2} - 2x + 9 = 0\)
\(x^{2} + 2x - 9 = 0\)
\(x^{2} - 2x - 7 = 0\)
\(x^{2} + 9x - 2 = 0\)
2010013303
Level:
B
Find the quadratic equation with the solution
\(x_{1, 2} =\pm 3\mathrm{i}\).
\(x^{2} + 9 = 0\)
\(x^{2} - 9\mathrm{i} = 0\)
\(x^{2} - 9 = 0\)
\(x^{2} + 9\mathrm{i} = 0\)
2010013304
Level:
B
Find the values of the parameter \(p\in \mathbb{R}\)
which guarantee that the equation
\[
x^{2} - 2px + 4 = 0
\]
has solution with a nonzero imaginary part.
\(p\in (-2;2)\)
\(p\in (-\infty ;-2)\)
\(p\in (2;\infty )\)
\(p\in \emptyset\)
2010013305
Level:
B
The number \(\sqrt{2}\left(\cos \frac{3\pi}
{4} + \mathrm{i}\sin \frac{3\pi}
{4}\right) \)
is a solution of a quadratic equation with real valued coefficients. Find the second
solution.
\(\sqrt{2}\left(\cos \frac{5\pi}
{4} + \mathrm{i}\sin \frac{5\pi}
{4}\right) \)
\(\sqrt{2}\left(\cos \frac{\pi}
{4} + \mathrm{i}\sin \frac{\pi}
{4}\right) \)
\(\sqrt{2}\left(\cos \frac{7\pi}
{4} + \mathrm{i}\sin \frac{7\pi}
{4}\right) \)
\(\sqrt{2}\left(\cos \frac{3\pi}
{4} + \mathrm{i}\sin \frac{3\pi}
{4}\right) \)
2010013306
Level:
B
Find the set of all values of the parameter \(p\in \mathbb{R}\) for which the following quadratic equation has solutions with nonzero imaginary part.
\[
9px^{2} + 5x + p = 0
\]
\(\left (-\infty ;-\frac{5}
{6}\right )\cup \left (\frac{5}
{6};\infty \right )\)
\(\left (-\frac{5}
{6}; \frac{5}
{6}\right )\)
\(\left (\frac{5}
{6};\infty \right )\)
\(\left \{-\frac{5}
{6}; \frac{5}
{6}\right \}\)
\(\mathbb{R}\setminus \left \{-\frac{5}
{6}; \frac{5}
{6}\right \}\)
2010013307
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 \(x_{1, 2} = \frac12\pm \mathrm{i}\).
\(a = 4,\ b = -4,\ c = 5\)
\(a = 4,\ b = 4,\ c = 5\)
\(a = 5,\ b = -5,\ c = 4\)
\(a = -4,\ b = 4,\ c = 5\)
2010013310
Level:
B
Find the quadratic equation with real coefficients such that one of the solutions is the complex number \(x_1=2\left(\cos\frac{2\pi}{3} + \mathrm{i}\sin \frac{2\pi}{3}\right)\).
\(x^2+2x+4=0\)
\(x^2-2x+4=0\)
\(x^2+4x+2=0\)
\(x^2+2\sqrt{3}x+4=0\)
2010013311
Level:
B
Find the quadratic equation with real coefficients such that one of the solutions is the complex number \(x_1=2\left(\cos\frac{11\pi}{6} + \mathrm{i}\sin \frac{11\pi}{6}\right)\).
\(x^2-2\sqrt{3}x+4=0\)
\(x^2+2\sqrt{3}x+4=0\)
\(x^2+4x+2\sqrt{3}=0\)
\(x^2-2x+4=0\)