Quadratic equations with complex roots

2010013308

Level: 
C
Find the solution set of the following quadratic equation in the complex plane. \[ 2x^2-(2-4\mathrm{i})x + 3 - 2\mathrm{i}= 0 \]
\( \left\{ \frac12+\frac12\mathrm{i}; \frac12-\frac52\mathrm{i} \right\} \)
\( \left\{ \frac12+\frac12\mathrm{i}; \frac12-\frac12\mathrm{i} \right\} \)
\( \emptyset \)
\( \left\{ -\frac12-\frac12\mathrm{i}; -\frac12+\frac52\mathrm{i} \right\} \)
\( \left\{ 1+\mathrm{i}; 1-5\mathrm{i} \right\} \)

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 \(x_{1, 2} = \frac12\pm \mathrm{i}\).
\(a = 4,\ b = -4,\ c = 5\)
\(a = 4,\ b = 4,\ c = 5\)
\(a = 5,\ b = -5,\ c = 4\)
\(a = -4,\ b = 4,\ c = 5\)

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. \[ 9px^{2} + 5x + p = 0 \]
\(\left (-\infty ;-\frac{5} {6}\right )\cup \left (\frac{5} {6};\infty \right )\)
\(\left (-\frac{5} {6}; \frac{5} {6}\right )\)
\(\left (\frac{5} {6};\infty \right )\)
\(\left \{-\frac{5} {6}; \frac{5} {6}\right \}\)
\(\mathbb{R}\setminus \left \{-\frac{5} {6}; \frac{5} {6}\right \}\)

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.
\(\sqrt{2}\left(\cos \frac{5\pi} {4} + \mathrm{i}\sin \frac{5\pi} {4}\right) \)
\(\sqrt{2}\left(\cos \frac{\pi} {4} + \mathrm{i}\sin \frac{\pi} {4}\right) \)
\(\sqrt{2}\left(\cos \frac{7\pi} {4} + \mathrm{i}\sin \frac{7\pi} {4}\right) \)
\(\sqrt{2}\left(\cos \frac{3\pi} {4} + \mathrm{i}\sin \frac{3\pi} {4}\right) \)