Quadratic Equations with Complex Roots

9000019808

Level: 
B
Assuming \(x\in \mathbb{C}\), find the solution set of the following equation. \[ x\left (x + 1\right )\left (x^{2} + 1\right ) = 0 \]
\(\left \{-1,0,-\mathrm{i},\mathrm{i}\right \}\)
\(\left \{-1,0,1,-\mathrm{i},\mathrm{i}\right \}\)
\(\left \{-1,1,-\mathrm{i},\mathrm{i}\right \}\)
\(\left \{-1,0,-\mathrm{i}\right \}\)

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.
\(\left (-4,8\right )\)
\(\left [ -4,8\right ] \)
\(\left (-\infty ,-4\right )\cup \left (8,\infty \right )\)
\(\left (-\infty ,-4\right ] \cup \left [ 8,\infty \right )\)

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. \[ px^{2} - 3x + 4p = 0 \]
\(p\in \left (-\infty ,-\frac{3} {4}\right )\cup \left (\frac{3} {4},\infty \right )\)
\(p\in\left (-\frac{3} {4}, \frac{3} {4}\right )\)
\(p\in\left (\frac{3} {4},\infty \right )\)
\(p\in\left \{-\frac{3} {4}, \frac{3} {4}\right \}\)
\(p\in\mathbb{R}\setminus \left \{-\frac{3} {4}, \frac{3} {4}\right \}\)

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.
\(\cos \frac{5} {6}\pi + \mathrm{i}\sin \frac{5} {6}\pi \)
\(\cos \frac{1} {6}\pi + \mathrm{i}\sin \frac{1} {6}\pi \)
\(\cos \frac{7} {6}\pi + \mathrm{i}\sin \frac{7} {6}\pi \)
\(\cos \frac{11} {6} \pi + \mathrm{i}\sin \frac{11} {6} \pi \)

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

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 \(x_{1, 2} =\pm \mathrm{i}\frac{\sqrt{5}} {3} \).
\(a = 9\text{, }b = 0\text{, }c = 5\)
\(a = 5\text{, }b = 0\text{, }c = 9\)
\(a = 9\text{, }b = 0\text{, }c = -5\)
\(a = 5\text{, }b = 0\text{, }c = -9\)