9000035607
Level:
C
Identify the quadratic equation with solutions
\(x_{1} = 2\mathrm{i}\),
\(x_{2} = -\mathrm{i}\).
\(x^{2} -\mathrm{i}x + 2 = 0\)
\(x^{2} + \mathrm{i}x + 2 = 0\)
\(x^{2} + \mathrm{i}x - 2 = 0\)
\(x^{2} -\mathrm{i}x - 2 = 0\)
Quadratic Equations with Complex Roots
9000035608
Level:
C
The equation
\[
x^{2} - 2\mathrm{i}x + q = 0
\]
with a parameter \(q\in \mathbb{C}\)
has a solution \(x_{1} = 1 + 2\mathrm{i}\). Find
the second solution \(x_{2}\)
and the parameter \(q\).
\(x_{2} = -1,\ q = -1 - 2\mathrm{i}\)
\(x_{2} = -1 - 4\mathrm{i},\ q = 9 - 6\mathrm{i}\)
\(x_{2} = 1 - 4\mathrm{i},\ q = 7 - 4\mathrm{i}\)
\(x_{2} = 1,\ q = -1 - 2\mathrm{i}\)
\(x_{2} = -1,\ q = 1 + 2\mathrm{i}\)
9000035602
Level:
C
Find the values of the parameter \(m\in \mathbb{C}\)
which guarantee that the following quadratic equation has a double solution.
\[
mx^{2} - 2x - 1 + \mathrm{i} = 0
\]
\(m = -\frac{1}
{2} -\frac{1}
{2}\mathrm{i}\)
\(m = -1\)
\(m = -1 + \mathrm{i}\)
\(m = -\frac{1}
{2} + \frac{1}
{2}\mathrm{i}\)
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
\(x_{1} = -1 + \mathrm{i}\sqrt{3}\).
\(x^{2} + 2x + 4 = 0\)
\(x^{2} - 2x + 4 = 0\)
\(x^{2} + 2x - 4 = 0\)
\(x^{2} - 2x - 2 = 0\)
\(x^{2} + 2x + 2 = 0\)
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.
\[
x\left (x + 1\right )\left (x^{2} + 1\right ) = 0
\]
\(\left \{-1;0;-\mathrm{i};\mathrm{i}\right \}\)
\(\left \{-1;0;1;-\mathrm{i};\mathrm{i}\right \}\)
\(\left \{-1;1;-\mathrm{i};\mathrm{i}\right \}\)
\(\left \{-1;0;-\mathrm{i}\right \}\)