Complex numbers in algebraic and polar form

1003082306

Level: 
C
Let \( [x;y]\in\mathbb{R}\times\mathbb{R} \). Find all \( [x;y] \) that satisfy \[ (3x + 2y\,\mathrm{i})\cdot(3x - 2y\,\mathrm{i}) + y^2\,\mathrm{i} = 97 + 4\,\mathrm{i}. \]
\( [x;y]\in\left\{[3;2], [-3;2], [3;-2], [-3;-2]\right\} \)
\( [x;y]\in\left\{[3;2], [-3;2]\right\} \)
\( [x;y]\in\left\{[3;2], [3;-2]\right\}\)
\( [x;y]\in\left\{[3;2], [-3;-2]\right\} \)

1003082307

Level: 
C
Let \( z_1 = x^2 + 9y\,\mathrm{i}-20\,\mathrm{i} \) and \( z_2 = 7x-12+ y^2\,\mathrm{i} \). Find all \( [x;y] \in \mathbb{R}\times\mathbb{R} \) such that \( z_1= z_2 \).
\( [x;y]\in\left\{[3;4], [3;5], [4;4], [4;5]\right\} \)
\( [x;y]\in\left\{[4;3], [4;4], [5;3], [5;4]\right\} \)
\( [x;y]\in\left\{[-3;-4], [-3;-5], [-4;-4], [-4;-5]\right\} \)
\( [x;y]\in\left\{[-4;-3], [-4;-4], [-5;-3], [-5;-4]\right\} \)

1003082308

Level: 
C
Let \( [x;y]\in\mathbb{N}\times\mathbb{N} \). Find all \( [x;y] \), that satisfy \[ x(8 + 4\,\mathrm{i}) + y(1 - 4\,\mathrm{i}) + 5 = x(3 +\mathrm{i}) + 6(y - 2\,\mathrm{i}) + 9\,\mathrm{i}. \]
There is no \( [x;y] \). (There is no solution.)
\( [1;0] \)
\( [0;1] \)
\( [-1; 0] \)
\( [0;-1] \)

2010013107

Level: 
C
Let \( z_1 = x^2 + 9y\,\mathrm{i}-10\,\mathrm{i} \) and \( z_2 = 8x-15+ y^2\,\mathrm{i} \). Find all \( [x;y] \in \mathbb{R}\times\mathbb{R} \) such that \( z_1= \overline{z_2} \).
\( [x;y]\in\left\{[3;-10], [3;1], [5;-10], [5;1]\right\} \)
\( [x;y]\in\left\{[-10;3], [1;3], [-10;5], [1;5]\right\} \)
\( [x;y]\in\left\{[3;10], [3;-1], [5;10], [5;-1]\right\} \)
\( [x;y]\in\left\{[-3;-10], [-3;1], [-5;-10], [-5;1]\right\} \)

9000039108

Level: 
C
Assuming \(z\in \mathbb{C}\), solve the following equation. By \(\overline{z }\) the complex conjugate of \(z \) is denoted. \[ 2z -\mathrm{i}\, \overline{z} = 1 -\mathrm{i} \]
\(z = \frac{1} {3} -\frac{1} {3}\mathrm{i}\)
\(z = 1 + \mathrm{i}\)
\(z = -\frac{3} {5} + \frac{6} {5}\mathrm{i}\)
\(z = -\frac{1} {5} -\frac{3} {5}\mathrm{i}\)