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. \[ 2z -\mathrm{i}\, \overline{z} = 1 -\mathrm{i} \]
\(z = \frac{1} {3} -\frac{1} {3}\mathrm{i}\)
\(z = 1 + \mathrm{i}\)
\(z = -\frac{3} {5} + \frac{6} {5}\mathrm{i}\)
\(z = -\frac{1} {5} -\frac{3} {5}\mathrm{i}\)