Complex numbers in algebraic and polar form

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.)
\( [1;0] \)
\( [0;1] \)
\( [-1; 0] \)
\( [0;-1] \)

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 \( z_1= z_2 \).
\( [x;y]\in\left\{[3;4], [3;5], [4;4], [4;5]\right\} \)
\( [x;y]\in\left\{[4;3], [4;4], [5;3], [5;4]\right\} \)
\( [x;y]\in\left\{[-3;-4], [-3;-5], [-4;-4], [-4;-5]\right\} \)
\( [x;y]\in\left\{[-4;-3], [-4;-4], [-5;-3], [-5;-4]\right\} \)

1003082306

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

1003082305

Level: 
B
Let \( [x;y]\in\mathbb{R}\times\mathbb{R} \), \( z_1 = 5 + xy\,\mathrm{i} \) and \( z_2 = x + y - 4\,\mathrm{i} \). Find all \( [x;y] \) such that \( z_1 \) and \( z_2 \) are the complex conjugates.
\( [x;y] \in\left\{[4;1],[1;4]\right\} \)
\( [x;y]\in\left\{[6;1],[9;4]\right\} \)
\( [x;y]\in\left\{[4;9],[1;6]\right\} \)
\([x;y]\in\left\{[-4;9],[-1;6]\right\} \)
\( [x;y]\in\left\{[6;-1],[9;-4]\right\} \)

1003082303

Level: 
B
Given the complex numbers \( a=6\sqrt2\left(\cos\frac{\pi}3+\mathrm{i}\cdot\sin\frac{\pi}3\right) \), \( b=3\sqrt2\left(\cos\frac56\pi+\mathrm{i}\cdot\sin\frac56\pi\right) \) and \( c=2\left(\cos240^{\circ}+\mathrm{i}\cdot\sin240^{\circ}\right) \), evaluate \( \frac a{b\cdot c} \).
\( \cos\frac{\pi}6+\mathrm{i}\cdot\sin\frac{\pi}6 \)
\( \cos\frac{11}6\pi+\mathrm{i}\cdot\sin\frac{11}6\pi \)
\( 4\left(\cos\frac{\pi}6\pi+\mathrm{i}\cdot\sin\frac{\pi}6\pi\right) \)
\( 4\left(\cos⁡\frac{11}6\pi+\mathrm{i}\cdot\sin\frac{11}6\pi\right) \)

1003082302

Level: 
B
Given the complex numbers \( a=10\left(\cos\frac43\pi+\mathrm{i}\cdot\sin\frac43\pi\right) \), \( b=7\left(\cos150^{\circ}+\mathrm{i}\cdot\sin150^{\circ}\right) \) and \( c=5\left(\cos⁡\frac74\pi+\mathrm{i}\cdot\sin\frac74\pi \right) \), evaluate \( \frac{a\cdot b}c \).
\( 14\left(\cos⁡\frac5{12}\pi+\mathrm{i}\cdot\sin\frac5{12}\pi\right) \)
\( 14\left(\cos\frac14\pi+\mathrm{i}\cdot\sin\frac14\pi\right) \)
\( 14\left(\cos\frac{23}{12}\pi+\mathrm{i}\cdot\sin\frac{23}{12}\pi\right) \)
\( 14\left(\cos\frac54\pi+\mathrm{i}\cdot\sin\frac54\pi\right) \)