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 \( \frac{a\cdot b}c \).
\( 14\left(\cos⁡\frac5{12}\pi+\mathrm{i}\cdot\sin\frac5{12}\pi\right) \)
\( 14\left(\cos\frac14\pi+\mathrm{i}\cdot\sin\frac14\pi\right) \)
\( 14\left(\cos\frac{23}{12}\pi+\mathrm{i}\cdot\sin\frac{23}{12}\pi\right) \)
\( 14\left(\cos\frac54\pi+\mathrm{i}\cdot\sin\frac54\pi\right) \)

1003082301

Level: 
B
Given the complex numbers \( a=\sqrt2\left(\cos⁡160^{\circ}+\mathrm{i}\cdot\sin⁡160^{\circ}\right) \), \( b=3\sqrt2\left(\cos⁡150^{\circ}+\mathrm{i}\cdot\sin150^{\circ}\right) \) and \( c=2\left(\cos240^{\circ}+\mathrm{i}\cdot\sin240^{\circ}\right) \), evaluate \( a\cdot b\cdot c \).
\(12\left(\cos190^{\circ}+\mathrm{i}\cdot\sin190^{\circ}\right) \)
\(12\left(\cos10^{\circ}+\mathrm{i}\cdot\sin10^{\circ}\right) \)
\(12\left(\cos⁡10^{\circ}-\mathrm{i}\cdot\sin10^{\circ}\right) \)
\(12\left(\cos⁡190^{\circ}-\mathrm{i}\cdot\sin190^{\circ}\right) \)