Complex numbers in algebraic and polar form

9000037507

Level: 
A
Given complex numbers \[ a = \sqrt{3} + 2\mathrm{i}\text{, }\quad b = \sqrt{2} -\mathrm{i}\text{, } \] find the quotient \(\frac{a} {b}\).
\(\frac{\sqrt{6}-2} {3} + \mathrm{i}\frac{2\sqrt{2}+\sqrt{3}} {3} \)
\(\frac{\sqrt{6}-2} {3} -\mathrm{i}\frac{2\sqrt{2}+\sqrt{3}} {3} \)
\(\frac{\sqrt{6}-3} {2} + \mathrm{i}\frac{2\sqrt{2}+\sqrt{3}} {2} \)
\(\frac{\sqrt{6}-2} {2} -\mathrm{i}\frac{2\sqrt{2}+\sqrt{3}} {2} \)

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 \( a\cdot b\cdot c \).
\(12\left(\cos190^{\circ}+\mathrm{i}\cdot\sin190^{\circ}\right) \)
\(12\left(\cos10^{\circ}+\mathrm{i}\cdot\sin10^{\circ}\right) \)
\(12\left(\cos⁡10^{\circ}-\mathrm{i}\cdot\sin10^{\circ}\right) \)
\(12\left(\cos⁡190^{\circ}-\mathrm{i}\cdot\sin190^{\circ}\right) \)

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

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 \( \frac a{b\cdot c} \).
\( \cos\frac{\pi}6+\mathrm{i}\cdot\sin\frac{\pi}6 \)
\( \cos\frac{11}6\pi+\mathrm{i}\cdot\sin\frac{11}6\pi \)
\( 4\left(\cos\frac{\pi}6\pi+\mathrm{i}\cdot\sin\frac{\pi}6\pi\right) \)
\( 4\left(\cos⁡\frac{11}6\pi+\mathrm{i}\cdot\sin\frac{11}6\pi\right) \)

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.
\( [x;y] \in\left\{[4;1],[1;4]\right\} \)
\( [x;y]\in\left\{[6;1],[9;4]\right\} \)
\( [x;y]\in\left\{[4;9],[1;6]\right\} \)
\([x;y]\in\left\{[-4;9],[-1;6]\right\} \)
\( [x;y]\in\left\{[6;-1],[9;-4]\right\} \)