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9000046402

Level:

Consider the rectangle

A B C D

with length and height

| A B | = 6 cm

and

| B C | = 2 \sqrt{3} cm

. Let

S

be the intersection of diagonals. Find the angle

∡ A S B

120^{\circ}

60^{\circ}

90^{\circ}

9000046503

Level:

Identify the optimal first step convenient to solve the following trigonometric equation. Do not consider the step which is possible but does not help to solve the equation.

tg (- x + \frac{π}{6}) = \sqrt{3}

substitution

- x + \frac{π}{6} = z

tg (- x) = \sqrt{3} - \frac{π}{6}

{tg}^{2} (- x + \frac{π}{6}) = 3

\frac{\sin (- x + \frac{π}{6})}{\cos (- x + \frac{π}{6})} = \sqrt{3}

9000046504

Level:

Identify the optimal first step convenient to solve the following trigonometric equation. Do not consider the step which is possible but does not help to solve the equation.

\cos (x + \frac{π}{3}) = \frac{\sqrt{3}}{2}

substitution

x + \frac{π}{3} = z

\cos^{2} (x + \frac{π}{3}) = \frac{3}{4}

substitution

\frac{\sqrt{3}}{2} = z

\cos x \cdot \cos \frac{π}{3} - \sin x \cdot \sin \frac{π}{3} = \frac{\sqrt{3}}{2}

9000046510

Level:

Identify the optimal first step convenient to solve the following trigonometric equation. Do not consider the step which is possible but does not help to solve the equation.

2 \sin^{2} x - \sin x - 1 = 0

substitution

\sin x = z

substitution

\sin^{2} x = z

2 \sin^{2} x - \sin x = 1

2 \sin^{2} x - \sin x = \sin^{2} x + \cos^{2} x

9000037505

Level:

Find the complex conjugate of the following complex number.

- 2 \sqrt{3} - i

- 2 \sqrt{3} + i

2 \sqrt{3} - i

11

10 i

9000037404

Level:

Given

z = \sqrt{2} (\cos \frac{π}{3} - i \sin \frac{π}{3})

, find

z^{2}

- 1 - i \sqrt{3}

1 + i \sqrt{3}

- 2 - i \sqrt{2}

2 + i \sqrt{2}

9000037506

Level:

Given complex numbers

a = 3 + 5 i, b = 2 - i,

find the quotient

\frac{a}{b}

\frac{1}{5} + i \frac{13}{5}

\frac{1}{3} + i \frac{13}{3}

\frac{1}{5} + i \frac{7}{5}

\frac{1}{3} + i \frac{7}{3}

9000037503

Level:

Given complex numbers

a = \sqrt{2} + \sqrt{3} i, b = \sqrt{2} - \sqrt{3} i,

find the product

a b

5

2

\sqrt{2} + i \sqrt{3}

\sqrt{2} - i \sqrt{3}

9000037507

Level:

Given complex numbers

a = \sqrt{3} + 2 i, b = \sqrt{2} - i,

find the quotient

\frac{a}{b}

\frac{\sqrt{6} - 2}{3} + i \frac{2 \sqrt{2} + \sqrt{3}}{3}

\frac{\sqrt{6} - 2}{3} - i \frac{2 \sqrt{2} + \sqrt{3}}{3}

\frac{\sqrt{6} - 3}{2} + i \frac{2 \sqrt{2} + \sqrt{3}}{2}

\frac{\sqrt{6} - 2}{2} - i \frac{2 \sqrt{2} + \sqrt{3}}{2}

9000037403

Level:

Given

z = \sqrt{3} (\cos \frac{π}{3} + i \sin \frac{π}{3})

, find

z^{4}

- \frac{9}{2} - \frac{9 i \sqrt{3}}{2}

\frac{9}{2} + \frac{9 i \sqrt{3}}{2}

\frac{3}{2}

- \frac{3}{2}

A

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