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1003085702

Level:

A

Solve

2 \sin^{2} x = \sqrt{2} \sin x

for

x

, where

x \in R

.

x \in ⋃_{k \in Z} [{k π} \cup {\frac{π}{4} + 2 k π} \cup {\frac{3 π}{4} + 2 k π}]

x \in ⋃_{k \in Z} [{\frac{π}{4} + 2 k π} \cup {\frac{3 π}{4} + 2 k π}]

x \in ⋃_{k \in Z} [{\frac{π}{4} + k π} \cup {\frac{3 π}{4} + k π}]

x \in ⋃_{k \in Z} [{2 k π} \cup {\frac{π}{4} + k π} \cup {\frac{3 π}{4} + k π}]

1003085701

Level:

A

Find all

x

,

x \in R

, such that

{cotg}^{2} x = - cotg x

.

x \in ⋃_{k \in Z} [{\frac{3 π}{4} + k π} \cup {\frac{π}{2} + k π}]

x \in ⋃_{k \in Z} {\frac{3 π}{4} + 2 k π}

x \in ⋃_{k \in Z} {\frac{3 π}{4} + k π}

x \in ⋃_{k \in Z} {\frac{π}{2} + k π}

1003099509

Level:

A

Given the numbers

x = 4 + 2 \sqrt{5}

and

y = 6 - 2 \sqrt{5}

, the fraction

\frac{x}{y}

can be written in the form:

\frac{11 + 5 \sqrt{5}}{4}

\frac{7 \sqrt{5} - 9}{4}

\frac{- 5 \sqrt{5}}{2}

8 \sqrt{5}

1003099508

Level:

A

Evaluate the expression

\frac{2 - x}{x - 2}

for

x = 2 - \sqrt{2}

.

- 1

\sqrt{2} - 2

2 - \sqrt{2}

1

1003099505

Level:

A

Rewrite

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

by rationalizing the denominator.

7 - 4 \sqrt{3}

(2 - \sqrt{3}) (2 + \sqrt{3})

\frac{7 - 4 \sqrt{3}}{5}

\frac{7 - 4 \sqrt{3}}{7}

1003099504

Level:

A

Rationalizing the denominator

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

we get:

\frac{\sqrt{7} - \sqrt{5}}{2}

\frac{\sqrt{5} + \sqrt{7}}{2}

\frac{\sqrt{5} - \sqrt{7}}{2}

\frac{- \sqrt{7} - \sqrt{5}}{2}

1003099607

Level:

A

Let

\frac{m}{6 - \sqrt{6}} = \frac{6 + \sqrt{6}}{6}

, determine

m

.

m = 5

m = 6

m = 1

m = - 5

1003099604

Level:

A

Express

{(\sqrt{2} + 3)}^{2}

in the simplest form:

11 + 6 \sqrt{2}

11

6 \sqrt{2}

5

1003099603

Level:

A

Calculate

{(2 \sqrt{75} - 3 \sqrt{48} + 2 \sqrt{27})}^{2}

.

48

192

12

60

1003099602

Level:

A

Simplifying

\frac{3}{2} \sqrt{8} + \sqrt{16} + \sqrt{32} - \frac{1}{3} \sqrt{18}

you get:

4 + 6 \sqrt{2}

4 + \sqrt{12}

2 + \sqrt{56}

4 + \sqrt{40}