1003124906
Level:
C
The number \( \left(\sqrt7+1\right)^4-\left(\sqrt7-1\right)^4 \) equals:
\( 64\sqrt7 \)
\( 32\sqrt7 \)
\( \sqrt7 \)
\( 2 \)
C
1003124905
Level:
C
Numbers \( \left(2^2\right)^{\left(2^2\right)} \), \( \left(2\right)^{\left({2^2}^2\right)} \), \( \left({2^2}^2\right)^2 \), \( \left(2\right)^{\left(2^2\right)^2} \) were written on the board. How many different numbers are represented by these records?
\( 2 \)
\( 3 \)
\( 4 \)
\( 1 \)
1003124904
Level:
C
The number\( \sqrt[4]{\left( \sqrt3-\sqrt2 \right)^4}+\sqrt[4]{\left( \sqrt2-\sqrt5 \right)^4}+\sqrt[3]{\left( \sqrt3-\sqrt5 \right)^3} \) equals:
\( 2\sqrt3 - 2\sqrt2 \)
\( 2\sqrt5 - 2\sqrt2 \)
\( 2\sqrt3 - 2\sqrt5 \)
\( 2\sqrt5 - 2\sqrt3 \)
1003124903
Level:
C
The number \( \left( \sqrt2+1 \right)^4 \) is greater than the number \( \left( \sqrt2-1 \right)^4 \) by:
\( 24\sqrt2 \)
\( 12\sqrt2 \)
\( 2 \)
\( 4\sqrt2 \)
1003124902
Level:
C
Rationalizing the denominator \( \frac1{\left(2-\sqrt3\right)^3} \) you get:
\( 26+15\sqrt3 \)
\( 27+30\sqrt3 \)
\( 14+7\sqrt3 \)
\( 27+24\sqrt3 \)
1003107511
Level:
C
Find the solution set of the following quadratic equation in the complex plane.
\[ 2x^2-x=3\mathrm{i}x+2 \]
\( \left\{-\frac12+\frac12\mathrm{i}, 1+\mathrm{i} \right\} \)
\( \left\{-\frac12+\mathrm{i}, 1+\frac12\mathrm{i} \right\} \)
\( \left\{\frac12-\frac12\mathrm{i}, -1-\mathrm{i} \right\} \)
\( \left\{\frac12-\mathrm{i}, -1-\frac12\mathrm{i} \right\} \)
\( \left\{-\frac12-\frac12\mathrm{i}, 1-\mathrm{i} \right\} \)
\( \left\{\frac12+\mathrm{i}, -1+\frac12\mathrm{i} \right\} \)
1003107510
Level:
C
Find the solution set of the following quadratic equation in the complex plane.
\[ x+\frac{\mathrm{i}}2=\frac1x+\frac2{\mathrm{i}} \]
\( \left\{-\frac12\mathrm{i},-2\mathrm{i} \right\} \)
\( \left\{\frac12\mathrm{i},2\mathrm{i} \right\} \)
\( \left\{\frac12\mathrm{i},-2\mathrm{i} \right\} \)
\( \left\{-\frac12\mathrm{i},2\mathrm{i} \right\} \)
\( \emptyset \)
1003107509
Level:
C
Find the solution set of the following quadratic equation in the complex plane.
\[ (3-2\mathrm{i})x^2-(2-4\mathrm{i})x + 2 = 0 \]
\( \left\{ 1-\mathrm{i}, \frac1{13}+\frac5{13}\mathrm{i} \right\} \)
\( \left\{ -1+\mathrm{i}, -\frac1{13}-\frac5{13}\mathrm{i} \right\} \)
\( \emptyset \)
\( \left\{ -1+\mathrm{i}, \frac1{13}+\frac5{13}\mathrm{i} \right\} \)
\( \left\{ 1-\mathrm{i}, -\frac1{13}-\frac5{13}\mathrm{i} \right\} \)
1003107508
Level:
C
Find the complex roots of the following quadratic equation.
\[ x^2 + (2 + 2\mathrm{i})x + 2\mathrm{i} = 0 \]
\( x_{1,2}=-1-\mathrm{i} \)
\( x_{1,2}=1+\mathrm{i} \)
\( x_1=1+\mathrm{i}\text{, }\ x_2=-1-\mathrm{i} \)
\( x_1=1-\mathrm{i}\text{, }\ x_2=-1+\mathrm{i} \)
1003107507
Level:
C
Find the solution set of the following quadratic equation in the complex plane.
\[ x^2 +3\mathrm{i}x + 10 = 0 \]
\( \{-5\mathrm{i}, 2\mathrm{i}\} \)
\( \{-2\mathrm{i}, 5\mathrm{i}\} \)
\( \{2-5\mathrm{i}, 5-2\mathrm{i}\} \)
\( \{-2+5\mathrm{i}, -5+2\mathrm{i}\} \)
\( \emptyset \)