1003099601
Level:
A
Given the numbers \( x=1+2\sqrt2 \) and \( y=\sqrt2-1 \), calculate \( xy \).
\( 3-\sqrt2 \)
\( 4-\sqrt2 \)
\( 3 \)
\( -\sqrt2 \)
A
1003123402
Level:
A
Given the complex number \( b=\sqrt[3]2\cdot\left(\cos\frac56\pi+\mathrm{i}\cdot\sin\frac56\pi\right) \), find the polar form of \( b^9 \).
\( 8\cdot\left(\cos\frac32\pi+\mathrm{i}\cdot\sin\frac32\pi\right) \)
\( 64\cdot\left(\cos\frac12\pi-\mathrm{i}\cdot\sin\frac12\pi\right) \)
\( 8\cdot\left(\cos\frac12\pi-\mathrm{i}\cdot\sin\frac12\pi\right) \)
\( 64\cdot\left(\cos\frac32\pi+\mathrm{i}\cdot\sin\frac32\pi\right) \)
1003123401
Level:
A
Given the complex number \( a =\sqrt3\cdot\left( \cos 225^{\circ} + \mathrm{i}\cdot\sin 225^{\circ}\right) \), find the polar form of \( a^6 \).
\( 27\cdot\left(\cos270^{\circ}+\mathrm{i}\cdot\sin270^{\circ}\right) \)
\( 9\cdot\left(\cos90^{\circ}+\mathrm{i}\cdot\sin90^{\circ}\right) \)
\( 27\cdot\left(\cos90^{\circ}+\mathrm{i}\cdot\sin90^{\circ}\right) \)
\( 9\cdot\left(\cos270^{\circ}+\mathrm{i}\cdot\sin270^{\circ}\right) \)
1003085104
Level:
A
The \( n \)th term of an arithmetic sequence is \( 1-3n \). Find the \( 5 \)th term and the common difference.
\( a_5=-14;\ d=-3 \)
\( a_5=-2;\ d=-3 \)
\( a_5=14;\ d=-3 \)
\( a_5=-14;\ d=3 \)
\( a_5=-2;\ d=3 \)
1003085103
Level:
A
The third term of an arithmetic sequence is \( 3 \) and the common difference is \( 3 \). Find the \(n\)th term.
\( a_n=3n-6 \text{ for all } n\in\mathbb{N} \)
\( a_n=3n-3 \text{ for all } n\in\mathbb{N} \)
\( a_n=3n \text{ for all } n\in\mathbb{N} \)
\( a_n=3n+3 \text{ for all } n\in\mathbb{N} \)
\( a_n=3n+6 \text{ for all } n\in\mathbb{N} \)
1003085102
Level:
A
The first term of an arithmetic sequence is \( 6 \) and the sixth term is \( 1 \). Find the recursive formula for the sequence.
\( a_1=6;\ a_{n+1}=a_n-1 \text{ for all } n\in\mathbb{N} \)
\( a_1=6;\ a_{n+1}=a_n+1 \text{ for all } n\in\mathbb{N} \)
\(a_1=1;\ a_{n+1}=a_n+5 \text{ for all } n\in\mathbb{N} \)
\( a_1=1;\ a_{n+1}=a_n-5 \text{ for all } n\in\mathbb{N} \)
1003085101
Level:
A
The second term of an arithmetic sequence is \( 3 \) and the fourth term is \( -1 \). Find the recursive formula for the sequence.
\( a_1=5;\ a_{n+1}=a_n-2 \text{ for all } n\in\mathbb{N} \)
\( a_1=2;\ a_{n+1}=a_n-2 \text{ for all } n\in\mathbb{N} \)
\( a_1=3;\ a_{n+1}=a_n-1 \text{ for all } n\in\mathbb{N} \)
\( a_1=5;\ a_{n+1}=a_n-4 \text{ for all } n\in\mathbb{N} \)
\( a_1=3;\ a_{n+1}=a_n-4 \text{ for all } n\in\mathbb{N} \)
1003123807
Level:
A
Choose the interval in which at least one root of the following equation lies.
\[ x^2-8=0 \]
\( [ 2;3 ] \)
\( [ 3;4 ] \)
\( [ -1;1 ] \)
\( [ -8;-5 ] \)
1003123806
Level:
A
Choose the interval in which all the roots of the following equation lie.
\[ 5x^2-7x=0 \]
\( \left[-\frac75;\frac75\right] \)
\( \left[-1;1\right] \)
\( ( 0;2] \)
\( \left[-\frac75;0\right] \)
1003123805
Level:
A
Choose the equation which has the double root equal to zero.
\( 2x^2=0 \)
\( x^2+x=0 \)
\( x^2-x=0 \)
\( -x^2+x=0 \)