9000064507
Level:
A
Solve the following quadratic equation in the complex plane.
\[
4x^{2} + 12 = 0
\]
\(x_{1, 2} =\pm \mathrm{i}\sqrt{3}\)
\(x_{1, 2} =\pm 3\)
\(x_{1, 2} =\pm 3\mathrm{i}\)
\(x_{1, 2} =\pm \sqrt{3}\)
A
9000064508
Level:
A
Solve the following quadratic equation in the complex plane.
\[
2x^{2} + x + 1 = 0
\]
\(x_{1, 2} = \frac{-1\pm \mathrm{i}\sqrt{7}}
{4} \)
\(x_{1, 2} = \frac{-1\pm \mathrm{i}\sqrt{7}}
{2} \)
\(x_{1, 2} = \frac{1\pm \mathrm{i}\sqrt{7}}
{4} \)
\(x_{1, 2} = \frac{1\pm \mathrm{i}\sqrt{7}}
{2} \)
9000065301
Level:
A
Find the recurrence equations for the arithmetic sequence with the first term
\(a_{1} = 4\) and the common
difference \(d = -2\).
\(a_{1} = 4;\ a_{n+1} = a_{n} - 2,\ n\in\mathbb{N}\)
\(a_{1} = 4;\ a_{n+1} = a_{1} - 2,\ n\in\mathbb{N}\)
\(a_{n} = 4 + a_{n+2},\ n\in\mathbb{N}\)
\(a_{n+1} = a_{n} + 2,\ n\in\mathbb{N}\)
9000065608
Level:
A
Using integrals write formula for the area of the shaded region.
\(\int _{a}^{b}(f(x) - g(x))\, \mathrm{d}x +\int _{ b}^{c}(g(x) - f(x))\, \mathrm{d}x\)
\(\int _{a}^{b}(g(x) - f(x))\, \mathrm{d}x +\int _{ b}^{c}(g(x) - f(x))\, \mathrm{d}x\)
\(\int _{a}^{b}(f(x) - g(x))\, \mathrm{d}x +\int _{ b}^{c}(f(x) - g(x))\, \mathrm{d}x\)
\(\int _{a}^{b}(f(x) + g(x))\, \mathrm{d}x +\int _{ b}^{c}(f(x) - g(x))\, \mathrm{d}x\)
9000065302
Level:
A
Find the formula for the \(n\)-th
term of an arithmetic sequence with the first term
\(a_{1} = 1\) and the
second term \(a_{2} = -2\).
\(a_{n} = 4 - 3n,\ n\in\mathbb{N}\)
\(a_{n} = 1 - 2n,\ n\in\mathbb{N}\)
\(a_{n} = -2 + n,\ n\in\mathbb{N}\)
\(a_{n} = 3 + 2n,\ n\in\mathbb{N}\)
9000064506
Level:
A
Find the factorization of the following quadratic polynomial in the set of polynomial
with complex valued coefficients.
\[
2x^{2} + 4x + 5
\]
\(2\! \left (x + 1 + \frac{\sqrt{6}}
{2} \mathrm{i}\right )\! \! \left (x + 1 -\frac{\sqrt{6}}
{2} \mathrm{i}\right )\)
\(2\! \left (x - 1 + \frac{\sqrt{6}}
{2} \mathrm{i}\right )\! \! \left (x - 1 -\frac{\sqrt{6}}
{2} \mathrm{i}\right )\)
\(\left (x + 1 -\frac{\sqrt{6}}
{2} \mathrm{i}\right )\! \! \left (x + 1 + \frac{\sqrt{6}}
{2} \mathrm{i}\right )\)
\(\left (x - 1 -\frac{\sqrt{6}}
{2} \mathrm{i}\right )\! \! \left (x - 1 + \frac{\sqrt{6}}
{2} \mathrm{i}\right )\)
9000065303
Level:
A
Find the recurrence equations for the arithmetic sequence with the second term
\(a_{2} = 7\) and the common
difference \(d = 4\).
\(a_{1} = 3;\ a_{n} = a_{n-1} + 4,\ n\in\mathbb{N}\)
\(a_{1} = 7;\ a_{n+1} = a_{n} + 4,\ n\in\mathbb{N}\)
\(a_{n} = 7 + a_{n+4},\ n\in\mathbb{N}\)
\(a_{n+1} = a_{n} + 7,\ n\in\mathbb{N}\)
9000065306
Level:
A
In the arithmetic sequence given by the second term
\(a_{2} = -3\) and the
fifth term \(a_{5} = 3\)
find \(a_{11}\).
\(a_{11} = 15\)
\(a_{11} = 22\)
\(a_{11} = 19\)
\(a_{11} = 27\)
9000065304
Level:
A
Find the first term \(a_{1}\) and
the common difference \(d\) of the
arithmetic sequence \((5 + 2n)_{n=1}^{\infty }\).
\(a_{1} = 7;\ d = 2\)
\(a_{1} = 5;\ d = 2\)
\(a_{1} = 3;\ d = -2\)
\(a_{1} = 2;\ d = 5\)
9000065305
Level:
A
In the arithmetic sequence given by the relations
\(a_{1} =\pi \),
\(a_{n+1} = a_{n} + 2\pi \) find
\(a_{13}\).
\(a_{13} = 25\pi \)
\(a_{13} = 27\pi \)
\(a_{13} = 26\pi \)
\(a_{13} = 24\pi \)