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 \(d = 4\).
\(a_{1} = 3;\ a_{n} = a_{n-1} + 4,\ n\in\mathbb{N}\)
\(a_{1} = 7;\ a_{n+1} = a_{n} + 4,\ n\in\mathbb{N}\)
\(a_{n} = 7 + a_{n+4},\ n\in\mathbb{N}\)
\(a_{n+1} = a_{n} + 7,\ n\in\mathbb{N}\)

9000065501

Level: 
A
Evaluate the following integral on \(\mathbb{R}\). \[ \int (x^{3} + x^{2} - 2x)\, \mathrm{d}x \]
\(\frac{1} {4}x^{4} + \frac{1} {3}x^{3} - x^{2} + c,\ c\in \mathbb{R}\)
\(\frac{1} {4}x^{4} -\frac{1} {3}x^{3} + x^{2} + c,\ c\in \mathbb{R}\)
\(3x^{2} + 2x - 2 + c,\ c\in \mathbb{R}\)
\(3x^{2} - 2x + 2 + c,\ c\in \mathbb{R}\)