2010001401
Level:
B
Given the sets \(A =\mathbb{Z}\)
and \(B = \{x\in \mathbb{N}\ \colon x < 5\}\),
find the union \(A\cup B\).
\(\mathbb{Z}\)
\(\mathbb{N}\)
\(\emptyset \)
\( \{ 1;2;3;4\}\)
B
2010000707
Level:
B
We are given a sequence \( \left( a_n \right)^{6}_{n=1} \) defined by the following graph. Find the recursive formula of such sequence.
\( a_1=2, \ a_{n+1}=-a_n, \ n \in \{1;2;3;4;5\}\)
\( a_1=-2, \ a_{n+1}=-a_n, \ n \in \{1;2;3;4;5\}\)
\( a_1=-2, \ a_{n+1}=-2a_n, \ n \in \{1;2;3;4;5\}\)
\( a_1=2, \ a_{n+1}=a_n, \ n \in \{1;2;3;4;5\}\)
2010000706
Level:
B
We are given a sequence \( \left( a_n \right)^{6}_{n=1} \) defined by the following graph. Find the recursive formula of such sequence.
\( a_1=-2, \ a_{n+1}=-a_n, \ n \in \{1;2;3;4;5\}\)
\( a_1=2, \ a_{n+1}=-a_n, \ n \in \{1;2;3;4;5\}\)
\( a_1=-2, \ a_{n+1}=-2a_n, \ n \in \{1;2;3;4;5\}\)
\( a_1=2, \ a_{n+1}=a_n, \ n \in \{1;2;3;4;5\}\)
2010001305
Level:
B
Factor the following polynomial expression.
\[
36b^{2}c^{2} - 9a^{2}b^{2} - 36c^{2}d^{2} + 9a^{2}d^{2}
\]
\(9\left (b - d\right )\left (b + d\right )\left (2c + a\right )\left (2c - a\right )\)
\(\left (b^2 + d^2\right )\left (36c^2 + 9a^2\right )\)
\(9\left (a - d\right )\left (a + d\right )\left (2b + c\right )\left (2b - c\right )\)
\(\left (a^2 + d^2\right )\left (36b^2 + 9c^2\right )\)
2010001304
Level:
B
Factor the following polynomial.
\[
20xy + 12y - 5x - 3
\]
\(\left (4y - 1\right )\left (5x +3\right )\)
\(4y\left (5x +3\right )\)
\(\left (1-4y\right )\left (5x +3\right )\)
\(- 4y\left (5x +3\right )\)
2010001303
Level:
B
Expand \(\left (x + y\right )^{3} - y\left (x -y\right )^{2}\).
\(x^{3} + 2x^{2}y + 5xy^{2}\)
\(x^{3} + 2x^{2}y + xy^{2}\)
\(x^{3} + 2x^{2}y + xy^{2}+2y^3\)
\(x^{3} + 2x^{2}y + 5xy^{2}+2y^3\)
2010001204
Level:
B
Evaluate the definite integral.
\[
\int _{-1}^{2} \frac{-2x}
{4+x^{2}}\, \text{d}x
\]
\(\ln \frac{5}
{8}\)
\(\ln 40\)
\(\ln \frac{8}
{5}\)
\(-\ln 40\)
2010001203
Level:
B
Evaluate the definite integral.
\[
\int _{\frac{2}{3}}^{3} \frac{1}
{3x -1}\, \text{d}x
\]
\(\ln 2\)
\(\ln 1\)
\(\ln 8\)
\(3\ln 8\)
2010001202
Level:
B
Evaluate the definite integral.
\[ \int\limits_{-\pi}^{\frac{\pi}2}x\cdot\cos x\,\mathrm{d}x \]
\( \frac{\pi+2}{2} \)
\(\pi +1\)
\(\frac{\pi}2\)
\( \frac{\pi}2-1\)
2010001201
Level:
B
Evaluate the definite integral \( \int\limits_{\frac{\pi}6}^{\frac{\pi}3}\frac{\mathrm{cotg}\,b}{\sin2b}\,\mathrm{d}b \).
\( \frac{\sqrt3}3 \)
\( -2\sqrt3 \)
\( 2\sqrt3 \)
\( -\sqrt3 \)