2010000701
Level:
B
We are given the sequence \(\left (an + b\right )_{n=1}^{\infty }\).
This sequence satisfies \(a_{7} - a_{2} = -10\). Use this information to find \(a\).
\(a = -2\)
\(a = 2\)
\(a = -1\)
\(a = 1\)
B
2010000601
Level:
B
Given the sets \(A = \{x\in \mathbb{Z}:x> - 1\}\)
and \(B = \{x\in \mathbb{N}: x\leq 3\}\) find the intersection \(A \cap B\).
\(\{1,2,3\}\)
\(\{0,1,2,3\}\)
\(\{1,2\}\)
\(\{-2,-1,0,1,2,3\}\)
2010000502
Level:
B
Suppose three numbers form three consecutive terms of an arithmetic sequence. The sum of these numbers is \(36\) and the product is \(-972\). Find the biggest of these three numbers.
\( 27\)
\( 12\)
\( 17\)
\( -3\)
\( 15\)
2010000501
Level:
B
Find the sum of the first $15$ negative terms of the arithmetic sequence: $\left(7-2n\right)_{n=1}^{\infty}$.
\( -225\)
\( -135\)
\( -240\)
\( -180\)
\( -450\)
2010000402
Level:
B
We are given the sequence \( \left( \frac{n}{n+1} \right)^{\infty}_{n=1} \). Find the recursive formula of such sequence.
\( a_1=\frac{1}{2}\,,\ a_{n+1}=a_n\frac{(n+1)^2}{n(n+2)},\ n\in\mathbb{N} \)
\( a_1={2}\,,\ a_{n+1}=a_n\frac{(n+1)^2}{n(n+2)},\ n\in\mathbb{N} \)
\( a_1=\frac{1}{2}\,,\ a_{n+1}=a_n\frac{n(n+1)}{(n+1)(n+2)},\ n\in\mathbb{N} \)
\( a_1={2}\,,\ a_{n+1}=a_n\frac{n(n+1)}{(n+1)(n+2)},\ n\in\mathbb{N} \)
2010000303
Level:
B
Evaluate the following integral on the interval \(\left(\frac54,+\infty\right)\).
\[
\int \frac{3}
{5 - 4x}\, \mathrm{d}x
\]
\(-\frac{3}
{4}\ln |5 - 4x| + c,\ c\in \mathbb{R}\)
\(-\frac{3}
{4\cdot \ln |5-4x|} + c,\ c\in \mathbb{R}\)
\(\frac{3}
{4}\ln |5 - 4x| + c,\ c\in \mathbb{R}\)
\( \frac{3}
{4\cdot \ln |5-4x|} + c,\ c\in \mathbb{R}\)
2010000302
Level:
B
Evaluate the following integral on the interval \(\left(\sqrt{\frac34},+\infty\right)\).
\[
\int \frac{8x}
{(4x^{2} - 3)^{2}}\, \mathrm{d}x
\]
\(\frac{1}
{3-4x^{2}} + c,\ c\in \mathbb{R}\)
\(\frac{4x^{2}}
{\frac{16}{5}x^{5}-8x^{3}+9x} + c,\ c\in \mathbb{R}\)
\(\frac{1}
{4x^{2}-3} + c,\ c\in \mathbb{R}\)
2010000301
Level:
B
Evaluate the following integral on the interval \(\left(0,\frac{\pi}2\right)\).
\[
\int \frac{\cos 2x}
{\cos ^{2}x}\, \mathrm{d}x
\]
\(2x -\mathop{\mathrm{tg}}\nolimits x + c,\ c\in \mathbb{R}\)
\(\frac{\sin 2x}
{\frac{1}
{3} \sin ^{3}x} + c,\ c\in \mathbb{R}\)
\(2x +\mathop{\mathrm{cotg}}\nolimits x + c,\ c\in \mathbb{R}\)
2010000208
Level:
B
The picture shows a part of the graph of the arithmetic sequence. Find the sum of the first twenty-four terms of the sequence.
\( 192\)
\( 560\)
\( 576\)
\( 200\)
2010000207
Level:
B
The picture shows a part of the graph of the arithmetic sequence. Find the sum of the first twenty-five terms of the sequence.
\( -350\)
\( -32\)
\( -68\)
\( -800\)