B

2010000702

Level: 
B
We are given a sequence \( \left( a_n \right)^{\infty}_{n=1} \) defined recursively by: \( a_1=-1,\ a_2=0\) and \(\ a_{n+2}=a_{n}-a_{n+1}-d\), where \(\ n\in\mathbb{N} \). Find the value of an unknown constant \( d\in\mathbb{R} \) and of the term \( a_5 \) if \( a_3 = -4 \).
\( d=3,\ a_5=-8 \)
\( d=5,\ a_5=-10 \)
\( d=3,\ a_5=1\)
\( d=5,\ a_5=-9 \)

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 \(\left(0;\frac{\pi}2\right)\). \[ \int \frac{\cos 2x} {\cos ^{2}x}\, \mathrm{d}x \]
\(2x -\mathop{\mathrm{tg}}\nolimits x + c,\ c\in \mathbb{R}\)
\(\frac{\sin 2x} {\frac{1} {3} \sin ^{3}x} + c,\ c\in \mathbb{R}\)
\(2x +\mathop{\mathrm{cotg}}\nolimits x + c,\ c\in \mathbb{R}\)