2010001105
Level:
B
Evaluate the definite integral \( \int\limits_{-1}^1 (x+1)^4\,\mathrm{d}x \).
\( 6.4 \)
\( 32\)
\( 0.2 \)
\( 8\)
B
2010001104
Level:
B
Evaluate the definite integral \( \int\limits_1^4\frac{\sqrt x+x^2-x^3}{x^2}\,\mathrm{d}x \).
\( -3.5 \)
\( -11 \)
\( -5.5 \)
\( -3\)
2010000905
Level:
B
Suppose we are given the following equality of two fractions with nonzero denominators. From the given expressions, choose the one that by substituting to the starred position makes the equality true.
\[
\frac{2- 3x}
{x +2} = \frac{2(9x^{2} - 12x + 4)}
{*}
\]
\((2x +4)(2 - 3x)\)
\((x +2)(2 - 3x)\)
\((x +2)(4 - 9x)\)
\((2x +4)(3x - 2)\)
2010000902
Level:
B
Assuming \(x\neq \pm y\)
and \(x\neq 0\),
simplify the expression:
\[
\left ( \frac{y}
{y-x} - \frac{2x}
{y+x} - \frac{y^{2}}
{y^{2} - x^{2}}\right ) : \left ( \frac{1}
{x + y} - \frac{y}
{y^{2} - x^{2}}\right )
\]
\( y-2x\)
\(2x-y\)
\(\frac{2x-y}
{x}\)
\( 0\)
2010000901
Level:
B
Assuming \(xy\neq 1\),
simplify the expression:
\[
\frac{ \frac{x+y}
{1-xy} -x}
{1 +\frac{x(x+y)}
{1-xy} }
\]
\( y\)
\(\frac{y(1+x^{2})}
{1-x^{2}} \)
\(\frac{y}
{1+x^{2}} \)
\( y(1+x^2)\)
2010000814
Level:
B
Assuming \(x\neq 0\), \(y\neq 0\), \(y\neq \pm 1\), simplify the expression:
\[\left [\left ( \frac{y-1} {y}\right )^{2} : \left (\frac{x} {y+1} \right )^{2}\right ] : \frac{2(y^2-1)} {xy}\]
\(\frac{y^2-1}
{2xy}\)
\( 2\)
\(\frac{y^2-1}
{2}\)
\(\frac{y-1}
{2}\)
2010000705
Level:
B
The \(n\)th term of a sequence is given by \(a_n = \frac{n^2-4}{n+4}\). Which term of this sequence is equal to \(5\)?
the eighth term
the third term
the ninth term
the fifth term
2010000704
Level:
B
The \(n\)th term of a sequence is given by \(a_n = \frac{n^2-1}{n+5}\). Which term of this sequence is equal to \(4\)?
the seventh term
the third term
the twenty-first term
the fourth term
2010000703
Level:
B
Consider the sequence \(2a_{n} = a_{n+1} - a_{n-1}\)
with \(a_{3} = 2\)
and \(a_{4} = 5\).
Find \(a_{2} - a_{1}\).
\( 1\)
\(4\)
\(-20\)
\(-25\)
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 \( a_3 = -4 \).
\( d=3,\ a_5=-8 \)
\( d=5,\ a_5=-10 \)
\( d=3,\ a_5=1\)
\( d=5,\ a_5=-9 \)