2110005106
Level:
A
In which picture can you see a pair of graphs of functions \( f_1 \) and \( f_2 \) that are primitive to the same function?
A
2010005105
Level:
A
Given the function \(f(x) =\cos x -\sin x\), find its primitive function F so that the graph of \(F\) passes through the point \(A = \left [ \pi,3\right ]\).
\(F(x) =\sin x +\cos x + 4\)
\(F(x) =\cos x -\sin x + 4\)
\(F(x) = \sin x -\cos x + 2\)
2010005104
Level:
A
Evaluate the following integral on the interval \((0,+\infty)\).
\[
\int \left (2x^{-1}+\frac{2}
{x^2} - 3x^{-3} \right )\, \mathrm{d}x
\]
\(2\ln |x| - \frac{2}
{x} +\frac{3}
{2x^{2}} + c,\ c\in \mathbb{R}\)
\(2\ln |x| - \frac{2}
{x} -\frac{3}
{2x^{2}} + c,\ c\in \mathbb{R}\)
\(2\ln |x| + \frac{2}
{x} +\frac{3}
{2x^{2}} + c,\ c\in \mathbb{R}\)
\(2\ln |x| + \frac{2}
{x} -\frac{3}
{2x^{2}} + c,\ c\in \mathbb{R}\)
2010005103
Level:
A
Evaluate the following integral on \( (0,\infty) \).
\[ \int\left(6\sqrt x-5\sqrt[3]{x^2}+10\sqrt[4]{x}\right)\mathrm{d}x \]
\( 4x\sqrt x-3x\sqrt[3]{x^2}+8x\sqrt[4]{x}+c,\ c\in\mathbb{R} \)
\( 9 x\sqrt x-\frac{25}3 x^3\sqrt[3]{x^2}+\frac{25}2 x\sqrt[4]{x}+c,\ c\in\mathbb{R} \)
\( 4\sqrt x-3\sqrt[3]{x^2}+8\sqrt[4]{x}+c,\ c\in\mathbb{R} \)
\(x+20\sqrt{x}+c,\ c\in\mathbb{R} \)
2010005102
Level:
A
Evaluate the following integral on \( \left(\frac{\pi}2,\pi\right) \).
\[ \int\left(5 \sin x-\frac3{\cos^2x}-\frac7{\sin^2x}\right)\mathrm{d}x \]
\( -5\cos x-3\,\mathrm{tg}\,x+7\,\mathrm{cotg}\,x+c,\ c\in\mathbb{R} \)
\( 5\cos x+3\,\mathrm{tg}\,x+7\,\mathrm{cotg}\,x+c,\ c\in\mathbb{R} \)
\( -5\cos x-3\,\mathrm{tg}\,x-7\,\mathrm{cotg}\,x+c,\ c\in\mathbb{R} \)
\( 5\cos x+3\,\mathrm{tg}\,x+7\,\mathrm{cotg}\,x+c,\ c\in\mathbb{R} \)
2010005101
Level:
A
Evaluate the following integral on \( \mathbb{R} \).
\[ \int\left(2^3+2x^3+\mathrm{e}^x-2^x-2^{\mathrm{e}}\right)\mathrm{d}x \]
\( 8x+0.5x^4+\mathrm{e}^x-\frac{2^x}{\ln2} -2^{\mathrm{e}} x+c,\ c\in\mathbb{R} \)
\( -0.5x^4+\mathrm{e}^x-\frac{2^x}{\ln2} +c,\ c\in\mathbb{R} \)
\( 8x-2x^4+\mathrm{e}^x-2^x-\frac{2^{\mathrm{e}+1}}{\mathrm{e}+1}+c,\ c\in\mathbb{R} \)
\( 4-6x^4+\mathrm{e}^x -\frac{2^x}{\ln2} -2^\mathrm{e} x+c,\ c\in\mathbb{R} \)
2010005008
Level:
A
Determine whether the following planes \(\alpha \) and
\(\beta\) are parallel, identical or intersecting.
\[
\begin{aligned}[t] \alpha \colon &x = 1-m+2n, &
\\&y =2m-n,
\\&z = 2-m+n,\ m,n\in \mathbb{R},
\\ \end{aligned}\qquad \beta \colon x-y-3z+5 = 0
\]
identical
intersecting
parallel, not identical
2010005003
Level:
A
Find all the values of the real parameter \(p\)
so that the lines \(a\)
and \(b\)
are skew lines.
\[
\begin{aligned}a\colon x& =- 1 + 2m, &
\\y & = 1 - pm,
\\z & = 2 - m,\ m\in \mathbb{R}
\\ \end{aligned}\qquad \qquad \begin{aligned}b\colon x& = 3+2n, &
\\y & = 1-n,
\\z & = 5+4n,\ n\in \mathbb{R}
\\ \end{aligned}
\]
\(p\in\mathbb{R}\setminus\{-1\}\)
\(p = -1\)
No solution exists.
The lines are skew for every real \(p\).
2010005002
Level:
A
Find the intersection of the line \(KL\)
and the line \(q\),
where \(K = [1,3,5]\),
\(L = [3,-2,4]\) and
\[
\begin{aligned}q\colon x& = 1 + r, &
\\y & = 5 - 2r,
\\z & = 3 - r,\ r\in \mathbb{R}.
\\ \end{aligned}
\]
\([-3,13,7]\)
\([5,-7,3]\)
\([5,-3,-1]\)
There is no intersection.
2010005001
Level:
A
Determine whether two lines $a$ and $b$ are identical, parallel, intersecting or skew.
\[\begin{aligned}
a\colon x & = 3 -2m, & &
\\y & = 4 - 3m, & &
\\z & = 4+m,\ m\in \mathbb{R} & &
\end{aligned}\] \[\begin{aligned}
b\colon x & = - n, & &
\\y & = -5, & &
\\z & = 4-3n,\ n\in \mathbb{R} & &
\end{aligned}\]
skew lines
identical lines
intersecting lines
parallel lines, not identical