1003134601
Level:
A
The fifth term of a geometric sequence is $4\sqrt2$ and the second term is $2$. Find the fourth term.
$4$
$2\sqrt2$
$3\sqrt2$
$\sqrt2$
$2$
A
1003107807
Level:
A
Find a function $F(x)$, that is primitive to the function $f(x)=2^x\cdot\ln2+4^x\cdot2\ln2+8^x\cdot3\ln2$ on $\mathbb{R}$, and satisfies the condition $F(0)=5$.
$F(x)=2^x+4^x+8^x+2$
$F(x)=\frac{2^x}{\ln 2}+\frac{4^x}{\ln 4}+\frac{8^x}{\ln 8}+2x$
$F(x)=2^x+4^x+8^x+5$
$F(x)=2^x\cdot\ln2+2^{x+1}\cdot\ln2+2^{x+3}\cdot\ln2+5$
1003107806
Level:
A
Define the function $f(x)$ so that the following holds: $f''(x)=\mathrm{e}^x+x^5$ on $\mathbb{R}$, $f(0)=1$, and $f(1)=\frac{43}{42}$.
$f(x)=\mathrm{e}^x+\frac{x^7}{42}+(1-\mathrm{e})x$
$f(x)=\mathrm{e}^x+\frac{x^7}{42}+(-\mathrm{e}-1)x$
$f(x)=\mathrm{e}^x+\frac{7}{6}x^7+x-\mathrm{e}x$
$f(x)=\mathrm{e}^x+\frac{x^7}{42}+\frac{43}{42}$
1003107805
Level:
A
Define the function $f(x)$ so that it holds: $f'(x)=x^5-\sqrt[4]x$ on $(0;\infty)$, $f(1)=-1$.
$f(x)=\frac{x^6}6-\frac45x\sqrt[4]x-\frac{11}{30}$
$f(x)=\frac{x^6}6-\frac45\sqrt[4]{x^5}+\frac{11}{30}$
$f(x)=\frac{x^6}6-\frac54x\sqrt[4]x-\frac{11}{30}$
$f(x)=\frac{x^6}6-\frac54x\sqrt[4]x+\frac{11}{30}$
1003107801
Level:
A
Identify the difference of the functions $F(x)=\frac{\sin^2x}2$ and $G(x)=-0.25\cdot\cos(2x)$, that are primitive to the same function $f(x)$.
$\frac14$
$\frac18$
$\frac12$
$1$
1003108908
Level:
A
What is the radius of a circle inscribed in a square, if the area of the square is given by the expression $\int\limits_1^6(6-x)\,\mathrm{d}x$ ?
$\frac{5\sqrt2}4$
$3\sqrt2$
$2\sqrt2$
$\frac{7\sqrt2}2$
1003108907
Level:
A
What are the lengths of a right triangle legs, if the triangle area is given by the expression $\int\limits_{-4}^1(0.8x+4.2)\,\mathrm{d}x-5$?
$5$ and $4$
$6$ and $3$
$4$ and $6$
$6$ and $5$
1103108906
Level:
A
Find the area of the triangle $ABC$ highlighted in yellow (see the picture).
$20.5$
$22.5$
$20$
$21$
1003108905
Level:
A
Find the area of the plain region bounded by the curves $f(x)=x^5$ and $g(x)=x^9$ on the interval $[1;2]$.
$91.8$
$113.2$
$91.6$
$100.4$
1003108904
Level:
A
What is the area of the plain region bounded by the lines $x=2$, $x=3$ and the graphs of functions $f(x)=\sin x$ and $g(x)=\frac1{x^2}$? Round the result to three decimal digits.
$0.407$
$0.167$
$1.573$
$0.623$