1003085202
Level:
B
What is the sum of all two-digit numbers divisible by five?
$945$
$950$
$1050$
$1045$
$940$
B
1003085201
Level:
B
Find the sum of all even numbers which are greater than $6$ and smaller than $122$.
$3648$
$3584$
$3654$
$3712$
$3776$
1003134604
Level:
B
Find the $5$th term of a geometric sequence $\{a_n\}_{n=1}^{\infty}$, where $a_1\cdot a_3\cdot a_5=x^3$, and $\frac{a_4}{a_2} =y^4$.
$xy^4$
$\frac x{y^4}$
$\frac x{y^2}$
$x$
$xy^2$
1003134603
Level:
B
The sum of the first five terms of a geometric sequence is less than $1$ and the common ratio is $10$. Determine all possible values of the first term.
$ a_1 < \frac1{11111}$, $a_1\in\mathbb{R}$
$ -\frac1{10^5} < a_1 < \frac1{10^5}$, $a_1\in\mathbb{R}$
$a_1 < \frac1{99999}$, $a_1\in\mathbb{R}$
$a_1 < 10^{-4}$, $a_1\in\mathbb{R}$
$a_1 < 10^{-5}$, $a_1\in\mathbb{R}$
1003134602
Level:
B
The sum of the second and the third term of a geometric sequence is $6\log3$ and the common ratio is $2$. Find the sum of the first three terms.
$7\log3$
$2\log3^6$
$2\log6$
$\log9$
$3\log6$
1003107809
Level:
B
Solve an indefinite integral $\int\frac{8x^7-30x^5}{x^8-5x^6+2}\,\mathrm{d}x$ on $(3;\infty)$.
$\ln\left|x^8-5x^6+2\right|+c$, $c\in\mathbb{R}$
$\ln\left|8x^7-30x^5+2x\right|+c$, $c\in\mathbb{R}$
$\log\left|x^8-5x^6+2\right|+c$, $c\in\mathbb{R}$
$\log\left|8x^7-30x^5+2x\right|+c$, $c\in\mathbb{R}$
1003107808
Level:
B
Find the indefinite integral $\int\frac{\ln^5x}x\,\mathrm{d}x$ on $(0;\infty)$. Use a substitution $a=\ln x$.
$\frac{\ln^6x}6+c$, $c\in\mathbb{R}$
$5\ln^4x+c$, $c\in\mathbb{R}$
$\frac{\ln^2x}2+c$, $c\in\mathbb{R}$
$\frac12\ln^5x+c$, $c\in\mathbb{R}$
1003107804
Level:
B
Four girls evaluated the integral $I=\int\sin x\cdot\cos x\,\mathrm{d}x$ on $\mathbb{R}$. Ann started to integrate by parts like this: $I=\int\sin x\cdot\cos x\,\mathrm{d}x=\sin^2x-\int\cos x\cdot\sin x\,\mathrm{d}x$. Beth started to integrate by parts like this: $I=\int\sin x\cdot\cos x\,\mathrm{d}x=-\cos^2 x-\int\sin x\cdot\cos x\,\mathrm{d}x$. Claire used substitution $a=\sin x$ like this: $I=\int\sin x\cdot\cos x\,\mathrm{d}x=\int a\,\mathrm{d}a$. Diana integrated $\int\sin x\cdot\cos x\,\mathrm{d}x=-\cos x\cdot\sin x+c$, $c\in\mathbb{R}$. Which of the girls made a mistake?
Diana
Ann
Beth
Claire
1103107803
Level:
B
Which of the following pictures shows the graph of a function $F(x)$, that is primitive to the function $f(x)=\frac{x^2-11x+28}{x-7}$ on $(-\infty;7)$?
1103107802
Level:
B
What is the color of the graph of a function $F(x)$, that is primitive to the function $f(x)=\frac2{x+1}$ on $(-1;\infty)$? (See the picture.)
yellow
green
blue
red