1003138502
Level:
A
Solve.
\[ \log_{\frac13}\!(3-x)=0\]
\( x=2 \)
The equation has no solution.
\( x =3 \)
\( x=-4 \)
A
1003138501
Level:
A
Solve.
\[ \log_2(3x-5)=4\]
\( x=7 \)
\( x=3 \)
\( x=\frac{11}3 \)
\( x=-\frac13 \)
1003206001
Level:
A
We are given three quadratic functions:
\[ \begin{aligned}
f_1(x)&=-x^2-2, \\
f_2(x)&=-x^2-2x-4, \\
f_3(x)&=x^2+2.
\end{aligned} \]
Which of the given functions are increasing on the interval \( (-2;0) \)?
only \( f_1 \)
only \( f_2 \)
\( f_1 \) and \( f_2 \)
all three given functions
1003206202
Level:
A
Given \( f(x)=-\frac12x^2+x+\frac32 \), find all input values of \( f \) such that the output values of \( f \) are positive.
\( x\in(-1;3) \)
\( x\in(-\infty;-1)\cup(3;+\infty) \)
\( x\in(-3;1) \)
\( x\in(-\infty;-3)\cup(1;+\infty) \)
1003206201
Level:
A
Given \( f(x)=2x^2-6x+8 \), find all input value(s) of \( f \) such that the output value of \( f \) is \( 5.5 \).
\( x_1=\frac52 \), \( x_2=\frac12 \)
\( x=35.5 \)
\( x_1=13 \), \( x_2=11 \)
\( x_1=-\frac52 \), \( x_2=-\frac12 \)
1003163909
Level:
A
Evaluate the following limit. (Apply L'Hospital's rule repeatedly.)
\[ \lim_{x\to0}\frac{x-\sin x}{x^3} \]
\( \frac16 \)
\( -\frac16 \)
\( \frac13 \)
\( -\frac13 \)
\( 0 \)
1003163908
Level:
A
Evaluate the following limit. (Apply L'Hospital's rule repeatedly.)
\[ \lim_{x\to1}\frac{\cos(\pi x)+1}{(x-1)^2} \]
\( \frac{\pi^2}2 \)
\( -\frac{\pi^2}2 \)
\( \frac{\pi}2 \)
\( -\frac{\pi}2 \)
1003163907
Level:
A
Evaluate the following limit. (Apply L'Hospital's rule repeatedly.)
\[ \lim_{x\to\infty}\frac{\mathrm{e}^x-2}{x^2} \]
\( \infty \)
\( 0 \)
\( 1 \)
\( \frac12 \)
1003163906
Level:
A
Evaluate the following limit. (Apply L'Hospital's rule repeatedly.)
\[ \lim_{x\to\infty}\frac{x^2-1}{x^3-2x^2+x} \]
\( 0 \)
\( \infty \)
\( \frac23 \)
\( 1 \)
1003163905
Level:
A
Use L'Hospital's rule to find the following limit.
\[ \lim_{x\to1}\frac{\sqrt{x+3}-2}{\ln x} \]
\( \frac14 \)
\( \frac12 \)
\( 0 \)
\( 1 \)
\( \frac{\sqrt2}2 \)