1003163904
Level:
A
Use L'Hospital's rule to find the following limit.
\[ \lim_{x\to\frac{\pi}3}\frac{1-2\cos x}{\pi-3x} \]
\( -\frac{\sqrt3}3 \)
\( -\frac{\sqrt3}6 \)
\( \frac{\sqrt3}3 \)
\( \frac{\sqrt3}6 \)
\( -\frac13 \)
A
1003163903
Level:
A
Use L'Hospital's rule to find the following limit.
\[ \lim_{x\to0}\frac{\sin2x}{\mathrm{tg}\,3x} \]
\( \frac23 \)
\( 1 \)
\( 2 \)
\( 0 \)
\( 6 \)
1003163902
Level:
A
Use L'Hospital's rule to find the following limit.
\[ \lim_{x\to0}\frac{\mathrm{e}^x-1}{\sin2x} \]
\( \frac12 \)
\( -\frac12 \)
\( 1 \)
\( 0 \)
\( -1 \)
1003163901
Level:
A
Use L'Hospital's rule to find the following limit.
\[ \lim_{x\to2}\frac{2x^3-3x^2-4}{x^2+x-6} \]
\( \frac{12}5 \)
\( \frac{18}5 \)
\( \frac{12}3 \)
\( 0 \)
\( \infty \)
1003160803
Level:
A
Use the substitution method to find the solution \( [x;y] \) of the following system of equations.
\[ \begin{aligned}
\frac{x+y}x+\frac1{x+y}=1 \\
\frac{2\cdot(x+y)}x-\frac1{x+y}=-7
\end{aligned} \]
\( \left[-\frac16;\frac12\right] \)
\( [-2;3] \)
\( \left[-\frac12;-\frac12\right] \)
\( \left[\frac12;\frac3{-2}\right] \)
1003160802
Level:
A
Use the substitution method to find the solution \( [x;y] \) of the following system of equations.
\[ \begin{aligned}
\frac{2x}{x+3}-3\cdot\frac{y+2}y=2 \\
\frac x{x+3}+2\cdot\frac{y+2}y=8
\end{aligned} \]
\( [-4;2] \)
\( [4;2] \)
\( [2;-4] \)
\( [-1;2] \)
1003160801
Level:
A
Use the substitution method to find the solution \( [x;y] \) of the following system of equations.
\[ \begin{aligned}
\frac2{x+4}-\frac1{2-y}=-6 \\
\frac1{x+4}+\frac5{2-y}=8
\end{aligned} \]
\( \left[-\frac92;\frac32\right] \)
\( [-2;2] \)
\( [2;10] \)
\( \left[-\frac92;3\right] \)
1003086104
Level:
A
Which of the following equations has exactly two solutions in the interval \( [0;\pi] \)?
\( 3\sin x - 2 = 0 \)
\( 2\sin x - 3 = 0 \)
\( 3\cos x + 2 = 0 \)
\( 3\sin x + 2 = 0 \)
1003032308
Level:
A
Consider polynomials \( p(x)=(m-2)x^3+3mx^2-x+m \) and \( q(x)=x^3+m^2x^2+x+3 \).
Polynomials \( p \) and \( q \) are different for every \( m \).
Polynomials \( p \) and \( q \) are equal for \( m=3 \).
Polynomials \( p \) and \( q \) are equal for \( m=-3 \).
Polynomials \( p \) and \( q \) are equal for \( m=3 \) and for \( m=0 \).
1003032307
Level:
A
The sum of the polynomials \( -x^3 y^2+6xy+5xy^4 \) and \( x^3-4xy^4+y^2 x^3+2xy \) is:
\( x^3+xy^4+8xy \)
\( -y^2+8xy+xy^4+y^2 x^3 \)
\( -x^3 y^2+8xy+xy^4+y^2 x^3+3x \)
\( x^3+xy^4+8x^2 y^2 \)