A

1003261908

Level: 
A
Determine all the values of \( t \), \( t\in\mathbb{R} \), such that the function \[ f(x)=tx^3+(t+1)x^2-(t-2)x+3 \] has local extrema.
\( t\in\mathbb{R}\setminus\left\{\frac12\right\} \)
\( t\in\mathbb{R} \)
\( t\in\left(-\frac12;\frac12\right) \)
\( t\in\left(-\infty;-\frac12\right)\cup\left(\frac12;\infty\right) \)

1003261905

Level: 
A
Find the local extrema of the function \[ f(x)=x-\ln⁡(1+x)\text{ .} \]
the local minimum at \( x=0 \)
the local minimum at \( x=0 \), the local maximum at \( x=-1 \)
the local maximum at \( x=0 \)
the local maximum at \( x=0 \), the local minimum at \( x=-1 \)
do not exist

1103061207

Level: 
A
We are given the straight line \( m= \left\{[3-t;t]\text{, } t\in\mathbb{R} \right\} \) which intersects lines \( a \), \( b \), \( c \) in points \( A \), \( B \), \( C \) consecutively (see the picture). Find the values of a parameter \( t \) corresponding to these line intersections.
\( t_A=1; t_B=\frac32;\ t_C=2 \)
\( t_A=-1; t_B=-2;\ t_C=-3 \)
\( t_A=2; t_B=\frac32;\ t_C=1 \)
\( t_A=2; t_B=\frac52;\ t_C=3 \)

1103061205

Level: 
A
From the following list choose the equation of a straight line that passes through the given point \( K \) and is not perpendicular to the given line \( m \) (see the picture).
\( r\colon y=\frac23x-\frac{13}3 \)
\( p\colon 3x+2y-13=0 \)
\( s\colon y=-\frac32x+\frac{13}2 \)
$\begin{aligned} q\colon x&=5+2t, \\ y&=-1-3t;\ t\in\mathbb{R} \end{aligned}$

1103061204

Level: 
A
From the following list choose the equation of a straight line that passes through the given point \( K \) and is not parallel to the given line \( m \) (see the picture).
\( g\colon y=-\frac32x+\frac{13}2 \)
\( b\colon 2x-3y-13=0 \)
\( f\colon y=\frac23x-\frac{13}3 \)
$\begin{aligned} q\colon x&=5+3t, \\ y&=-1+2t;\ t\in\mathbb{R} \end{aligned}$

1103061203

Level: 
A
A straight line \( p \) is given by the point \( A \) and the direction angle \( \varphi \) (see the picture). Choose the equation of the line \( p \) in the slope-intercept form.
\( p\colon y=-\sqrt3x+3 \)
\( p\colon y=\sqrt3x+3 \)
\( p\colon y=1.7x+3 \)
\( p\colon y=-1.7x+3 \)