1003162306
Level:
C
Find all the values of the real parameter \( p \) such that \( f(x)=2x^2+px+p \) is positive for all \( x\in\mathbb{R} \).
\( p\in(0;8) \)
\( p\in(-\infty;0)\cup(8;+\infty) \)
\( p\in(-\infty;0) \)
\( p\in(0;\infty) \)
C
1003162305
Level:
C
Find all the values of the real parameter \( p \) such that \( f(x)=3(x-2)^2+p \) is nonnegative for all \( x\in\mathbb{R} \).
\( p\in[0;\infty) \)
\( p\in(-\infty;0) \)
\( p=0 \)
\( p\in(0;\infty) \)
1003162304
Level:
C
Find all the values of the real parameter \( m \) such that \( f(x)=-x^2+2xm-m^2+2 \) is increasing on \( (-\infty;0) \).
\( m\in[0;\infty) \)
\( m\in(-\infty;0) \)
\( m\in(-\infty;0] \)
\( m\in(-\infty;2] \)
1003162303
Level:
C
Find all the values of the real parameter \( m \) such that \( f(x)=3(x+m)^2-2 \) is increasing on \( (0;\infty) \).
\( m\in[0;\infty) \)
\( m\in(-\infty;0) \)
\( m\in(-\infty;0] \)
\( m\in(-\infty;2] \)
1003162302
Level:
C
Find all the values of the real parameter \( m \) such that \( f(x)=-2(x-m)^2+3 \) is an even function.
\( m=0 \)
\( m=3 \)
\( m=-3 \)
\( m\in(-\infty;\infty) \)
1003162301
Level:
C
Find all the values of the real parameter \( a \) such that \( f(x)=ax^2-2 \) is decreasing on \( (0;\infty) \).
\( a\in(-\infty;0) \)
\( a\in(0;\infty) \)
\( a\in[2;+\infty) \)
\( a\in(-\infty;2] \)
1003160903
Level:
C
The graph of a linear function \( f \) passes through the point \( \left[4\sqrt3;2\right] \) and it makes an angle of \( 30^{\circ} \) with the positive direction of \( x \)-axis measured anticlockwise. Choose the correct form of the function \( f \), such that the given property is preserved.
\( f(x)=\frac{\sqrt3}3x-2 \)
\( f(x)=\sqrt3x-10 \)
\( f(x)=\frac{\sqrt3}3x+2 \)
\( f(x)=\sqrt3x+10 \)
1003160902
Level:
C
Suppose \( f \) is a linear function. If the value of the independent variable \( x \) increases by \( 4 \), the function value decreases by \( 12 \). Choose the correct form of the function \( f \), such that the given property is preserved.
\( f(x)=-3x \)
\( f(x)=3x \)
\( f(x)=3x-12 \)
\( f(x)=-\frac13x \)
1003160901
Level:
C
Suppose \( f \) is a linear function. If the value of the independent variable \( x \) increases by \( 6 \), the function value increases by \( 18 \). Choose the correct form of the function \( f \), such that the given property is preserved.
\( f(x)=3x+1 \)
\( f(x)=-3x \)
\( f(x)=\frac13x+18 \)
\( f(x)=\frac13x \)
1003163004
Level:
C
Find the sum of all the roots of the given equation.
\[ \Bigl| |x-4|-9\Bigr|=0 \]
\( 8 \)
\( 18 \)
\( 6 \)
\( 4 \)