2010013906
Level:
B
Suppose a function is convex up on the interval \([-2;1]\). Which one of the offered functions has that property?
\(h(x)=-\sqrt{x+5}\)
\(f(x)=\frac{x-2}{(x+5)^2}\)
\(g(x)=\frac{x^2-2}{2x}\)
\(k(x)=-\frac15x^3-2x^2+x+1\)
B
2010013905
Level:
B
Suppose a function is concave down on the interval \([-1;2]\). Which one of the offered functions has that property?
\(g(x)=\sqrt{-2x+8}\)
\(f(x)=\frac{-x+2}{(x+3)^2}\)
\(h(x)=\frac{x^2-1}{x}\)
\(k(x)=\frac12x^3+3x^2-x+2\)
2000019101
Level:
B
Determine the set of all values of the parameter \( a \in \mathbb{R} \setminus \{0\}\) for which the given equation has no solution.
\[
\frac{x-1}{x} = \frac{2-a}{3a}
\]
\(\left\{ \frac12\right\}\)
\(\left\{ \frac12; 2\right\}\)
\( \{ 1 \}\)
\(\left\{ \frac12; 1\right\}\)
2000019004
Level:
B
The system of equations is given by:
\[\begin{aligned}
2 x-y +z=5 & &
\\x +2y-3z =17& &
\\x +y -2z= 12& &
\end{aligned}\]
When solving the system using Cramer's rule, we evaluate determinants of four matrices. What is the sum of all these determinants?
\(-14\)
\(12\)
\(0\)
\(-20\)
2000019010
Level:
B
The system of equations is given by:
\[\begin{aligned}
x+y -z=3 & &
\\3x -y-7z =-7& &
\\ax +3y +z= 11& &
\end{aligned}\]
Find the value of a real parameter \(a\) for which the system has infinitely many solutions.
\(1\)
\(-1\)
\(2\)
\(-2\)
2000019009
Level:
B
How many solutions does the following system have?
\[\begin{aligned}
x+y -z=3 & &
\\3x -y-7z =-7& &
\\x +3y +z= 11& &
\end{aligned}\]
infinity
one
three
two
2000019008
Level:
B
The system of equations is given by:
\[\begin{aligned}
x-2y +3z= 2 & &
\\ax +y =5& &
\\-2x +6y -2z= 4& &
\end{aligned}\]
Find the value of a real parameter \(a\) for which the system has no solution.
\(-\frac27
\)
\(-\frac37
\)
\(-\frac25
\)
\(\frac27
\)
2000019007
Level:
B
The system of equations is given by:
\[\begin{aligned}
x+2z= 3 & &
\\2x -y+ z = 2& &
\\3x -2 y -z= 1 & &
\end{aligned}\]
When solving the system using Cramer's rule, we evaluate determinants of four matrices. What is the arithmetic mean of all these determinants?
\(2
\)
\(3.5
\)
\(\frac73
\)
\(\frac83
\)
2000019006
Level:
B
The coefficient matrix of a system of three linear equations with three unknowns is:
\[
\left (\array{
1& 2& 1\cr
3& -5& 2\cr
1& 0& -3} \right ).~
\]
What is the column of the right sides if the solution is the ordered triple \([−7; 2;−1]\)?
\(
\left (\array{
-4\cr
-33\cr
-4} \right )
\)
\(
\left (\array{
-2\cr
-33\cr
-4} \right )
\)
\(
\left (\array{
-4\cr
-31\cr
-4} \right )
\)
\(
\left (\array{
-4\cr
-33\cr
-10} \right )
\)
2000019005
Level:
B
To solve the system of three linear equations with three unknowns, it is necessary to calculate the determinants of the matrices:
\[
\left (\array{
1& -2& 3\cr
2& 1& -7\cr
-3& 1& -5} \right ),~
\left (\array{
1& 3& -1\cr
2& -7& -3\cr
-3& -5& 1} \right ).
\]
Which of the given ordered triples is the solution to this system?
\( [2,-2,3]\)
\( [2,2,3]\)
\( [-2,2,3]\)
\( [3,-2,2]\)