9000146706
Level:
A
Expand the following polynomial.
\[
(2x + 3)^{2} - (2 - x)^{2}
\]
\(3x^{2} + 16x + 5\)
\(3x^{2} + 8x + 13\)
\(3x^{2} + 13\)
\(3x^{2} + 5\)
A
9000148902
Level:
A
There are four paths from a city to the top of nearby mountain. Find the number of
possible treks from the city to the mountain and back, if it is required to use one
path up and another one down.
\(4\cdot 3=12\)
\(4\cdot 4=16\)
\(4 + 3=7\)
\(2\cdot 4=8\)
9000149304
Level:
A
How many points of symmetry there exist for a rhombus? A rhombus -- opposite sides are parallel and all sides have equal length. (A point is a point of
symmetry of the rhombus if the reflection through this point maps the rhombus into
itself.)
one
none
two
four
9000149301
Level:
A
Which of the following is true for a reflection through a line?
It is a congruent mapping (preserves shapes and dimensions).
It is a similar mapping (preserves shapes, may change dimensions and position).
It is an identity mapping (objects are mapped to itself).
9000149302
Level:
A
Given a line in a plane, find how many points are mapped onto itself with reflection
through this line.
infinitely many
none
one
two
9000141901
Level:
A
Given the function \(f\),
find \(\lim _{x\to 1}f(x)\).
\[
f(x)=\begin{cases}
x^3+1 & \text{if } x\neq 1,\\
3 & \text{if } x = 1
\end{cases}
\]
\(2\)
\(3\)
\(1\)
Does not exist
9000141906
Level:
A
Given the function \(g\) (see the picture),
find \(\lim _{x\to \infty }g(x)\).
\[
g(x)=\begin{cases}
-\frac12(x-1)^2+2 & \text{if } x < 1,\\
\frac2{x^2}+1 & \text{if } x \geq 1
\end{cases}
\]
\(1\)
\(0\)
\(\infty \)
\(-\infty \)
This limit does not exist.
9000141902
Level:
A
Given the function \(f\),
find \(\lim _{x\to \infty }f(x)\).
\[
f(x)=\begin{cases}
x^3+1 & \text{if } x\neq 1,\\
3 & \text{if } x = 1
\end{cases}
\]
\(\infty \)
\(-\infty \)
\(4\)
Does not exist
9000141907
Level:
A
Given the function \(h\) (see the picture),
find \(\lim _{x\to 1^{-}}h(x)\).
\[
h(x)=\begin{cases}
-\frac1{x-1} & \text{if } x< 1,\\
-(x-1)^2+2 & \text{if } x\geq 1
\end{cases}
\]
\(\infty \)
\(1\)
\(2\)
\(-\infty \)
This limit does not exist.
9000141903
Level:
A
Given the function \(g\),
find \(\lim _{x\to 1^{-}}g(x)\).
\[
g(x)=\begin{cases}
-\frac12(x-1)^2+2 & \text{if } x < 1,\\
\frac2{x^2}+1 & \text{if } x \geq 1
\end{cases}
\]
\(2\)
\(3\)
\(1\)
Does not exist