2010008214
Level:
C
How many solutions in the set of positive integers does the following inequality have?
\[ 2^{3x}\cdot 7^{x-2}\leq 4^{x+1}\]
\( 2\)
\( 3\)
\( 1\)
\( 4\)
C
2010008209
Level:
C
Solve the following inequality.
\[ \left(\frac12\right)^{2x}- 7\cdot \left(\frac12\right)^x-8 < 0\]
\(x \in (-3;+\infty) \)
\(x \in (-3;0) \)
\(x \in (-\infty;-1) \)
\(x \in (-\infty;-3) \)
2010008208
Level:
C
Solve the following inequality.
\[ \left(\frac15\right)^{2x}- 3\cdot \left(\frac15\right)^x-10 >0\]
\(x \in (-\infty;-1 ) \)
\(x \in (-1;+\infty) \)
\(x \in (-2;-1) \)
\(x \in (-\infty;0) \)
2010008207
Level:
C
Solve the following inequality.
\[ 2^{x-2}-2^{x+1}-2^{x+4}\leq -142\]
\(x \in [ 3;+\infty) \)
\(x \in [ 2;+\infty) \)
\(x \in (-\infty;2 ] \)
\(x \in (-\infty;3 ] \)
2010008206
Level:
C
Solve the following inequality.
\[ 3^{x-1}-3^{x+2}-3^{x+1}\geq -105\]
\(x \in (-\infty;2 ] \)
\(x \in [ 2;+\infty) \)
\(x \in [ 3;+\infty) \)
\(x \in (-\infty;3 ] \)
2010007603
Level:
C
Find the constant term in the expansion of \( \left(2x^2-3\right)^{25} \).
\( -3^{25} \)
\( 3^{25} \)
\( 2^{25} \)
\( -2^{25} \)
2010007602
Level:
C
What is the coefficient at \( x^5 \) in the expansion of \( (1-2x)^{11} \)?
\( -14\:784 \)
\( 14\:784 \)
\( 7\:374 \)
\( -7\:374 \)
2010007304
Level:
C
Which of the given sets contains exactly all non-negative integers that satisfy the inequality \( \sqrt{(3x+6)^2} \leq 12 \)?
\( \{0;1;2\} \)
\( \{0;1;2;3;4;5;6\} \)
\( \{2;3;4;5\} \)
\( \{1;2\} \)
2010007302
Level:
C
The surface area of a cuboid is \( 19\,942\,\mathrm{cm}^2 \). Its dimensions are in the ratio \( 2:5:7 \). Find the volume of the cuboid.
\( 153\,790\,\mathrm{cm}^3 \)
\( 615\,160\,\mathrm{cm}^3 \)
\( 76\,895\,\mathrm{cm}^3 \)
\( 175\,760\,\mathrm{cm}^3 \)
2010007003
Level:
C
The shop offers a selection of \(8\) types of ice-lollies? (All types in numbers over \(4\) pieces.). In how many ways can we choose \(4\) ice-lollies?
\(\frac{11!}
{4!\, 7!}=330\)
\(\frac{8!}
{4!\, 4!}=70\)
\(\frac{8!}
{4!}=1680\)
\(4^8=65536\)