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Entrance Exam from Mathematics – online testing

There is a multiple choice of 5 answers in each of the 15 assignments of the following online test. There is always only one correct answer.

Please mark the answer you consider correct in the form on the left side.

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1 point
1. The line ${ax-5y+c=0}$ and the line $AB$, where $A[-2,\,-1]$, $B[3,\,1]$, are identical if and only if
 
   
 
2 points
2. An equation of the ellipse with the axes parallel to the coordinate axes and tangent to both coordinate axes at the points ${M[-4,\,0]}$, ${N[0,\,3]}$ is
 
   
 
1 point
3. The term $a_4$ of the sequence $(a_n)_{n=1}^{\infty}$, that is given by the recurrent formula ${a_{n+1}=(n+1)a_n-5}$ and by the term ${a_1=0}$, is
 
   
 
1 point
4. If ${\cos2\alpha=-1}$, then
 
   
 
2 points
5. In the cube $ABCDEFGH$ with the side $a$ denote $L$ as a midpoint of side $EH$. The volume of pyramid $BCDL$ is
 
   
 
1 point
6. Find the set of all solutions of the inequality ${\dfrac1{|2x+3|}<1}$ with the unknown ${x\in\R}$.
 
   
 
1 point
7. The algebraic form of the complex number ${z=(1+2\i)[\cos(\frac14\pi)-\i\sin(\frac14\pi)]}$ is
 
   
 
1 point
8. The expression ${\left(\dfrac{2x^2-4x+2}{x^2+1}:\dfrac{6x-6}{x^4-1}\right):\dfrac{x+1}3}$ equals to
 
   
 
1 point
9. The set of all solutions of the equation ${\cos^2x+3\sin x+3=0}$ in the interval $\langle -\pi,\,2\pi)$ is
 
   
 
1 point
10. If ${\log_2y=2-\frac12\log_2(2x+1)-\log_2(x+3)}$, then the number $y$ equals to
 
   
 
1 point
11. The graph of the function ${f(x)=\sqrt{\dfrac{x+3}{x^2+5x+6}}}$ is
 
   
 
1 point
12. The maximal domain of the function ${f(x)=\log^{-1}|x|}$ is
 
   
 
2 points
13. The set of all solutions of the inequality ${\left|\dfrac{{-5}}{x+2}\right|<\left|\dfrac{10}{x-1}\right|}$ with the unknown ${x\in\R}$ is
 
   
 
2 points
14. The area ratio of a regular hexagon and the circle inscribed in the hexagon is
 
   
 
2 points
15. The equation ${x^2+x+m^2+4m-5=0}$ (with the unknown $x$) has one root equal to zero, if and only if

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