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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 ${2x+by+1=0}$ and the line $AB$, where $A[-3,\,c]$, $B[2,\,-1]$, are identical if and only if
 
   
 
2 points
2. An equation of the tangent line to the ellipse ${9x^2+54x+16y^2-63=0}$ at the point $[1,\,0]$ is
 
   
 
1 point
3. The sum of the first three numbers of the sequence $(a_n)_{n=1}^{\infty}$, that is given by the recurrent formula ${a_{n+1}=10a_n-n}$ and by the term ${a_1=10}$, is
 
   
 
1 point
4. If ${\cotg\alpha=1}$, then $\sin2\alpha$ equals to the number
 
   
 
2 points
5. The volume of the cube inscribed to the sphere with the radius $r$ is
 
   
 
1 point
6. Find the set of all solutions of the inequality ${\dfrac2{|3x-2|}<\dfrac14}$ with the unknown ${x\in\R}$.
 
   
 
1 point
7. The complex conjugate of ${z=(1+2\i)(5-3\i)-5+2\i}$ is
 
   
 
1 point
8. The expression ${\dfrac{a^2+b^2}{a^2+ab}:\left(\dfrac a{a-b}-\dfrac b{a+b}\right)}$ equals to
 
   
 
1 point
9. The equation ${2\cos^2x-\sqrt 3\sin x-2=0}$ has in the interval $\langle 0,\,2\pi)$
 
   
 
1 point
10. If ${\log_4y=1-2\log_4(x^2+1)+\frac {1}{2}\log_4(x+1)}$, then the number $y$ equals to
 
   
 
1 point
11. The graph of the function ${f(x)=\dfrac{1}{|x|-3}\,}$ is
 
   
 
1 point
12. The maximal domain of the function ${f(x)=3^{\tfrac{x}{x^2-6x+8}}}$ is
 
   
 
2 points
13. The set of all solutions of the inequality ${\left|\dfrac{2x+1}{x-3}+1\right|<1}$ with the unknown ${x\in\R}$ is
 
   
 
2 points
14. An equilateral triangle $DEF$ is inscribed in the equilateral triangle $ABC$ with the side $a$ satisfying ${D\in AB}$, ${E\in BC}$, ${F\in CA}$. If the area of the triangle $DEF$ is equal to a third of the area of the triangle $ABC$, then its side is equal to
 
   
 
2 points
15. The equation ${x^2+(2m+4)x+m-1=0}$ (with the unknown $x$)

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