CTU FCE Department of Mathematics
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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.

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 a b c d e
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 a) $a=5\lland c=5$ b) $a=2\lland c=1$ c) $a=-5\lland c=-15$ d) $a=-2\lland c=-9$ e) $a=2\lland c=-1$
 a b c d e
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 a) ${3x^2+4y^2+36x-48y+96=0}$ b) ${9x^2+16y^2+72x-96y+144=0}$ c) ${3x^2+4y^2+18x-20y+24=0}$ d) ${x^2+2y^2+7x-10y+12=0}$ e) ${9x^2+16y^2+39x-52y+12=0}$
 a b c d e
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 a) $-20$ b) $-430$ c) $-48$ d) $-85$ e) $-80$
 a b c d e
1 point
 4. If ${\cos2\alpha=-1}$, then a) $\tg\alpha=0$ b) $\tg\alpha$ is undefined c) $\tg\alpha=\sqrt3$ d) $\tg\alpha=-1$ e) $\tg\alpha=1$
 a b c d e
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 a) $\dfrac{a^3}2$ b) $\dfrac{a^3}{3\sqrt2}$ c) $\dfrac{a^3}3$ d) $\dfrac{a^3}6$ e) $\dfrac{a^3}8$
 a b c d e
1 point
 6. Find the set of all solutions of the inequality ${\dfrac1{|2x+3|}<1}$ with the unknown ${x\in\R}$. a) $(-\infty,\,2\rangle$ b) $(-1,\,\infty)$ c) $(-\infty,\,-2)\cup(-1,\,\infty)$ d) $(-2,\,-\frac32)\cup(-\frac32,\,2)$ e) $(-\infty,\,-\frac32)\cup(-\frac32,\,\infty)$
 a b c d e
1 point
 7. The algebraic form of the complex number ${z=(1+2\i)[\cos(\frac14\pi)-\i\sin(\frac14\pi)]}$ is a) $\frac{3\sqrt2}2+\frac{\sqrt2}2\,\i$ b) $3\sqrt2+\sqrt2\,\i$ c) $3\sqrt2-\sqrt2\,\i$ d) $\frac32+\frac12\,\i$ e) $\frac{3\sqrt2}2-\frac{\sqrt2}2\,\i$
 a b c d e
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 a) ${x+1}$, if ${x\ne1}$ b) ${(x-1)^2}$, if ${x\ne1\lland x\ne-1}$ c) ${\dfrac1{(x-1)(x+1)}}$, if ${x\ne1\lland x\ne-1}$ d) ${(x+1)^2}$, if ${x\ne1\lland x\ne-1}$ e) ${x-1}$, if ${x\ne1\lland x\ne-1}$
 a b c d e
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 a) $\{-\frac12\pi,\,\frac12\pi,\,\frac32\pi\}$ b) $\{\frac12\pi,\,\pi\}$ c) $\{-\frac12\pi,\,0,\,\pi,\,\frac32\pi\}$ d) $\emptyset$ e) $\{-\frac12\pi,\,\frac32\pi\}$
 a b c d e
1 point
 10. If ${\log_2y=2-\frac12\log_2(2x+1)-\log_2(x+3)}$, then the number $y$ equals to a) $\dfrac{(x+3)\sqrt{2x+1}}2$ b) $-2x-\frac32$ c) $\dfrac1{(x+3)\sqrt{2x+1}}$ d) $\dfrac{\sqrt{2x+1}}{x+3}$ e) $\dfrac4{(x+3)\sqrt{2x+1}}$
 a b c d e
1 point
 11. The graph of the function ${f(x)=\sqrt{\dfrac{x+3}{x^2+5x+6}}}$ is a) b) c) d) e) not in any of the figures
 a b c d e
1 point
 12. The maximal domain of the function ${f(x)=\log^{-1}|x|}$ is a) $(0,\,\infty)$ b) $(-1,\,0)\cup(0,\,1)$ c) $\R-\{-1,\,0,\,1\}$ d) $(0,1)\cup(1,\,\infty)$ e) $\R-\{0\}$
 a b c d e
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 a) $(-\infty,\,-5)\cup(-1,\,1)\cup(1,\,\infty)$ b) $(-\infty,\,-2)\cup(1,\,\infty)$ c) $(-\infty,\,-5)\cup(1,\,\infty)$ d) $(-2,\,1)$ e) $(-5,\,\infty)$
 a b c d e
2 points
 14. The area ratio of a regular hexagon and the circle inscribed in the hexagon is a) $a^2:\pi$ b) $\sqrt3:\pi$ c) $\sqrt2:\pi$ d) $2\sqrt3:\pi$ e) $a:\pi$
 a b c d e
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 a) $m=1\llor m=-5$ b) $m=-4$ c) $m=\frac54$ d) $m=-\sqrt5\llor m=\sqrt5$ e) $m=0\llor m=1$