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 ${2x+by+1=0}$ and the line $AB$, where $A[-3,\,c]$, $B[2,\,-1]$, are identical if and only if a) $b=0\lland c=-6$ b) $b=5\lland c=1$ c) $b=5\lland c=-1$ d) $b=-5\lland c=-3$ e) $b=-3\lland c=-1$
 a b c d e
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 a) ${y=0}$ b) ${x-y-1=0}$ c) ${x-1=0}$ d) ${x+y-1=0}$ e) ${y-1=0}$
 a b c d e
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 a) $988$ b) $1\,097$ c) $1\,087$ d) $100$ e) $998$
 a b c d e
1 point
 4. If ${\cotg\alpha=1}$, then $\sin2\alpha$ equals to the number a) $0$ b) $\frac12$ c) $1$ d) $-1$ e) $\frac{\sqrt2}2$
 a b c d e
2 points
 5. The volume of the cube inscribed to the sphere with the radius $r$ is a) ${\dfrac{4\sqrt3}9\,r^3}$ b) ${\dfrac{2\sqrt2}9\,r^3}$ c) ${\dfrac{8\sqrt2}9\,r^3}$ d) ${\dfrac{3\sqrt3}5\,r^3}$ e) ${\dfrac{8\sqrt3}9\,r^3}$
 a b c d e
1 point
 6. Find the set of all solutions of the inequality ${\dfrac2{|3x-2|}<\dfrac14}$ with the unknown ${x\in\R}$. a) $(-4,\,\frac23)\cup(\frac23,\,4)$ b) $(-\infty,\,-\frac83)\cup(\frac83,\,\infty)$ c) $(-\infty,\,-2)\cup(\frac{10}3,\,\infty)$ d) $(\frac23,\,\infty)$ e) $(-2,\,\frac23)\cup(\frac23,\,\frac{10}3)$
 a b c d e
1 point
 7. The complex conjugate of ${z=(1+2\i)(5-3\i)-5+2\i}$ is a) $-6+9\i$ b) $-6-9\i$ c) $9-6\i$ d) $6-9\i$ e) $6+9\i$
 a b c d e
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 a) $\dfrac{a-b}{a+b}$, if ${a\ne0\lland a\ne b\lland a\ne-b}$ b) $\dfrac{a+b}{a-b}$, if ${a\ne0\lland a\ne b\lland a\ne-b}$ c) $\dfrac{a+b}a$, if ${a\ne0}$ d) $\dfrac{a-b}a$, if ${a\ne0\lland a\ne b\lland a\ne-b}$ e) $\dfrac a{a-b}$, if ${a\ne0\lland a\ne b\lland a\ne-b}$
 a b c d e
1 point
 9. The equation ${2\cos^2x-\sqrt 3\sin x-2=0}$ has in the interval $\langle 0,\,2\pi)$ a) no solution b) exactly three solutions c) exactly two solutions d) exactly one solution e) exactly four solutions
 a b c d e
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 a) $\dfrac{3+x}{4(x^2+1)}$ b) $\dfrac{\sqrt{x+1}}{2(x^2+1)}$ c) $\dfrac{4\sqrt{x+1}}{(x^2+1)^2}$ d) $\frac12(x-1)-2x^2$ e) $\dfrac{\sqrt{x+1}}{(x^2+1)^2}$
 a b c d e
1 point
 11. The graph of the function ${f(x)=\dfrac{1}{|x|-3}\,}$ is a) b) c) not in any of the figures d) e) a b c d e
1 point
 12. The maximal domain of the function ${f(x)=3^{\tfrac{x}{x^2-6x+8}}}$ is a) $\R-\{0,\,2,\,4\}$ b) $(2,\,4)\cup(4,\,\infty)$ c) $\R-\{2,\,4\}$ d) $(0,\,\infty)$ e) $(4,\,\infty)$
 a b c d e
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 a) $(\frac32,\,3)$ b) $(-\frac12,\,\frac54)$ c) $(-\infty,\,-\frac12)$ d) $(3,\,\infty)$ e) $(-\frac12,\,\frac32)$
 a b c d e
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 a) ${\dfrac a4\,}$ b) ${\dfrac a{\sqrt6}\,}$ c) ${\dfrac a{\sqrt2}\,}$ d) ${\dfrac a2\,}$ e) ${\dfrac a{\sqrt3}\,}$
 a b c d e
2 points
 15. The equation ${x^2+(2m+4)x+m-1=0}$ (with the unknown $x$) a) has two different real roots, if and only if ${m=-1\llor m=0\llor m=1}$ b) has two different real roots, if and only if ${m=0}$ c) has two different real roots, if and only if ${m\in\langle -1,\,1\rangle }$ d) has two different real roots for all ${m\in\R}$ e) has no real root for all ${m\in\R}$