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Square Pyramid -- Angles

Submitted by ladislav.foltyn on Sat, 06/01/2019 - 12:23
Question: 
\vspace{-2em} \begin{minipage}{0.55\linewidth} The base $ABCD$ of a square pyramid $ABCDV$ has an edge of length $a$, and the lateral face is an equilateral triangle (see the picture). Let $S$ be the midpoint of the base $ABCD$ and let $P$ be the midpoint of the edge $AV$. Find the angle between \end{minipage} \hfill \begin{minipage}{0.4\linewidth} \obrMsr[x=3cm,y=3cm,z=0.3cm]{-1}2{-1}2 { \footnotesize \pgfmathsetmacro{\cubex}{1} \pgfmathsetmacro{\cubey}{1} \pgfmathsetmacro{\cubez}{2} \coordinate (A) at (0,0,0); \coordinate (B) at (\cubex,0,0); \coordinate (C) at (\cubex.2,0,\cubez); \coordinate (D) at (0.2,0,\cubez); \coordinate (V) at (0.6,0.7,1); \coordinate (P) at ($(A)!0.5!(V)$); \draw[thick,dashed] (A) -- (D) node [yshift=4pt,xshift=-6pt]{$D$} -- (C) node [yshift=-5pt,xshift=5pt]{$C$}; \draw[dashed] (A) -- (C); \draw[dashed] (B) -- (D); \draw (0.6,0,1) node [below,xshift=-2pt,yshift=1pt]{$S$}; \draw[thick] (A) node [yshift=-5pt,xshift=-5pt]{$A$} -- (B) node [yshift=-6pt,xshift=3pt]{$B$} --(C); \draw[thick] (A) -- (V) node [above]{$V$}; \draw[thick] (B) -- (V); \draw[thick] (C) -- (V); \draw[thick,dashed] (D) -- (V); \draw[dashed] (0.6,0,1) -- (V); \begin{scope}[thick] \obrKrizek[2pt]{P}{above left}{P} \end{scope} } \end{minipage}

Relative Position of Points and Lines in a Plane

Submitted by ladislav.foltyn on Fri, 05/31/2019 - 12:44
Question: 
Suppose a line $p$ passes through the point $A=[2;3]$ and additionally has the property described in the first column of the table. In each row mark the line’s equation. \begin{align*} p_1\colon y&=-3x+9 & p_2\colon y&=x+1 & p_3\colon y&=2x-1 \\ p_4\colon y&=-2x+7 & p_5\colon y&=3 & p_6\colon x&=2 \end{align*}

Relative Position of Circles

Submitted by ladislav.foltyn on Sun, 05/05/2019 - 16:52
Question: 
\kern -2em Let $p$ be a line containing the points $S_4$, $S_1$, $S_3$, $S_2$ in this order, where $|S_4 S_1|= 1\,\mathrm{cm}$, $|S_1 S_3|= 1.5\,\mathrm{cm}$, and $|S_3 S_2|= 3.5\,\mathrm{cm}$. \smallskip Further, let $k_1$, $k_2$, $k_3$, $k_4$, and $k_5$ be circles with the centers $S_1$, $S_2$, $S_3$, $S_4$, and $S_2$ (again) and radii $r_1=3\,\mathrm{cm}$, $r_2=2\,\mathrm{cm}$, $r_3=1.5\,\mathrm{cm}$, $r_4=1.5\,\mathrm{cm}$, and $r_5=8\,\mathrm{cm}$ respectively. Determine the relative position of the circles. \kern 6em

Parallel Lines

Submitted by ladislav.foltyn on Thu, 05/02/2019 - 14:03
Question: 
In the table, mark a cell, if the two corresponding lines are parallel to each other. \begin{align*} a\colon&\, \left\{\begin{array}{ll} x=3+t\text{, } & \\ y=-3-t; & t\in\mathbb{R}\end{array}\right. & b\colon&\, y=3x-2 & c\colon&\, 4x-2y+5=0 \\ d\colon&\, y=\frac23x-7 & e\colon&\, 2x+y-6=0 & f\colon&\, \left\{\begin{array}{ll} x=3+4t\text{, } & \\ y=\phantom{3\,}-6t; & t\in\mathbb{R}\end{array}\right. \end{align*}

Limits of Function from its Graph II

Submitted by ladislav.foltyn on Wed, 05/01/2019 - 12:19
Question: 
\vspace{-1em} \begin{minipage}{0.3\linewidth} The figure shows a~graph of a~function $f$. Mark the true values of the~given limits of $f$. \end{minipage} \hfill \begin{minipage}{0.3\linewidth} \obrA \end{minipage} \hfill \begin{minipage}{0.35\linewidth} \footnotesize $$\def\arraystretch{2.2} f(x)=\left\{\begin{array}{ll}-\frac1{x-\frac12}-1; & x < 0 \\ \mathrm{cotg}\!\left(\pi\frac x2\right); & 0\leq x < 2 \\ -\frac1{x-2}-6x+20; & 2\leq x < 3.2 \\ \sin \left(2\pi(x+0.3)\right) & x \geq 3.2 \end{array}\right.$$ \end{minipage}