1003021108
Level:
B
Evaluate the following limit.
\[ \lim_{x\to\frac{\pi}4}\frac{\cos x-\sin x}{1-\mathrm{cotg}\,x} \]
\( -\frac{\sqrt2}{2} \)
\( \frac{\sqrt2}{2} \)
\( 0 \)
\( -\sqrt2 \)
B
1003021107
Level:
B
Evaluate the following limit.
\[ \lim_{x\to\pi}\frac{\mathrm{tg}\,x}{\sin2x} \]
\( \frac12 \)
\( -\frac12 \)
\( 0 \)
\( -2 \)
1003021102
Level:
B
Evaluate the following limit.
\[ \lim_{x\to\frac{\pi}4}\frac{1+\sin2x}{1-\cos4x} \]
\( 1 \)
\( 0 \)
\( 2 \)
\( \frac12 \)
1003021101
Level:
B
Evaluate the following limit.
\[ \lim_{x\to -2}\frac{2-\sqrt{x+6}}{x-2} \]
\( 0 \)
\( \frac14 \)
\( -\frac14 \)
\( 1 \)
1003030605
Level:
B
Let \( \overrightarrow{a}=(3;-5) \) and \( \overrightarrow{b}=(6;-10) \). Find all the vectors \( \overrightarrow{c} \) such that
\[ \overrightarrow{a}\cdot\overrightarrow{c}=11\ \text{ and }\ \overrightarrow{b}\cdot\overrightarrow{c}=22\text{ .} \]
\( \overrightarrow{c}=(2+5k;-1+3k);\ k\in\mathbb{R} \)
\( \overrightarrow{c}_1=(7;2);\ \overrightarrow{c}_2=(-7;-2) \)
\( \overrightarrow{c}=(2k;-k);\ k\in\mathbb{R} \)
\( \overrightarrow{c}_1=(2;-1);\ \overrightarrow{c}_2=(-2;1) \)
1003030604
Level:
B
Let \( \overrightarrow{a}=(2;- 3) \) and \( \overrightarrow{b}=(3;-2) \). Find all the vectors \( \overrightarrow{c} \) such that
\[ \overrightarrow{a}\cdot\overrightarrow{c}=8\ \text{ and }\ \overrightarrow{b}\cdot\overrightarrow{c}=27. \]
\( \overrightarrow{c}=(13;6) \)
\( \overrightarrow{c_1}=(13;6);\ \overrightarrow{c_2}=(-13;-6) \)
\( \overrightarrow{c}=(13k;6k);\ k\in\mathbb{R} \)
\( \overrightarrow{c}=(-13;-6) \)
1003030603
Level:
B
Let \( \overrightarrow{v}=(12;5) \). Find all the vectors \( \overrightarrow{u} \) that are perpendicular to the vector \( \overrightarrow{v} \) and have the length of \( 26 \).
\( \overrightarrow{u_1} =(10;-24);\ \overrightarrow{u_2}=(-10; 24) \)
\( \overrightarrow{u}=(10;-24) \)
\( \overrightarrow{u_1}=\frac12 (5;-12);\ \overrightarrow{u_2}=\frac12 (-5; 12) \)
\( \overrightarrow{u_1}=26\cdot(5;-12);\ \overrightarrow{u_2}=26\cdot(-5; 12) \)
1103030602
Level:
B
Let \( ABCDV \) be a right pyramid with a square base, such that its opposite edges contain a right angle (see the picture).
Specify the missing coordinate of the apex \( V \).
\( m=2\sqrt2 \)
\( m=-2\sqrt2 \)
\( m=4\sqrt2 \)
\( m=\sqrt2 \)
1103030601
Level:
B
In the cube \( ABCDEFGH \) find the angle \( \varphi \) between the vectors \( \overrightarrow{b}=\overrightarrow{EB} \) and \( \overrightarrow{a}=\overrightarrow{AK} \), where \( K \) is the midpoint of \( HG \). Round \( \varphi \) to the nearest degree.
Help: Choose the appropriate coordinate system.
\( \varphi\doteq 104^{\circ} \)
\( \varphi\doteq 76^{\circ} \)
\( \varphi\doteq 100^{\circ} \)
\( \varphi\doteq 80^{\circ} \)
1003025201
Level:
B
Two hunters, Adam and Boris, competed in target shooting. Adam hit the target points \( \{10;10;9;8;7\}\), and Boris \( \{10;10;9;9;6\} \). Who is the winner? In the case of the same sum of gained points the shooting accuracy is decisive. Which of the following statements is true, if the accuracy is quantified by the variance of the points? (The variance is rounded to two decimal places.)
Adam won with the variance of \( 1{.}36\,\mathrm{points}^2 \).
Adam won with the variance of \( 1{.}17\,\mathrm{points}^2 \).
Boris won with the variance of \( 2{.}16\,\mathrm{points}^2 \).
Adam won with the variance of \( 1{.}36\,\mathrm{points} \).
Adam won with the variance of \( 1{.}17\,\mathrm{points} \).
Boris won with the variance of \( 2{.}16\,\mathrm{points} \).