B

1003230205

Level: 
B
Which of the statements A, B, C given bellow are correct? \[ \begin{array}{l} \text{A: } \left(\frac{2x-1}{2-x}\right)'=\frac{5-4x}{(2-x)^2},\ x\neq2 \\ \text{B: } \left(\frac{\mathrm{e}^x-1}{x}\right)'=\frac{\mathrm{e}^x(x-1)-1}{x^2},\ x\neq0 \\ \text{C: } \left(\frac{\cos⁡ x}{1-\sin ⁡x}\right)'=\frac1{1-\sin ⁡x},\ x\neq\frac{\pi}2+2k\pi,\ k\in\mathbb{Z} \end{array}\] The only correct statements are:
C
A, C
A, B
B
A
B, C

1003230204

Level: 
B
Which of the statements A, B, C given bellow are correct? \[ \begin{array}{l} \text{A: }\left(\frac1{x^3}\cdot\cos ⁡x\right)' =-\frac{\cos x+\sin ⁡x}{x^4},\ x\in\mathbb{R}\setminus\{0\} \\ \text{B: }\bigl(\left(1-x^3\right)\cdot\ln x \bigr)'=-3x^2\ln x+\frac1x - x^2,\ x\in\mathbb{R}^+ \\ \text{C: } \left(5^x\cdot\sqrt[5]x\right)'=5^{x-1}\sqrt[5]x\left(5\ln⁡5+\frac1x\right),\ x\in\mathbb{R}\setminus\{0\} \end{array} \] The only correct statements are:
B, C
A, C
A, B
B
A
C

1003230203

Level: 
B
Given the function \( f(x)=\frac{\sqrt x}{\ln ⁡x} \), find the set of all \( x \), \( x\in\mathbb{R} \), for which \( f'(x)=0 \).
\( \left\{ \mathrm{e}^2 \right\} \)
\( \{ \mathrm{e} \} \)
\( \left\{ \sqrt{\mathrm{e}} \right\} \)
\( \left\{ \frac1{\mathrm{e}};\mathrm{e} \right\} \)
\( \{ 2 \} \)
\( \left\{ 1;\mathrm{e}^2 \right\} \)

1003189005

Level: 
B
We are given a straight line \( p \) by parametric equations \begin{align*} x&=1+t, \\ y&= 1+2t, \\ z&= 4-t;\ t\in\mathbb{R}. \end{align*} Find the parametric equations of the line \( p' \) that is an orthogonal projection of the line \( p \) into the coordinate \(xy\)-plane .
$\begin{aligned} p'\colon x&=5+s, \\ y&= 9+2s, \\ z&= 0;\ s\in\mathbb{R} \end{aligned}$
$\begin{aligned} p'\colon x&=5+s, \\ y&= 9-2s, \\ z&=0;\ s\in\mathbb{R} \end{aligned}$
$\begin{aligned} p'\colon x&=1+s, \\ y&=1+2s, \\ z&= 4;\ s\in\mathbb{R} \end{aligned}$
$\begin{aligned} p'\colon x&=5+2s, \\ y&=9+s, \\ z&= 0;\ s\in\mathbb{R} \end{aligned}$

1103189004

Level: 
B
We are given the point \( A=[2;-1;-4] \) and planes \( \rho \) by \( x-y+3z-5=0 \) and \( \sigma \) by \( 2x-y-z-8=0 \). Find the general form of the equation of the plane \( \alpha \) which passes through the point \( A \) and is perpendicular to both planes (see the picture).
\( \alpha\colon 4x+7y+z+3=0 \)
\( \alpha\colon -2x+5y-3z-3=0 \)
\( \alpha\colon 4x-7y+z+3=0 \)
\( \alpha\colon 2x-5y+3z+3=0 \)

1103189003

Level: 
B
Find the general form of the equation of the plane \( \beta \) that passes through the straight line \( p \) given by parametric equations \begin{align*} x&=1+2t, \\ y&=-2t, \\ z&=1+t;\ t\in\mathbb{R}, \end{align*} and is perpendicular to the plane \( \alpha \) given by \( x+3y-z-7=0 \) (see the picture).
\( \beta\colon x-3y-8z+7=0 \)
\( \beta\colon 2x-2y+z-3=0 \)
\( \beta\colon x-3y-8z-7=0 \)
\( \beta\colon 2x-2y+z+3=0 \)

1103189002

Level: 
B
Find the general form of the equation of the plane \( \beta \) that passes through the points \( M=[-1;1;-3] \) and \( N=[0;2;-1] \) and is perpendicular to the plane \( \alpha \) given by \( 3x-y+2=0 \) (see the picture).
\( \beta\colon x+3y-2z-8=0 \)
\( \beta\colon x+3z+10=0 \)
\( \beta\colon x+3z+3=0 \)
\( \beta\colon x+3y-2z+8=0 \)

1103189001

Level: 
B
Find the general form of the equation of the plane \( \alpha \) that is perpendicular to the straight line \( p \) given by: \begin{align*} x&=7+t, \\ y&=2t, \\ z&=4-t;\ t\in\mathbb{R}, \end{align*} and passes through the point \( A=[1;0;4] \). Consequently, find the coordinates of the point \( B \) which is the point of intersection of \( p \) and \( \alpha \) (see the picture).
\( \alpha\colon x+2y-z+3=0;\ B=[6;-2;5] \)
\( \alpha\colon x+2y-z-3;\ B=[6;-2;5] \)
\( \alpha\colon x+2y-z-3=0;\ B=[8;2;3] \)
\( \alpha\colon x+2y-z+3=0;\ B=[8;2;3] \)