B | math4u.vsb.cz

2010008601

Level:

Choose the equation which is, on the interval \( \left[ \frac12, 2 \right] \), equivalent to \[ |2x-1|+2|x+5|=1-|2-x|. \]

\( (2x-1)+2(x+5)=1-(2-x) \)

\( (2x-1)+2(x+5)=1+(2-x) \)

\( (2x-1)-2(x+5)=1+(2-x) \)

\( -(2x-1)+2(x+5)=1+(2-x) \)

2010008504

Level:

Give the number which satisfies the equation \( |x+2|-1=x+3\).

\(-3\)

\( -2 \)

\( 0 \)

\( 1 \)

2010008501

Level:

Find all \( t\in\mathbb{R} \), such that the following equation with the variable \( x \) has exactly two solutions. \[ |x+2|+3=t-1 \]

\( t\in(4,\infty) \)

\( t\in(-2,\infty) \)

\( t\in\{4\} \)

\( \mathbb{R}\)

2010008404

Level:

Identify the set of the values of the real parameter \(t\) for which the following equation has no solution in \(\mathbb{R}\). \[ x^{2} + (t + 2)x + 1 = 0 \]

\((-4,0)\)

\((-\infty ,-4)\cup (0,\infty )\)

\((-\infty ,-4)\)

\((0,\infty )\)

2010008403

Level:

Identify the set of the values of the real parameter \(d\) for which the following equation has two different real roots. \[ x^{2} - 2dx + 2d^{2} - 9 = 0 \]

\( (-3,3)\)

\((-\infty ,-3)\cup (3,\infty )\)

\((-\infty,-3 )\)

\( (3,\infty )\)

2010008212

Level:

Identify the solution of the equation. \[ 4^{\sqrt{x}}+5\cdot4^{\frac{\sqrt{x}-1}{2}}-9=0 \]

\( x=1\)

\( x_1=\frac12\), \( x_2=2\)

\( x_1=\frac14\), \( x_2=4\)

\( x_1=-1\), \( x_2=1\)

2010008211

Level:

Find the sum of all the solutions of the equation. \[ 3^x \cdot \left(\frac13\right)^{x^3-2x^2}=9\]

\( 2\)

\( 1\)

\( 0\)

\( -1\)

2010008210

Level:

Find the sum of all the solutions of the equation. \[ 2\cdot 2^{x^3-x^2}=\left(\frac14\right)^{-\frac12 x}\]

\( 0\)

\( 1\)

\( 2\)

\( -1\)

2010008201

Level:

How many solutions does the following equation have? \[ 49^x+4\cdot7^x=5 \]

Exactly one solution

Exactly two solutions

No solutions

Infinitely many solutions

2010008110

Level:

Evaluate the following integral on the interval \(\left(0,\frac{\pi}{2}\right)\). \[ \int \left(\cos 2x+ \frac{1}{\sin^2 2x}-\frac{1}{2x} \right) \mathrm{d}x \]

\( \frac12\left(\sin 2x- \mathrm{cotg}\, 2x-\ln x \right) +c,~c \in \mathbb{R}\)

\( \frac12\left( \sin 2x- \mathrm{cotg }\, 2x -\ln 2x\right)+c,~c \in \mathbb{R}\)

\( \sin 2x- \mathrm{cotg }\, 2x - \ln 2x +c,~c \in \mathbb{R}\)

\( \sin 2x+ \mathrm{cotg }\, 2x +\ln 2x +c,~c \in \mathbb{R}\)

B

2010008601

2010008504

2010008501

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2010008110

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