1003086110
Level:
B
Which of the following intervals contains four solutions of the equation \( \cos x = 0.5 \)?
\( [\pi,5\pi] \)
\( [-2\pi,\pi] \)
\( [-4\pi,-2\pi] \)
\( [5\pi,10\pi] \)
B
1003086109
Level:
B
The solution set of the equation \( \mathrm{tg}\,x + \mathrm{cotg}\,x = 2 \) for \( x\in[-2\pi,2\pi] \) is:
\( \left\{-\frac{7\pi}4,-\frac{3\pi}4,\frac{\pi}4,\frac{5\pi}4 \right\} \)
\( \left\{-\frac{7\pi}4,\frac{\pi}4,\right\} \)
\( \left\{-\frac{5\pi}4,-\frac{\pi}4,\frac{3\pi}4,\frac{7\pi}4 \right\} \)
\( \left\{-\frac{3\pi}4,\frac{\pi}4\right\} \)
1003086106
Level:
B
The solution set of the equation \( \sin 2x = \cos 3x \cdot \sin 2x \) for \( x\in\left[0^{\circ},180^{\circ}\right] \) is:
\( \left\{0^{\circ},90^{\circ},120^{\circ},180^{\circ}\right\} \)
\( \left\{90^{\circ},120^{\circ},180^{\circ}\right\} \)
\( \left\{90^{\circ},180^{\circ}\right\} \)
\( \left\{0^{\circ},90^{\circ}\right\} \)
1003086102
Level:
B
How many solutions does the equation \( \mathrm{cotg}\left(2x - \frac{\pi}4\right) = \frac1{\sqrt3} \) have in the interval \( [0,2\pi] \)?
\( 4 \)
\( 8 \)
\( 2 \)
\( 0 \)
1003086101
Level:
B
Which of the following equations has exactly four solutions in the interval \( [0,\pi] \)?
\( \mathrm{tg}\,4x = 2 \)
\( \mathrm{tg}\,2x = 4 \)
\( \mathrm{tg}\,\frac x4 = 2 \)
\( \mathrm{tg}\,\frac x4 = 4 \)
1003118805
Level:
B
Evaluate the integral \( \int\limits_{-5}^5 f(x)\,\mathrm{d}x \), where \( f(x)=x^2+2 \) for \( x\in[-5,1] \) and \( f(x)=3 \) for \( x\in[ 1,5] \).
\( 66 \)
\( \frac73 \)
\( \frac{25}3 \)
\( 42 \)
1003118804
Level:
B
Evaluate the integral \( \int\limits_{-3}^2 f(x)\,\mathrm{d}x \), where \( f(x)=x^{-4} \) for \( x\in\left[-3,-\frac12\right] \) and \( f(x)=12.8-6.4x \) for \( x\in\left[ -\frac12,2\right]\). (rounded to \( 2 \) decimal places)
\( 22.65 \)
\( 43.\overline{8} \)
\( 44.\overline{1} \)
\( 29.7\overline{1} \)
1103118803
Level:
B
The picture shows a graph of a power function \( f(x) \). Find the value of the definite integral of this function between \( -1 \) and \( -\frac12 \).
\( 1 \)
\( 3 \)
\( -1 \)
\( 2 \)
1003118802
Level:
B
Find positive real numbers \( a \) and \( b \), \( a \), \( b>0 \), such that \( b-a=6 \) and \( \int\limits_a^b(4x-6)\,\mathrm{d}x=60 \).
\( a=1 \), \( b=7 \)
\( a=7 \), \( b=1 \)
\( a=24 \), \( b=30 \)
\( a=1.5 \), \( b=7.5 \)
1003118801
Level:
B
Which of the following formulas is not equal to \( \int\limits_4^8\frac{3x+1}{x^2-x-6}\,\mathrm{d}x \)?
\( \int\limits_4^8\frac2{x-3}\,\mathrm{d}x -\int\limits_4^8\frac1{x+2}\,\mathrm{d}x \)
\( \int\limits_4^8\frac2{x-3}\,\mathrm{d}x + \int\limits_4^8\frac1{x+2}\,\mathrm{d}x \)
\( \int\limits_4^6\frac{3x+1}{x^2-x-6}\,\mathrm{d}x + \int\limits_6^8\frac{3x+1}{x^2-6-x}\,\mathrm{d}x \)
\( \int\limits_8^4\frac{-1-3x}{x^2-x-6}\,\mathrm{d}x \)