B

9000062907

Level: 
B
A quarter circle is an arc formed by one quarter of the full circle. An infinite spiral is built from quarter circles with an increasing radius. The radius of the first quarter circle is \(1\, \mathrm{cm}\). The radius of each quarter circle in the spiral is bigger by one half than the radius of the previous quarter circle. Find the total length of the spiral.
\(\infty \)
\(4\pi \)
\(\frac{2} {5}\pi \)
\(\frac{1} {3}\pi \)

9000062904

Level: 
B
A semicircle is a half of the full circle. An infinite spiral is built from semicircles with a decreasing radius. The radius of the first semicircle is \(3\, \mathrm{cm}\). The radius of each semicircle in the spiral is smaller by one third of the radius of the previous semicircle. Find the total length of the spiral.
\(9\pi \)
\(9\)
\(\frac{9} {5}\pi \)
\(\infty \)

9000046609

Level: 
B
Identify an inequality which is true for every \(x\) from the interval \(\left ( \frac{\pi }{4}, \frac{3\pi } {4}\right )\).
\(\sin x\geq \frac{\sqrt{2}} {2} \)
\(\mathop{\mathrm{tg}}\nolimits x > 1\)
\(\cos x > 0\)
\(\mathop{\mathrm{cotg}}\nolimits x\geq \frac{\sqrt{3}} {3} \)

9000046610

Level: 
B
Identify an inequality which is true for every \(x\) from the interval \(\left (\frac{5\pi } {6}, \frac{3\pi } {2}\right )\).
\(\cos x < \frac{1} {2}\)
\(\mathop{\mathrm{tg}}\nolimits x < 0\)
\(\sin x\geq -\frac{\sqrt{2}} {2} \)
\(\mathop{\mathrm{cotg}}\nolimits x < 1\)

9000063101

Level: 
B
Differentiate the following function. \[ f(x) = \frac{x^{2} - 1} {x^{2} + 1} \]
\(f'(x) = \frac{4x} {(x^{2}+1)^{2}} ,\ x\in \mathbb{R}\)
\(f'(x) = \frac{-4x} {x^{2}+1},\ x\in \mathbb{R}\)
\(f'(x) = \frac{4x^{3}} {(x^{2}+1)^{2}} ,\ x\in \mathbb{R}\)
\(f'(x) = \frac{4x} {x^{2}+1},\ x\in \mathbb{R}\)