B

9000062909

Level: 
B
Consider the square of the side \(4\, \mathrm{cm}\). The second square is inscribed into this first square by joining the centers of all sides. In a similar way, the third square is inscribed into the second square by joining the centers of the sides of the second square and this process continues up to infinity. Find the sum of the perimeters of all squares.
\(32 + 16\sqrt{2}\)
\(32 - 16\sqrt{2}\)
\(32\)
\(\infty \)

9000063105

Level: 
B
Differentiate the following function. \[ f(x) = \frac{\sqrt{x} - 1} {\sqrt{x} + 1} \]
\(f'(x) = \frac{1} {\sqrt{x}(\sqrt{x}+1)^{2}} ,\ x > 0\)
\(f'(x) = \frac{\sqrt{x}} {(\sqrt{x}+1)^{2}} ,\ x > 0\)
\(f'(x) = \frac{2} {x(\sqrt{x}+1)^{2}} ,\ x > 0\)
\(f'(x) = \frac{1} {(\sqrt{x}+1)^{2}} ,\ x > 0\)

9000062910

Level: 
B
Consider the square of the side \(4\, \mathrm{cm}\). The second square is inscribed into this first square by joining the centers of all sides. In a similar way, the third square is inscribed into the second square by joining the centers of the sides of the second square and this process continues up to infinity. Find the sum of the squares of all squares.
\(32\)
\(40\)
\(\frac{32} {3} \)
\(\infty \)

9000062903

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

9000038910

Level: 
B
Consider the function \(f\colon y =\mathop{\mathrm{cotg}}\nolimits x\). In the following list identify the function which has the same graph as the graph of the function \(f\).
\(k\colon y = -\mathop{\mathrm{tg}}\nolimits \left (x + \frac{\pi } {2}\right )\)
\(g\colon y = -\mathop{\mathrm{tg}}\nolimits x\)
\(b\colon y =\mathop{\mathrm{tg}}\nolimits \left (x + \frac{\pi } {2}\right )\)
\(h\colon y =\mathop{\mathrm{tg}}\nolimits \left (x - \frac{\pi } {2}\right )\)
\(m\colon y = -\mathop{\mathrm{tg}}\nolimits x - \frac{\pi } {2}\)

9000046506

Level: 
B
Identify the optimal first step convenient to solve the following trigonometric equation. Do not consider the step which is possible but does not help to solve the equation. \[ \sin 2x =\mathop{\mathrm{tg}}\nolimits x \]
\(2\sin x\cdot \cos x = \frac{\sin x} {\cos x}\)
substitution \( 2x = z\)
\(\sin x = \frac{\mathop{\mathrm{tg}}\nolimits x} {2} \)
\(\cos ^{2}x -\sin ^{2}x =\mathop{\mathrm{tg}}\nolimits x\)

9000045702

Level: 
B
Given a right triangle \(ABC\) (see the picture). Find the valid relation between the angle \(\alpha \) and the sides of the triangle.
\(\mathop{\mathrm{tg}}\nolimits \alpha = \frac{a} {c}\)
\(\sin \alpha = \frac{a} {c}\)
\(\cos \alpha = \frac{b} {a}\)
\(\mathop{\mathrm{cotg}}\nolimits \alpha = \frac{b} {a}\)

9000046509

Level: 
B
Identify the optimal first step convenient to solve the following trigonometric equation. Do not consider the step which is possible but does not help to solve the equation. \[ 2\cos ^{2}x =\sin x + 1 \]
\(2 - 2\sin ^{2}x =\sin x + 1\)
substitution \( \sin x + 1 = z\)
substitution \( \cos x = z\)
\(2\cos ^{2}x = \sqrt{1 -\sin ^{2 } x} + 1\)

9000045703

Level: 
B
Given a right triangle \(ABC\) with the right angle at $C$ and an altitude $v$ (see the picture). Find the valid relation between the angle \(\alpha \) and the lengths in the triangle.
\(\sin \alpha = \frac{v} {b}\)
\(\sin \alpha = \frac{v} {c}\)
\(\sin \alpha = \frac{a} {v}\)
\(\sin \alpha = \frac{c} {a}\)