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.
\(\infty \)
\(4\pi \)
\(\frac{2} {5}\pi \)
\(\frac{1} {3}\pi \)

9000063105

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

9000063110

Level: 
B
Differentiate the following function. \[ f(x) =\sin x(1 +\mathop{\mathrm{tg}}\nolimits x) \]
\(f'(x) =\cos x +\sin x + \frac{\sin x} {\cos ^{2}x},\ x\in \mathbb{R}\setminus\{\frac{\pi}{2}+k\pi; k\in \mathbb{Z}\}\)
\(f'(x) =\cos x +\sin x,\ x\in \mathbb{R}\setminus\{\frac{\pi}{2}+k\pi; k\in \mathbb{Z}\}\)
\(f'(x) = \frac{\sin x} {\cos ^{2}x},\ x\in \mathbb{R}\setminus\{\frac{\pi}{2}+k\pi; k\in \mathbb{Z}\}\)
\(f'(x) =\cos x + 2\sin x,\ x\in \mathbb{R}\setminus\{\frac{\pi}{2}+k\pi; k\in \mathbb{Z}\}\)

9000046403

Level: 
B
Consider an isosceles triangle, i.e. the triangle with two sides of equal length. The length of the third side is \(4\, \mathrm{cm}\). One of the interior angles is \(120^{\circ }\). Find the area of this triangle.
\(\frac{4\sqrt{3}} {3} \, \mathrm{cm}^{2}\)
\(4\sqrt{3}\, \mathrm{cm}^{2}\)
\(\frac{8\sqrt{3}} {3} \, \mathrm{cm}^{2}\)

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 \( 2x = z\)
\(\sin x = \frac{\mathop{\mathrm{tg}}\nolimits x} {2} \)
\(\cos ^{2}x -\sin ^{2}x =\mathop{\mathrm{tg}}\nolimits x\)

9000046404

Level: 
B
The parallelogram has sides of the length \(5\, \mathrm{cm}\) and \(4\, \mathrm{cm}\) (see the picture). The area of this parallelogram is \(S = 10\sqrt{2}\, \mathrm{cm}^{2}\). Find the measure of the smaller of the interior angles.
\(45^{\circ }\)
\(30^{\circ }\)
\(60^{\circ }\)

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 \( \cos x = z\)
\(2\cos ^{2}x = \sqrt{1 -\sin ^{2 } x} + 1\)