B

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

9000046406

Level: 
B
Find the area of the regular octagon of the perimeter \(16\, \mathrm{cm}\). Round the result to two decimal places. (The regular octagon is a polygon which has eight sides of equal length, see the picture. The perimeter of the octagon is the sum of the length of all eight sides.)
\(19.31\, \mathrm{cm}^{2}\)
\(3.31\, \mathrm{cm}^{2}\)
\(20.88\, \mathrm{cm}^{2}\)

9000039005

Level: 
B
Find all the values of \(x\) for which the following expression is positive. \[ \frac{2x - 3} {7 - 3x} \]
\(x\in \left (\frac{3} {2}; \frac{7} {3}\right )\)
\(x\in \left (\frac{3} {2};+\infty \right )\)
\(x\in \left (\frac{7} {3};+\infty \right )\)
\(x\in (0;+\infty )\)

9000046409

Level: 
B
The base of a pyramid is a square with the side of \(2\, \mathrm{cm}\). The height of the pyramid is \(4\, \mathrm{cm}\). Find the angle between the lateral side of the pyramid and the base. Round your result to two decimal places.
\(75.96^{\circ }\)
\(70.52^{\circ }\)
\(79.98^{\circ }\)

9000038902

Level: 
B
Consider the function \(f\colon y = A\cdot \sin (B\cdot x + C)\), with real nonzero parameters \(A\), \(B\) and \(C\). Which of the following operations makes the amplitude of the function five times bigger?
Decrease \(A\) by a factor \(5\).
Increase \(A\) by a factor \(5\).
Increase \(B\) by a factor \(5\).
Decrease \(B\) by a factor \(5\).
Increase \(C\) by a factor \(5\).
Decrease \(C\) by a factor \(5\).