9000039105
Level:
B
Find the quadratic equation with real coefficients such that one of the solutions is the complex
number \(x_{1} = 1 + 2\mathrm{i}\).
\(x^{2} - 2x + 5 = 0\)
\(x^{2} - 2x + 3 = 0\)
\(x^{2} + 2x + 5 = 0\)
\(x^{2} + 2x - 3 = 0\)
B
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.)
\(19.31\, \mathrm{cm}^{2}\)
\(3.31\, \mathrm{cm}^{2}\)
\(20.88\, \mathrm{cm}^{2}\)
9000046401
Level:
B
Consider the rectangle \(ABCD\)
with length and height \(|AB| = 6\, \mathrm{cm}\)
and \(|BC| = 2\sqrt{3}\, \mathrm{cm}\), respectively.
Find the measure of the angle \( CAB\).
\(30^{\circ }\)
\(45^{\circ }\)
\(60^{\circ }\)
9000039102
Level:
B
Identify a complex number with absolute value different from
\(1\).
\(1 + \mathrm{i}\)
\(\frac{1}
{2} -\frac{\sqrt{3}}
{2} \mathrm{i}\)
\(-\frac{3}
{5} -\frac{4}
{5}\mathrm{i}\)
\(-\mathrm{i}\)
9000046402
Level:
B
Consider the rectangle \(ABCD\)
with length and height \(|AB| = 6\, \mathrm{cm}\)
and \(|BC| = 2\sqrt{3}\, \mathrm{cm}\). Let
\(S\) be the intersection of
diagonals. Find the angle \(\measuredangle ASB\).
\(120^{\circ }\)
\(60^{\circ }\)
\(90^{\circ }\)
9000046501
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.
\[
\sin x\cdot \cos x = 0
\]
\(\sin 2x = 0\)
\(\cos 2x = 0\)
substitution \( \sin x = z\)
\(\sin ^{2}x\cdot \cos ^{2}x = 0\)
9000045706
Level:
B
Given a regular pentagon with the side
\(a\), find the
radius \(r\) of
the circle circumscribed to this pentagon.
\(r = \frac{a}
{2\cdot \cos 54^{\circ }}\)
\(r = \frac{2a}
{\cos 72^{\circ }}\)
\(r = \frac{2a}
{\cos 54^{\circ }}\)
\(r = \frac{a}
{2\cdot \cos 72^{\circ }}\)
9000039301
Level:
B
Given formula for an acceleration of the motion in the form
\[
a = \frac{v - v_{0}}
{t} ,
\]
find the initial velocity \(v_{0}\).
\(v_{0} = v - at\)
\(v_{0} = vat\)
\(v_{0} = v + at\)
\(v_{0} = at - v\)
9000045707
Level:
B
Given a regular pentagon with the side
\(a\), find the
radius \(\rho \) of
the circle inscribed to this pentagon.
\(\rho = \frac{a}
{2} \cdot \mathop{\mathrm{tg}}\nolimits 54^{\circ }\)
\(\rho = \frac{2a}
{\mathop{\mathrm{tg}}\nolimits 54^{\circ }}\)
\(\rho = \frac{a}
{2\cdot \mathop{\mathrm{tg}}\nolimits 54^{\circ }}\)
\(\rho = 2a\cdot \mathop{\mathrm{tg}}\nolimits 54^{\circ }\)
9000039302
Level:
B
Find \(N\),
the number of the turns, as a function of the other variables in the formula for the
magnetic induction of a solenoid.
\[
B =\mu \frac{NI}
{l}
\]
\(N = \frac{Bl}
{\mu I} \)
\(N = \frac{Bl\mu }
{I} \)
\(N = B -\mu \frac{I}
{l} \)
\(N = \frac{Bl}
{\mu } - I\)