9000091204
Level:
A
Consider the circle \(k\colon x^{2} + 4x + y^{2} - 8y + 11 = 0\).
Find the radius of the circle.
\(3\)
\(1\)
\(2\)
\(4\)
A
9000086702
Level:
A
Identify the optimal substitution which simplifies the following equation as much as
possible.
\[
\sin ^{2}2x -\sin 2x = 0
\]
\(\sin 2x = t\)
\(x = t\)
Equation is not convenient for a substitution.
\(\sin x = t\)
9000086703
Level:
A
Identify the optimal substitution which simplifies the following equation as much as
possible.
\[
5\mathop{\mathrm{cotg}}\nolimits (30^{\circ }- 4y) = 0
\]
\(30^{\circ }- 4y = t\)
\(\mathop{\mathrm{cotg}}\nolimits (30^{\circ }- 4y) = t\)
\(\mathop{\mathrm{cotg}}\nolimits (30^{\circ }- 4y) = 0\)
\(4y = 0\)
9000086704
Level:
A
Identify the optimal substitution which simplifies the following equation as much as
possible.
\[
\mathop{\mathrm{tg}}\nolimits ^{2}v -\mathop{\mathrm{cotg}}\nolimits ^{-1}v = 2
\]
\(\mathop{\mathrm{tg}}\nolimits v = t\)
\(v^{-1} = t\)
\(\mathop{\mathrm{cotg}}\nolimits v = t\)
Equation is not convenient for a substitution.
9000086705
Level:
A
Identify the optimal substitution which simplifies the following equation as much as
possible.
\[
2\cos (3x + 33^{\circ }) = -\sqrt{2}
\]
\(3x + 33^{\circ } = t\)
\(\sin 3x = t\)
\(3x = t\)
\(3x + 33^{\circ } = -\sqrt{2}\)
9000085601
Level:
A
Evaluate the following numbers and round to the nearest tens.
\[
10^{2} + 11^{2} + 12^{2} + 13^{2} + 14^{2}
\]
\(730\)
\(720\)
\(740\)
\(750\)
9000086706
Level:
A
Identify the equation which arises from the following equation using an optimal
substitution.
\[
2\cos ^{2}x - 7\cos x + 3 = 0
\]
\(2t^{2} - 7t + 3 = 0\)
\(2t^{2} = \frac{3}
{7}\)
\(\cos t = 0\)
\(7\cos t = 3\)
9000085607
Level:
A
Given numbers \(8\: 175\)
and \(3\: 926\),
find the difference between the sum of these numbers rounded to the nearest tens and the sum of these numbers rounded to the nearest hundreds.
\(10\)
\(20\)
\(100\)
\(0\)
9000086707
Level:
A
Identify the equation which arises from the following equation using an optimal
substitution.
\[
\mathop{\mathrm{tg}}\nolimits ^{2}y - 2\mathop{\mathrm{tg}}\nolimits y = 3
\]
\(t^{2} - 2t - 3 = 0\)
\(\mathop{\mathrm{tg}}\nolimits t = \frac{3}
{2}\)
\(t^{2} = \frac{3}
{2}\)
\(\mathop{\mathrm{tg}}\nolimits t = 3\)
9000085608
Level:
A
Given numbers \(456\: 138\)
and \(321\: 814\),
find the difference between the sum of these numbers rounded to the nearest tens and the sum of these numbers rounded to the nearest hundreds.
\(50\)
\(100\)
\(1\: 000\)
\(0\)