9000062409
Level:
A
Find the first derivative of the function
\(f(x) = x^{2} + x - 6\) at each of the
\(x\)-intercepts.
\(- 5;\ 5\)
\(- 3;\ 7\)
\(- 7;\ 6\)
\(- 9;\ 6\)
A
9000046407
Level:
A
Find the angle between the side of a cube and the cube’s space diagonal. Round the result to two decimal places.
\(35.26^{\circ }\)
\(45^{\circ }\)
\(54.76^{\circ }\)
9000046502
Level:
A
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.
\[
\cos 3x = 0.5
\]
substitution \( 3x = z\)
substitution \( \cos x = z\)
\(\cos ^{3}x -\sin ^{3}x = 0.5\)
\(\cos x = \frac{0.5}
{3} \)
9000045705
Level:
A
Find the formula for the radius \(r\)
of the circle circumscribed to the right triangle.
\(r = \frac{a}
{2\cdot \sin \alpha } \)
\(r = \frac{2a}
{\sin \alpha } \)
\(r = \frac{a}
{\sin 2\alpha } \)
\(r = \frac{a}
{\sin \alpha } \)
9000039107
Level:
A
Find the sum of the reciprocal values of the solutions of
\(5x^{2} - 2x + 1 = 0\).
\(2\)
\(\frac{2}
{5}\)
\(\frac{2}
{5} + \frac{4}
{5}\mathrm{i}\)
\(-\frac{2}
{5}\)
9000046503
Level:
A
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.
\[
\mathop{\mathrm{tg}}\nolimits \left (-x + \frac{\pi }
{6}\right ) = \sqrt{3}
\]
substitution \( - x + \frac{\pi } {6} = z\)
\(\mathop{\mathrm{tg}}\nolimits (-x) = \sqrt{3} - \frac{\pi } {6}\)
\(\mathop{\mathrm{tg}}\nolimits ^{2}\left (-x + \frac{\pi } {6}\right ) = 3\)
\(\frac{\sin \left (-x+ \frac{\pi }{6} \right )}
{\cos \left (-x+ \frac{\pi }{6} \right )} = \sqrt{3}\)
9000046504
Level:
A
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.
\[
\cos \left (x + \frac{\pi }
{3}\right ) = \frac{\sqrt{3}}
{2}
\]
substitution \( x + \frac{\pi } {3} = z\)
\(\cos ^{2}\left (x + \frac{\pi } {3}\right ) = \frac{3}
{4}\)
substitution \( \frac{\sqrt{3}}
{2} = z\)
\(\cos x\cdot \cos \frac{\pi }{3} -\sin x\cdot \sin \frac{\pi }{3} = \frac{\sqrt{3}}
{2} \)
9000046510
Level:
A
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.
\[
2\sin ^{2}x -\sin x - 1 = 0
\]
substitution \( \sin x = z\)
substitution \( \sin ^{2}x = z\)
\(2\sin ^{2}x -\sin x = 1\)
\(2\sin ^{2}x -\sin x =\sin ^{2}x +\cos ^{2}x\)
9000045709
Level:
A
Let \(\omega \) be the
angle between the solid diagonal of a box and the base of this box. Find the expression which
allows to find \(\omega \).
\(\mathop{\mathrm{tg}}\nolimits \omega = \frac{\sqrt{2}}
{2} \)
\(\cos \omega = \frac{\sqrt{2}}
{2} \)
\(\sin \omega = \frac{\sqrt{2}}
{2} \)
\(\mathop{\mathrm{cotg}}\nolimits \omega = \frac{\sqrt{2}}
{2} \)
9000037504
Level:
A
Given complex numbers
\[
a = 5 + 2\mathrm{i},\quad b = 3 -\mathrm{i},\quad c = \mathrm{i}\text{,}
\]
find the product \(abc\).
\(- 1 + 17\mathrm{i}\)
\(1 - 17\mathrm{i}\)
\(- 1 - 17\mathrm{i}\)
\(1 + 17\mathrm{i}\)