B

9000039101

Level: 
B
Find the polar form of the complex number \(z=\frac{\mathrm{i}^{14}-1} {\mathrm{i}^{9}+1} \).
\(\sqrt{2}\left (\cos \frac{3\pi } {4} + \mathrm{i}\sin \frac{3\pi } {4}\right )\)
\(\sqrt{2}\left (\cos \frac{5\pi } {4} + \mathrm{i}\sin \frac{5\pi } {4}\right )\)
\(\sqrt{2}\left (\cos \frac{\pi }{4} + \mathrm{i}\sin \frac{\pi }{4}\right )\)
\(\sqrt{2}\left (\cos \frac{7\pi } {4} + \mathrm{i}\sin \frac{7\pi } {4}\right )\)

9000039106

Level: 
B
Find the value of the parameter \(a\) which guarantees that the quadratic equation \[ x^{2} + 2ax + a = 0 \] has a pair of complex conjugate solutions with a nonzero imaginary part.
\(a\in (0;1)\)
\(a\in [ 0;1] \)
\(a\in (-\infty ;0)\cup (1;\infty )\)
Such an \(a\) does not exist

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 \( \sin x = z\)
\(\sin ^{2}x\cdot \cos ^{2}x = 0\)

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. \[ B =\mu \frac{NI} {l} \]
\(N = \frac{Bl} {\mu I} \)
\(N = \frac{Bl\mu } {I} \)
\(N = B -\mu \frac{I} {l} \)
\(N = \frac{Bl} {\mu } - I\)

9000045706

Level: 
B
Given a regular pentagon with the side \(a\), find the radius \(r\) of the circle circumscribed to this pentagon.
\(r = \frac{a} {2\cdot \cos 54^{\circ }}\)
\(r = \frac{2a} {\cos 72^{\circ }}\)
\(r = \frac{2a} {\cos 54^{\circ }}\)
\(r = \frac{a} {2\cdot \cos 72^{\circ }}\)