A

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 \(\omega \).
\(\mathop{\mathrm{tg}}\nolimits \omega = \frac{\sqrt{2}} {2} \)
\(\cos \omega = \frac{\sqrt{2}} {2} \)
\(\sin \omega = \frac{\sqrt{2}} {2} \)
\(\mathop{\mathrm{cotg}}\nolimits \omega = \frac{\sqrt{2}} {2} \)

9000037506

Level: 
A
Given complex numbers \[ a = 3 + 5\mathrm{i}\text{, }\quad b = 2 -\mathrm{i}\text{, } \] find the quotient \(\frac{a} {b}\).
\(\frac{1} {5} + \mathrm{i}\frac{13} {5} \)
\(\frac{1} {3} + \mathrm{i}\frac{13} {3} \)
\(\frac{1} {5} + \mathrm{i}\frac{7} {5}\)
\(\frac{1} {3} + \mathrm{i}\frac{7} {3}\)

9000037507

Level: 
A
Given complex numbers \[ a = \sqrt{3} + 2\mathrm{i}\text{, }\quad b = \sqrt{2} -\mathrm{i}\text{, } \] find the quotient \(\frac{a} {b}\).
\(\frac{\sqrt{6}-2} {3} + \mathrm{i}\frac{2\sqrt{2}+\sqrt{3}} {3} \)
\(\frac{\sqrt{6}-2} {3} -\mathrm{i}\frac{2\sqrt{2}+\sqrt{3}} {3} \)
\(\frac{\sqrt{6}-3} {2} + \mathrm{i}\frac{2\sqrt{2}+\sqrt{3}} {2} \)
\(\frac{\sqrt{6}-2} {2} -\mathrm{i}\frac{2\sqrt{2}+\sqrt{3}} {2} \)