БИЛЕТ №21 Блок А 1. Вычислите: 6cos(pi/4)*tg(pi/4)/(3sin(pi/4))

Блок А 1. $\frac{6 \cos \frac{\pi}{4} \cdot \operatorname{tg} \frac{\pi}{4}}{3 \sin \frac{\pi}{4}} = \frac{6 \cdot \frac{\sqrt{2}}{2} \cdot 1}{3 \cdot \frac{\sqrt{2}}{2}} = \frac{6}{3} = 2$ 2. $\frac{1000^{0.4} \cdot 100^{0.5}}{10000^{0.2} \cdot 10^{0.8} \cdot 10^{0.1}} = \frac{(10^3)^{0.4} \cdot (10^2)^{0.5}}{(10^4)^{0.2} \cdot 10^{0.8} \cdot 10^{0.1}} = \frac{10^{1.2} \cdot 10^1}{10^{0.8} \cdot 10^{0.8} \cdot 10^{0.1}} = \frac{10^{2.2}}{10^{1.7}} = 10^{0.5} = \sqrt{10}$ 3. $\log_7 343 - \log_7 7 = \log_7 (7^3) - \log_7 7 = 3 - 1 = 2$ 4. $\frac{\sqrt[3]{29} \cdot \sqrt[4]{29}}{\sqrt[6]{29} \cdot \sqrt[12]{29}} = \frac{29^{1/3} \cdot 29^{1/4}}{29^{1/6} \cdot 29^{1/12}} = \frac{29^{(4+3)/12}}{29^{(2+1)/12}} = \frac{29^{7/12}}{29^{3/12}} = 29^{4/12} = 29^{1/3} = \sqrt[3]{29}$ 5. $4^{2x} = 32^{x-1} \Rightarrow (2^2)^{2x} = (2^5)^{x-1} \Rightarrow 2^{4x} = 2^{5x-5} \Rightarrow 4x = 5x - 5 \Rightarrow x = 5$ 7. $\sin 3x = -1 \Rightarrow 3x = -\frac{\pi}{2} + 2\pi k \Rightarrow x = -\frac{\pi}{6} + \frac{2\pi k}{3}, k \in \mathbb{Z}$ 8. $4^{x+1} \ge 32 \Rightarrow (2^2)^{x+1} \ge 2^5 \Rightarrow 2^{2x+2} \ge 2^5 \Rightarrow 2x + 2 \ge 5 \Rightarrow 2x \ge 3 \Rightarrow x \ge 1.5$ Блок Б 1. $\cos (\frac{2x}{3} + \frac{\pi}{6}) = \frac{1}{2} \Rightarrow \frac{2x}{3} + \frac{\pi}{6} = \pm \frac{\pi}{3} + 2\pi k$ $\frac{2x}{3} = -\frac{\pi}{6} \pm \frac{\pi}{3} + 2\pi k$ $x = -\frac{\pi}{4} \pm \frac{\pi}{2} + 3\pi k, k \in \mathbb{Z}$ 2. $4^{x+1} - 5 \cdot 2^{x+1} + 4 = 0$. Пусть $t = 2^{x+1} (t>0)$, тогда $t^2 - 5t + 4 = 0$. По теореме Виета: $t_1 = 1, t_2 = 4$. $2^{x+1} = 1 \Rightarrow x+1=0 \Rightarrow x=-1$ $2^{x+1} = 4 \Rightarrow x+1=2 \Rightarrow x=1$ Ответ: $-1; 1$ 3. $\cos 6x \cos x + \sin 6x \sin x = -\frac{1}{2}$ По формуле косинуса разности: $\cos(6x - x) = -\frac{1}{2} \Rightarrow \cos 5x = -\frac{1}{2}$ $5x = \pm \frac{2\pi}{3} + 2\pi k \Rightarrow x = \pm \frac{2\pi}{15} + \frac{2\pi k}{5}, k \in \mathbb{Z}$