[tex] \it 6)\ \ cos^2B+cos^2C = \dfrac{b^2}{a^2} +\dfrac{c^2}{a^2} =\dfrac{b^2+c^2}{a^2} \ \stackrel{T. Pitagora}{=}\ \ \dfrac{a^2}{a^2} =1
\\ \\ \\4)\ 1+2+4+8+\ ...\ +2^{n+1} =1023\Leftrightarrow \dfrac{2^{n+2}-1}{2-1} =1023 \Leftrightarrow \\ \\ \\\Leftrightarrow 2^{n+2}-1=1023 \Leftrightarrow 2^{n+2} =1024 \Leftrightarrow 2^{n+2} =2^{10} \Leftrightarrow n+2=10\Leftrightarrow n=8[/tex]
[tex] \it 2)\ \ \bar{v} =\bar{i} (1-1+2) +\bar{j} (-1+3+1) \Rightarrow \bar{v} = 2\bar{i} +3\bar{j}
\\ \\\bar{d} = 4\bar{i} +6\bar{j} =2\cdot(2\bar{i} +3\bar{j}) =2\cdot\bar{v} \Rightarrow \bar{v},\ \ \bar{d}\ \ coliniari [/tex]
[tex] \it 3)\ \ D = \dfrac{25}{100} \cdot4000 \cdot5 = 25\cdot40\cdot5= 25\cdot200 = 5000\ lei [/tex]
5) Folosim relațiile lui Viète.
[tex] \it 2x_1+5x_1x_2+2x_2=0 \Leftrightarrow 2(x_1+x_2) +5x_1x_2 = 0 \Leftrightarrow 2\cdot3+5a=0\Leftrightarrow \\ \\\Leftrightarrow 5a=6 \Leftrightarrow a = \dfrac{6}{5} [/tex]
