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МатематикаРешебник по алгебре 9 класс Макарычев Задание 852
Задание 852
\[\boxed{\text{852\ (852).}\text{\ }\text{Еуроки\ -\ ДЗ\ без\ мороки}}\]
\[\textbf{а)}\ \frac{P_{6} - P_{4}}{P_{5}} = \frac{6! - 4!}{4! \cdot 5} =\]
\[= \frac{4!(5 \cdot 6 - 1)}{4! \cdot 5} = \frac{29}{5} = 5\frac{4}{5};\]
\[\textbf{б)}\ \frac{P_{12} + P_{13}}{P_{11}} = \frac{12! + 13!}{11!} =\]
\[= \frac{11! \cdot 12 + 11! \cdot 12 \cdot 13}{11!} =\]
\[= \frac{11! \cdot (12 + 12 \cdot 13)}{11!} =\]
\[= 12 + 12 \cdot 13 = 168;\]
\[\textbf{в)}\ \frac{A_{8}^{4} - A_{8}^{3}}{A_{7}^{3} - A_{7}^{2}} = \frac{\frac{8!}{4!} - \frac{8!}{5!}}{\frac{7!}{4!} - \frac{7!}{5!}} =\]
\[= \frac{\frac{8!}{4!} \cdot \ (1 - \frac{1}{5})}{\frac{7!}{4!} \cdot \left( 1 - \frac{1}{5} \right)} = 8;\]
\[\textbf{г)}\ \frac{C_{6}^{3} - C_{6}^{2}}{A_{6}^{2}} = \frac{\frac{6!}{3! \cdot 3!} - \frac{6!}{2! \cdot 4!}}{\frac{6!}{4!}} =\]
\[= \left( \frac{4 \cdot 5 \cdot 6}{2 \cdot 3} - \frac{5 \cdot 6}{2} \right) = \frac{30}{6} \cdot \frac{1}{30} =\]
\[= \frac{1}{6}.\]
\[\boxed{\text{852.}\text{\ }\text{ОК\ ГДЗ\ -\ домашка\ на\ 5}}\]
\[\textbf{а)}\ \left\{ \begin{matrix} \left( x^{2} + y^{2} \right)(x - y) = 447 \\ \text{xy}(x - y) = 210\ \ \ \ \ \ \ \ \ \ \ \ \\ \end{matrix} \right.\ \Longrightarrow\]
\[\Longrightarrow \left\{ \begin{matrix} x³ - y^{3} = 657\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \\ 210 \cdot \left( \frac{x}{y} + 1 + \frac{y}{x} \right) = 657. \\ \end{matrix} \right.\ \]
\[Пусть\ \frac{x}{y} = c,\]
\[210 \cdot \left( c + 1 + \frac{1}{c} \right) = 657\]
\[210c^{2} + 210c + 210 - 657c = 0\]
\[210c^{2} - 447c + 210 = 0\]
\[70c^{2} - 149c + 70 = 0\]
\[D = 22\ 201 - 19600 = 2601,\]
\[c_{1} = \frac{149 + 51}{140} = \frac{10}{7},\ \ \]
\[c_{2} = \frac{149 - 51}{140} = \frac{98}{140} = \frac{7}{10},\]
\[\Longrightarrow \frac{x}{y} = \frac{10}{7}\text{\ \ \ }или\ \ \frac{x}{y} = \frac{7}{10}.\]
\[1)\ \left\{ \begin{matrix} \frac{x}{y} = \frac{10}{7}\text{\ \ \ \ \ \ \ \ \ \ \ \ \ } \\ x^{3} - y^{3} = 657 \\ \end{matrix} \right.\ \Longrightarrow\]
\[\Longrightarrow \left\{ \begin{matrix} y = 0,7x\ \ \ \ \ \ \ \ \ \ \ \\ 0,657x^{3} = 657 \\ \end{matrix} \right.\ \Longrightarrow\]
\[\Longrightarrow \left\{ \begin{matrix} x = 10 \\ y = 7\ \ \ \ \\ \end{matrix} \right.\ \Longrightarrow (10;7).\]
\[2)\ \left\{ \begin{matrix} \frac{x}{y} = \frac{7}{10}\text{\ \ \ \ \ \ \ \ \ \ \ \ \ \ } \\ x^{3} - y^{3} = 657 \\ \end{matrix} \right.\ \Longrightarrow\]
\[\Longrightarrow \left\{ \begin{matrix} y = 0,7y\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \\ - 0,657y^{3} = 657 \\ \end{matrix} \right.\ \Longrightarrow\]
\[\Longrightarrow \left\{ \begin{matrix} y = - 10 \\ x = - 7.\ \ \\ \end{matrix} \right.\ \Longrightarrow ( - 7;\ - 10).\]
\[Ответ:(10;7);( - 7;\ - 10).\]
\[\textbf{б)}\ \left\{ \begin{matrix} \text{xy}(x + y) = 30 \\ x^{3} + y^{3} = 35\ \ \ \ \\ \end{matrix} \right.\ \Longrightarrow\]
\[\Longrightarrow \left\{ \begin{matrix} (x + y)\left( x^{2} - xy + y^{2} \right) = 35 \\ \text{xy}(x + y) = 30\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \\ \end{matrix} \right.\ \Longrightarrow\]
\[\Longrightarrow \left\{ \begin{matrix} x^{3} + y^{3} = 35\ \ \ \ \ \ \ \ \ \ \\ \frac{x^{2} - xy + y^{2}}{\text{xy}} = \frac{35}{30} \\ \end{matrix} \right.\ \Longrightarrow\]
\[\Longrightarrow \left\{ \begin{matrix} x^{3} + y^{3} = 35\ \ \\ \frac{x}{y} - 1 + \frac{y}{x} = \frac{7}{6} \\ \end{matrix} \right.\ ,\]
\[Пусть\ \ \ \frac{x}{y} = c,\ \]
\[c - 1 + \frac{1}{c} = \frac{7}{6}\]
\[6c^{2} - 6c + 6 - 7c = 0\]
\[6c^{2} - 13c + 6 = 0\]
\[D = 169 - 144 = 25,\]
\[c_{1} = \frac{13 + 5}{12} = \frac{18}{12} = \frac{3}{2},\ \ \]
\[c_{2} = \frac{13 - 5}{12} = \frac{8}{12} = \frac{2}{3},\]
\[\Longrightarrow \frac{x}{y} = \frac{3}{2}\text{\ \ \ }или\ \ \frac{x}{y} = \frac{2}{3}.\]
\[1)\ \left\{ \begin{matrix} x = \frac{3}{2}\text{y\ \ \ \ \ \ \ \ \ } \\ x^{3} + y^{3} = 35 \\ \end{matrix} \right.\ \Longrightarrow\]
\[\Longrightarrow \left\{ \begin{matrix} x = \frac{3}{2}\text{y\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ } \\ \frac{27}{8}y^{3} + y^{3} = 35 \\ \end{matrix} \right.\ \Longrightarrow\]
\[\Longrightarrow \left\{ \begin{matrix} \frac{35}{8}y³ = 35 \\ x = \frac{3}{2}\text{y\ \ \ \ \ \ } \\ \end{matrix} \right.\ \Longrightarrow \left\{ \begin{matrix} y = 2 \\ x = 3. \\ \end{matrix} \right.\ \]
\[2)\ \left\{ \begin{matrix} x = \frac{2}{3}\text{y\ \ \ \ \ \ \ \ \ } \\ x^{3} + y^{3} = 35 \\ \end{matrix} \right.\ \Longrightarrow\]
\[\Longrightarrow \left\{ \begin{matrix} x = \frac{2}{3}\text{y\ \ \ \ \ \ \ \ \ \ \ \ \ \ } \\ \frac{8}{27}y^{3} + y^{3} = 35 \\ \end{matrix} \right.\ \Longrightarrow\]
\[\Longrightarrow \left\{ \begin{matrix} \frac{35}{27}y^{3} = 35 \\ x = \frac{2}{3}\text{y\ \ \ \ \ \ \ \ } \\ \end{matrix} \right.\ \Longrightarrow \left\{ \begin{matrix} y = 3 \\ x = 2. \\ \end{matrix} \right.\ \]
\[Ответ:(2;3);\ \ \ (3;2).\]
