20А. Найдите все значения параметра a, при каждом из которых корни уравнения \({x^2} + \left( {{a^2}-1} \right)\,x-{a^2} = 0\) лежат на интервале \(\left( {-3a;\,a} \right)\).
ОТВЕТ: \(\left( {1;\,3} \right).\)
\({x^2} + \left( {{a^2}-1} \right)x-{a^2} = 0\,\,\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\,\,{x^2} + {a^2}\,x-x-{a^2} = 0\,\,\,\,\,\,\,\, \Leftrightarrow \) \( \Leftrightarrow \,\,\,\,\,\,\,x\left( {x-1} \right) + {a^2}\left( {x-1} \right) = 0\,\,\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\,\,\left( {x-1} \right)\left( {x + {a^2}} \right) = 0\,\,\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\,\,\left[ {\begin{array}{*{20}{c}}{x = 1,\,\,\,\,\,\,}\\{x = -{a^2}.}\end{array}} \right.\) Очевидно, что \(1 > -{a^2}\) при \(a \in R\), поэтому корни будут лежать в интервале \(\left( {-3a;\,a} \right)\) если: \(\left\{ {\begin{array}{*{20}{c}}{1 < a\,,\,\,\,\,\,\,\,\,\,\,\,}\\{-{a^2} > -3a}\end{array}\,\,\,\,\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\,\,\,\left\{ {\begin{array}{*{20}{c}}{a > 1,\,\,\,\,\,\,\,\,\,\,\,\,\,\,}\\{{a^2}-3a < 0}\end{array}\,\,\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\,\,\,\left\{ {\begin{array}{*{20}{c}}{a \in \left( {1;\infty } \right),}\\{a\, \in \,\left( {0;3} \right)\,\,\,\,}\end{array}} \right.} \right.} \right.\,\,\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\,\,a \in \,\left( {1;3} \right).\) ОТВЕТ: \(\left( {1;\,3} \right).\)