1)
\(\begin{array}{l}A \cap B = \emptyset \Leftrightarrow \left[ \begin{array}{l}3 \le m\\m + 5 \le - 2\end{array} \right. \Leftrightarrow \left[ \begin{array}{l}m \ge 3\\m \le - 7\end{array} \right.\\ \Rightarrow A \cap B \ne \emptyset \Leftrightarrow - 7 < m < 3\end{array}\)
2) Phương trình có hai nghiệm trái dấu \( \Leftrightarrow ac < 0 \Leftrightarrow 1.\left( {m - 1} \right) < 0 \Leftrightarrow m < 1\)
3) \(\overrightarrow {HB} .\overrightarrow {HC} = HB.HC.\cos \left( {\overrightarrow {HB} ,\overrightarrow {HC} } \right) = A{H^2}.\cos {180^0} = - A{H^2}\)
Mà \(AB = 3,AC = 4 \Rightarrow BC = \sqrt {{3^2} + {4^2}} = 5\)
\(\begin{array}{l} \Rightarrow AH.BC = AB.AC \Leftrightarrow AH = \dfrac{{AB.AC}}{{BC}} = \dfrac{{3.4}}{5} = \dfrac{{12}}{5}\\ \Rightarrow \overrightarrow {HB} .\overrightarrow {HC} = - A{H^2} = - {\left( {\dfrac{{12}}{5}} \right)^2} = - \dfrac{{144}}{{25}}\end{array}\)
4) Gọi \(D\left( {x;y} \right)\).
C là trọng tâm \(\Delta ABD\) \( \Leftrightarrow \left\{ \begin{array}{l}2 = \dfrac{{ - 4 + 2 + x}}{3}\\ - 2 = \dfrac{{1 + 4 + y}}{3}\end{array} \right. \Leftrightarrow \left\{ \begin{array}{l} - 2 + x = 6\\5 + y = - 6\end{array} \right. \Leftrightarrow \left\{ \begin{array}{l}x = 8\\y = - 11\end{array} \right.\)
Vậy \(D\left( {8; - 11} \right)\)
