Đáp án:
$\displaystyle \begin{array}{{>{\displaystyle}l}} Bài\ 1:\\ 1.\ A=-12\sqrt{a}\\ 2.\ GTLN\ M=-36.\\ Bài\ 2:\\ 1.\ ( a;b) =\left( -\frac{3}{2} ;-\frac{5}{2}\right)\\ 2.\ \ ( x;y) =( 1;-1)\\ Bài\ \ 3:\\ 1.\ S=\left\{4\pm 2\sqrt{3}\right\}\\ 2.\ m=\frac{1+\sqrt{73}}{6} \end{array}$
Giải thích các bước giải:
$\displaystyle \begin{array}{{>{\displaystyle}l}} Bài\ 1:\\ 1.\ A=\frac{\left( a-3\sqrt{a}\right)\left(\sqrt{a} -3\right) -\left( a+3\sqrt{a}\right)\left(\sqrt{a} +3\right)}{\left(\sqrt{a} -3\right)\left(\sqrt{a} +3\right)} .\frac{a-9}{\sqrt{a}}\\ A=\frac{a\sqrt{a} -3a-3a+9\sqrt{a} -a\sqrt{a} -3a-3a-9\sqrt{a}}{\sqrt{a}}\\ A=\frac{-3a-3a-3a-3a}{\sqrt{a}} =\frac{-12a}{\sqrt{a}} =-12\sqrt{a}\\ 2.M=a-12\sqrt{a} =\left(\sqrt{a} -6\right)^{2} -36\geqslant -36\\ Vậy\ GTLN\ M=-36.\\ Bài\ 2:\\ 1.\ ( d) \perp ( d') \Rightarrow a=-\frac{3}{2} .\\ A\in ( d) \Rightarrow -\frac{3}{2} .( -3) +b=2\Rightarrow b=-\frac{5}{2}\\ ( a;b) =\left( -\frac{3}{2} ;-\frac{5}{2}\right)\\ 2.\ \{_{-8x+9y=-17}^{x=y+2} \Leftrightarrow \{_{-8( y+2) +9y=-17}^{x=y+2} \Leftrightarrow x=1;\ y=-1\\ Vậy\ ( x;y) =( 1;-1)\\ Bài\ \ 3:\\ 1.\ x^{2} -8x+4=0\Leftrightarrow x=4\pm 2\sqrt{3}\\ S=\left\{4\pm 2\sqrt{3}\right\}\\ 2.\ \Delta '=1+m-1=m\\ Để\ PT\ có\ \ 2\ nghiệm\ \Leftrightarrow m >0\\ Theo\ \ Viet:\ x_{1} +x_{2} =2;\ x_{1} x_{2} =1-m\\ Ta\ có:\frac{x_{1}^{3} +mx_{2} -2x_{1}^{2}}{x_{2}^{2} +2} =\frac{x_{1}^{2} +2}{x_{2}^{3} +mx_{1} -2x_{2}^{2}}\\ \Leftrightarrow \left( x_{1}^{3} +mx_{2} -2x_{1}^{2}\right)\left( x_{2}^{3} +mx_{1} -2x_{2}^{2}\right) =\left( x_{1}^{2} +2\right)\left( x_{2}^{2} +2\right)\\ \Leftrightarrow \left[ x_{1}^{3} +( 1-x_{1} x_{2}) x_{2} -( \ x_{1} +x_{2}) x_{1}^{2}\right]\left[ x_{2}^{3} +( 1-x_{1} x_{2}) x_{1} -( \ x_{1} +x_{2}) x_{2}^{2}\right] =\left( x_{1}^{2} +2\right)\left( x_{2}^{2} +2\right)\\ \Leftrightarrow \left[ x_{1}^{3} +x_{2} -x_{1} x_{2}^{2} -x_{1}^{3} -x_{2} x_{1}^{2}\right]\left[ x_{2}^{3} +x_{1} -x_{1}^{2} x_{2} -x_{1} x_{2}^{2} -x_{2}^{3}\right] =\left( x_{1}^{2} +2\right)\left( x_{2}^{2} +2\right)\\ \Leftrightarrow \left[ x_{2} -x_{1} x_{2}^{2} -x_{2} x_{1}^{2}\right]\left[ x_{1} -x_{1}^{2} x_{2} -x_{1} x_{2}^{2}\right] =\left( x_{1}^{2} +2\right)\left( x_{2}^{2} +2\right)\\ \Leftrightarrow [ x_{2} -x_{1} x_{2}( x_{1} +x_{2})][ x_{1} -x_{1} x_{2}( x_{1} +x_{2})] =\left( x_{1}^{2} +2\right)\left( x_{2}^{2} +2\right)\\ \Leftrightarrow [ x_{2} -2( 1-m)][ x_{1} -2( 1-m)] =( x_{1} x_{2})^{2} +4+2x_{1} x_{2}\\ \Leftrightarrow x_{1} x_{2} +4( 1-m)^{2} -2( 1-m)( x_{1} +x_{2}) =( x_{1} x_{2})^{2} +4+2x_{1} x_{2}\\ \Leftrightarrow 1-m+4( 1-m)^{2} -4( 1-m) =( 1-m)^{2} +4+2( 1-m)\\ \Leftrightarrow 3( 1-m)^{2} -5( 1-m) -4=0\\ \Leftrightarrow 1-m=\frac{5\pm \sqrt{73}}{6}\\ \Leftrightarrow m=\frac{1-\sqrt{73}}{6} \ ( loại) ;\ m=\frac{1+\sqrt{73}}{6} \ ( chọn)\\ Vậy\ m=\frac{1+\sqrt{73}}{6} \end{array}$
