1.
Phương trình có 2 nghiệm khi:
\(\left\{{}\begin{matrix}m\ne0\\\Delta'\ge0\end{matrix}\right.\)
\(\Leftrightarrow\left\{{}\begin{matrix}m\ne0\\\left(m-2\right)^2-m\left(m-3\right)\ge0\end{matrix}\right.\)
\(\Leftrightarrow\left\{{}\begin{matrix}m\ne0\\-m+4\ge0\end{matrix}\right.\)
\(\Leftrightarrow\left\{{}\begin{matrix}m\ne0\\m\le4\end{matrix}\right.\)
Theo hệ thức Viet: \(\left\{{}\begin{matrix}x_1+x_2=\dfrac{2\left(m-2\right)}{m}\\x_1x_2=\dfrac{m-3}{m}\end{matrix}\right.\)
\(x_1+x_2+x_1x_2\ge2\)
\(\Leftrightarrow\dfrac{2\left(m-2\right)}{m}+\dfrac{m-3}{m}-2\ge0\)
\(\Leftrightarrow\dfrac{m-7}{m}\ge0\)
\(\Rightarrow\left[{}\begin{matrix}m\ge7\\m< 0\end{matrix}\right.\)
\(\Rightarrow m< 0\)
2.
\(T=\dfrac{\left(sinx+cosx\right)\left(sin^2x+cos^2x-sinx.cosx\right)}{sinx+cosx}+sinx.cosx\)
\(=1-sinx.cosx+sinx.cosx=1\)
3.
\(\dfrac{sinx}{cosx}+\dfrac{cosx}{sinx}=3\Leftrightarrow\dfrac{sin^2x+cos^2x}{sinx.cosx}=3\)
\(\Leftrightarrow\dfrac{1}{sinx.cosx}=3\Leftrightarrow sinx.cosx=\dfrac{1}{3}\Leftrightarrow2sinx.cosx=\dfrac{2}{3}\)
\(\Leftrightarrow sin2x=\dfrac{2}{3}\)
\(0< x< \dfrac{\pi}{4}\Rightarrow0< 2x< \dfrac{\pi}{2}\Rightarrow cos2x>0\)
\(\Rightarrow cos2x=\sqrt{1-sin^22x}=\dfrac{\sqrt{5}}{3}\)
