3)

a) Ta có:

\(\left(\dfrac{1}{a}+\dfrac{1}{b}+\dfrac{1}{c}\right)^2=\dfrac{1}{a^2}+\dfrac{1}{b^2}+\dfrac{1}{c^2}+\dfrac{2}{ab}+\dfrac{2}{ac}+\dfrac{2}{bc}\\ =\dfrac{1}{a^2}+\dfrac{1}{b^2}+\dfrac{1}{c^2}+\dfrac{2a+2b+2c}{abc}=\dfrac{1}{a^2}+\dfrac{1}{b^2}+\dfrac{1}{c^2}+\dfrac{2\left(a+b+c\right)}{abc}\\ =\dfrac{1}{a^2}+\dfrac{1}{b^2}+\dfrac{1}{c^2}\)

\(=>\sqrt{\dfrac{1}{a^2}+\dfrac{1}{b^2}+\dfrac{1}{c^2}}=\sqrt{\left(\dfrac{1}{a}+\dfrac{1}{b}+\dfrac{1}{c}\right)^2}=\left|\dfrac{1}{a}+\dfrac{1}{b}+\dfrac{1}{c}\right|\) 

b) 

\(\sqrt{1+\dfrac{1}{2^2}+\dfrac{1}{3^2}}+\sqrt{1+\dfrac{1}{3^2}+\dfrac{1}{4^2}}+...+\sqrt{\dfrac{1}{1^2}+\dfrac{1}{2024^2}+\dfrac{1}{2025^2}}\left(1\right)\\ =\sqrt{\dfrac{1}{1^2}+\dfrac{1}{2^2}+\dfrac{1}{\left(-3\right)^2}}+\sqrt{\dfrac{1}{1^2}+\dfrac{1}{3^2}+\dfrac{1}{\left(-4\right)^2}}+...+\sqrt{\dfrac{1}{1^2}+\dfrac{1}{2024^2}+\dfrac{1}{\left(-2025\right)^2}}\)

Theo câu a \(\dfrac{1}{a^2}+\dfrac{1}{b^2}+\dfrac{1}{c^2}=\left|\dfrac{1}{a}+\dfrac{1}{b}+\dfrac{1}{c}\right|\) khi \(a+b+c=0\)

Mà: \(\left\{{}\begin{matrix}1+2+\left(-3\right)=0\\1+3+\left(-4\right)=0\\....\\1+2024+\left(-2025\right)=0\end{matrix}\right.\)  

=> \(\left(1\right)=\left|1+\dfrac{1}{2}+\dfrac{1}{-3}\right|+\left|1+\dfrac{1}{3}+\dfrac{1}{-4}\right|+...+\left|1+\dfrac{1}{2024}+\dfrac{1}{-2025}\right|\)  

Mà: \(\left\{{}\begin{matrix}1+\dfrac{1}{2}\cdot\dfrac{1}{-3}>0\\1+\dfrac{1}{3}+\dfrac{1}{-4}>0\\...\\1+\dfrac{1}{2024}+\dfrac{1}{-2025}>0\end{matrix}\right.\) 

=> \(\left(1\right)=1+\dfrac{1}{2}+\dfrac{1}{-3}+1+\dfrac{1}{3}+\dfrac{1}{-4}+...+1+\dfrac{1}{2024}+\dfrac{1}{-2025}\\ =\left(1+1+...+1\right)+\dfrac{1}{2}+\left(\dfrac{1}{3}-\dfrac{1}{3}\right)+\left(\dfrac{1}{4}-\dfrac{1}{4}\right)+...+\left(\dfrac{1}{2024}-\dfrac{1}{2024}\right)-\dfrac{1}{2025}\\ =2023+\dfrac{1}{2}-\dfrac{1}{2025}\)

4)

\(\left(12-6\sqrt{3}\right)\sqrt{\dfrac{3}{14-8\sqrt{3}}}-3\sqrt{2\left(1-\sqrt{-2\sqrt{3}+4}\right)+2\sqrt{4+2\sqrt{3}}}\\ =\left(12-6\sqrt{3}\right)\sqrt{\dfrac{3}{\left(2\sqrt{2}\right)^2-2\cdot2\sqrt{2}\cdot\sqrt{6}+\left(\sqrt{6}\right)^2}}-3\sqrt{2\left(1-\sqrt{4-2\sqrt{3}}\right)+2\sqrt{\left(\sqrt{3}\right)^2+2\cdot\sqrt{3}\cdot1+1^2}}\\ =\left(12-6\sqrt{3}\right)\sqrt{\dfrac{3}{\left(2\sqrt{2}-\sqrt{6}\right)^2}}-3\sqrt{2\left(1-\sqrt{\left(\sqrt{3}\right)^2-2\cdot\sqrt{3}\cdot1+1^2}\right)+2\sqrt{\left(\sqrt{3}+1\right)^2}}\\ =\left(12-6\sqrt{3}\right)\dfrac{\sqrt{3}}{2\sqrt{2}-\sqrt{6}}-3\sqrt{2\left(1-\sqrt{\left(\sqrt{3}-1\right)^2}\right)+2\left(\sqrt{3}+1\right)}\\ =\left(12-6\sqrt{3}\right)\dfrac{\sqrt{3}}{2\sqrt{2}-\sqrt{6}}-3\sqrt{2\left(1-\sqrt{3}+1\right)+2\sqrt{3}+2}\\ =\left(12-6\sqrt{3}\right)\dfrac{\sqrt{3}}{2\sqrt{2}-\sqrt{6}}-3\sqrt{2-2\sqrt{3}+2+2\sqrt{3}+2}\\ =3\sqrt{2}\cdot\left(2\sqrt{2}-\sqrt{6}\right)\cdot\dfrac{\sqrt{3}}{2\sqrt{2}-\sqrt{6}}-3\sqrt{6}\\ =3\sqrt{2}\cdot\sqrt{3}-3\sqrt{6}\\ =3\sqrt{6}-3\sqrt{6}\\ =0\)

a: Xét ΔABC có AD là phân giác

nên \(AD=\dfrac{2\cdot AB\cdot AC}{AB+AC}\cdot cos\left(\dfrac{BAC}{2}\right)\)

=>\(AD=\dfrac{2\cdot AB\cdot AC}{AB+AC}\cdot cos\left(\dfrac{120}{2}\right)=\dfrac{2\cdot AB\cdot AC}{AB+AC}\cdot cos60\)

=>\(AD=\dfrac{2\cdot AB\cdot AC}{AB+AC}\cdot\dfrac{1}{2}=\dfrac{AB\cdot AC}{AB+AC}\)

=>\(\dfrac{1}{AD}=\dfrac{AB+AC}{AB\cdot AC}=\dfrac{1}{AB}+\dfrac{1}{AC}\)

b:  \(AD=\dfrac{2\cdot AB\cdot AC}{AB+AC}\cdot cos\left(\dfrac{BAC}{2}\right)\)

=>\(AD=\dfrac{2\cdot AB\cdot AC}{AB+AC}\cdot cos45=\dfrac{2\cdot AB\cdot AC\sqrt{2}}{2\left(AB+AC\right)}=\dfrac{AB\cdot AC\cdot\sqrt{2}}{AB+AC}\)

=>\(\dfrac{1}{AD}=\dfrac{AB+AC}{AB\cdot AC}\cdot\dfrac{1}{\sqrt{2}}\)

=>\(\dfrac{\sqrt{2}}{AD}=\dfrac{AB+AC}{AB\cdot AC}=\dfrac{1}{AB}+\dfrac{1}{AC}\)

c: \(AD=\dfrac{2\cdot AB\cdot AC}{AB+AC}\cdot cos\left(\dfrac{BAC}{2}\right)\)

=>\(AD=\dfrac{2\cdot AB\cdot AC}{AB+AC}\cdot cos\left(\dfrac{60}{2}\right)\)

=>\(AD=\dfrac{2\cdot AB\cdot AC}{AB+AC}\cdot cos30=\dfrac{2\cdot AB\cdot AC}{AB+AC}\cdot\dfrac{\sqrt{3}}{2}\)

=>\(\dfrac{AD}{\sqrt{3}}=\dfrac{AB\cdot AC}{AB+AC}\)

=>\(\dfrac{\sqrt{3}}{AD}=\dfrac{AB+AC}{AB\cdot AC}=\dfrac{1}{AB}+\dfrac{1}{AC}\)

Chúng đều được định nghĩa dựa trên các cạnh của tam giác vuông và góc nhọn trong tam giác đó. Sin: Tỷ số giữa cạnh đối diện với góc nhọn và cạnh huyền của tam giác vuông. Cos: Tỷ số giữa cạnh kề với góc nhọn và cạnh huyền của tam giác vuông. Tan: Tỷ số giữa cạnh đối diện và cạnh kề của góc nhọn trong tam giác vuông.

@ ánh lê Copy phải ghi Tk nhé!

Tk = Tham khảo

<=> \(\dfrac{\sqrt{x}+3}{\sqrt{x}-1}\) + 2 ≤ 0

<=> \(\dfrac{\sqrt{x}+3+2\sqrt{x}-2}{\sqrt{x}-1}\) ≤ 0

<=> \(\dfrac{3\sqrt{x}+1}{\sqrt{x}-1}\) ≤ 0

Mà ( \(3\sqrt{x}\) + 1 ) > 0

=> \(\sqrt{x}-1\) < 0

=> \(\sqrt{x}\) < 1

=> x ϵ [ 0 , 1 )

ĐKXĐ: x>=0

\(\dfrac{2\sqrt{x}-6}{x-\sqrt{x}+1}< 0\)

mà \(x-\sqrt{x}+1=\left(\sqrt{x}-\dfrac{1}{2}\right)^2+\dfrac{3}{4}>=\dfrac{3}{4}>0\forall x\) thỏa mãn ĐKXĐ

nên \(2\sqrt{x}-6< 0\)

=>\(\sqrt{x}< 3\)

=>0<=x<9

ĐKXĐ: \(\left\{{}\begin{matrix}x\ne-2\\y\ne-1\end{matrix}\right.\)

\(\left\{{}\begin{matrix}\dfrac{2x}{x+2}-\dfrac{3y}{y+1}=-4\\\dfrac{x}{x+2}+\dfrac{2y}{y+1}=\dfrac{1}{3}\end{matrix}\right.\)

=>\(\left\{{}\begin{matrix}\dfrac{2x+4-4}{x+2}-\dfrac{3y+3-3}{y+1}=-4\\\dfrac{x+2-2}{x+2}+\dfrac{2y+2-2}{y+1}=\dfrac{1}{3}\end{matrix}\right.\)

=>\(\left\{{}\begin{matrix}2-\dfrac{4}{x+2}-3+\dfrac{3}{y+1}=-4\\1-\dfrac{2}{x+2}+2-\dfrac{2}{y+1}=\dfrac{1}{3}\end{matrix}\right.\)

=>\(\left\{{}\begin{matrix}-\dfrac{4}{x+2}+\dfrac{3}{y+1}=-4-2+3=-6+3=-3\\-\dfrac{2}{x+2}-\dfrac{2}{y+1}=\dfrac{1}{3}-3=-\dfrac{8}{3}\end{matrix}\right.\)

=>\(\left\{{}\begin{matrix}\dfrac{-4}{x+2}+\dfrac{3}{y+1}=-3\\\dfrac{-4}{x+2}-\dfrac{4}{y+1}=-\dfrac{16}{3}\end{matrix}\right.\Leftrightarrow\left\{{}\begin{matrix}\dfrac{-4}{x+2}+\dfrac{3}{y+1}+\dfrac{4}{x+2}+\dfrac{4}{y+1}=-3+\dfrac{16}{3}\\\dfrac{-4}{x+2}-\dfrac{4}{y+1}=-\dfrac{16}{3}\end{matrix}\right.\)

=>\(\left\{{}\begin{matrix}\dfrac{7}{y+1}=\dfrac{7}{3}\\\dfrac{1}{x+2}+\dfrac{1}{y+1}=\dfrac{4}{3}\end{matrix}\right.\Leftrightarrow\left\{{}\begin{matrix}y+1=3\\\dfrac{1}{x+2}=\dfrac{4}{3}-\dfrac{1}{3}=1\end{matrix}\right.\)

=>\(\left\{{}\begin{matrix}y=2\\x=-1\end{matrix}\right.\left(nhận\right)\)

ĐKXĐ: x<>-2

\(\dfrac{x-3}{x+2}>=0\)

TH1: \(\left\{{}\begin{matrix}x-3>=0\\x+2>0\end{matrix}\right.\)

=>\(\left\{{}\begin{matrix}x>=3\\x>-2\end{matrix}\right.\)

=>x>=3

TH2: \(\left\{{}\begin{matrix}x-3< =0\\x+2< 0\end{matrix}\right.\)

=>\(\left\{{}\begin{matrix}x< =3\\x< -2\end{matrix}\right.\)

=>x<-2

\(\dfrac{5+2\sqrt{5}}{\sqrt{5}}+\dfrac{3+\sqrt{3}}{\sqrt{3}}-\left(\sqrt{5}+\sqrt{3}\right)\\ =\dfrac{\sqrt{5}\left(\sqrt{5}+2\right)}{\sqrt{5}}+\dfrac{\sqrt{3}\left(\sqrt{3}+1\right)}{\sqrt{3}}-\left(\sqrt{5}+\sqrt{3}\right)\\ =\left(\sqrt{5}+2\right)+\left(\sqrt{3}+1\right)-\left(\sqrt{5}+\sqrt{3}\right)\\ =\sqrt{5}+2+\sqrt{3}+1-\sqrt{5}-\sqrt{3}\\ =2+1=3\)

\(\sqrt{\dfrac{9}{4}}-\sqrt{2}+\sqrt{2}\\ =\dfrac{3}{2}-\left(\sqrt{2}-\sqrt{2}\right)\\ =\dfrac{3}{2}-0\\ =\dfrac{3}{2}\)

\(H=\dfrac{4}{1-\sqrt{3}}-\dfrac{\sqrt{15}+\sqrt{3}}{1+\sqrt{5}}\\ =\dfrac{4\left(1+\sqrt{3}\right)}{\left(1+\sqrt{3}\right)\left(1-\sqrt{3}\right)}-\dfrac{\sqrt{3}\left(\sqrt{5}+1\right)}{\sqrt{5}+1}\\ =\dfrac{4\left(1+\sqrt{3}\right)}{1-3}-\sqrt{3}\\ =\dfrac{4\left(1+\sqrt{3}\right)}{-2}-\sqrt{3}\\ =-2\left(1+\sqrt{3}\right)-\sqrt{3}\\ =-2-2\sqrt{3}-\sqrt{3}\\ =-2-3\sqrt{3}\)

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