Sửa đề: Cho ΔDEF đều

ΔDEF đều

=>\(\widehat{EFH}=60^0\)

=>\(sinEFH=sin60=\dfrac{\sqrt{3}}{2};cosEFH=cos60=\dfrac{1}{2}\)

\(tanEFH=tan60=\sqrt{3};cotEFH=cot60=\dfrac{1}{\sqrt{3}}\)

ΔDEF đều

mà EH là đường cao

nên EH là phân giác của góc DEF

=>\(\widehat{DEH}=30^0\)

=>\(sinDEH=sin30=\dfrac{1}{2};cosDEH=cos30=\dfrac{\sqrt{3}}{2}\)

\(tanDEH=tan30=\dfrac{1}{\sqrt{3}};cotDEH=cot30=\sqrt{3}\)

a) \(A=\left(\dfrac{2\sqrt{x}-9}{x-5\sqrt{x}+6}-\dfrac{\sqrt{x}+3}{\sqrt{x}-2}-\dfrac{2\sqrt{x}+1}{3-\sqrt{x}}\right)\cdot\dfrac{x-3\sqrt{x}}{x\sqrt{x}+1}\left(x\ne9;x\ne4;x\ge0\right)\) 

\(=\left[\dfrac{2\sqrt{x}-9}{\left(\sqrt{x}-2\right)\left(\sqrt{x}-3\right)}-\dfrac{\left(\sqrt{x}+3\right)\left(\sqrt{x}-3\right)}{\left(\sqrt{x}-2\right)\left(\sqrt{x}-3\right)}+\dfrac{2\sqrt{x}+1}{\sqrt{x}-3}\right]\cdot\dfrac{x-3\sqrt{x}}{x\sqrt{x}+1}\) 

\(=\left[\dfrac{2\sqrt{x}-9-x+9}{\left(\sqrt{x}-2\right)\left(\sqrt{x}-3\right)}+\dfrac{\left(2\sqrt{x}+1\right)\left(\sqrt{x}-2\right)}{\left(\sqrt{x}-2\right)\left(\sqrt{x}-3\right)}\right]\cdot\dfrac{\sqrt{x}\left(\sqrt{x}-3\right)}{\left(\sqrt{x}+1\right)\left(x-\sqrt{x}+1\right)}\)

\(=\dfrac{-x+2\sqrt{x}+2x-4\sqrt{x}+\sqrt{x}-2}{\left(\sqrt{x}-2\right)\left(\sqrt{x}-3\right)}\cdot\dfrac{\sqrt{x}\left(\sqrt{x}-3\right)}{\left(\sqrt{x}+1\right)\left(x-\sqrt{x}+1\right)}\)

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

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

\(=\dfrac{\sqrt{x}+1}{\sqrt{x}-3}\cdot\dfrac{\sqrt{x}\left(\sqrt{x}-3\right)}{\left(\sqrt{x}+1\right)\left(x-\sqrt{x}+1\right)}\)

\(=\dfrac{\sqrt{x}}{x-\sqrt{x}+1}\) 

b) \(x=\sqrt{3+2\sqrt{2}}+\sqrt{11-6\sqrt{2}}+12\)

\(=\sqrt{\left(\sqrt{2}\right)^2+2\cdot\sqrt{2}\cdot1+1^2}+\sqrt{3^2-2\cdot3\cdot\sqrt{2}+\left(\sqrt{2}\right)^2}+12\)

\(=\sqrt{\left(\sqrt{2}+1\right)^2}+\sqrt{\left(3-\sqrt{2}\right)^2}+12\)

\(=\sqrt{2}+1+3-\sqrt{2}+12\)

\(=16\)

Thay x=16 vào A ta có:

\(A=\dfrac{\sqrt{16}}{16-\sqrt{16}+1}=\dfrac{4}{16-4+1}=\dfrac{4}{13}\) 

c) \(A=\dfrac{\sqrt{x}}{x-\sqrt{x}+1}\Rightarrow\dfrac{1}{A}=\dfrac{x-\sqrt{x}+1}{\sqrt{x}}\)

\(\Rightarrow\dfrac{1}{A}=\dfrac{x}{\sqrt{x}}-\dfrac{\sqrt{x}}{\sqrt{x}}+\dfrac{1}{\sqrt{x}}=\sqrt{x}+\dfrac{1}{\sqrt{x}}-1\)

Vì \(\sqrt{x};\dfrac{1}{\sqrt{x}}>0\) nên áp dụng bđt cô si ta có: 

\(\dfrac{1}{A}\ge2\sqrt{\sqrt{x}\cdot\dfrac{1}{\sqrt{x}}}-1=2-1=1\)

\(\Leftrightarrow A\le1\)

Dấu "=" xảy ra khi: \(\sqrt{x}=\dfrac{1}{\sqrt{x}}\Leftrightarrow x=1\)

Vậy: ... 

sin 30° + sin 60°

= \(\dfrac{1}{2}+\dfrac{\sqrt{3}}{2}=\dfrac{1+\sqrt{3}}{2}\)

Lời giải:

Áp dụng BĐT AM-GM:

$A=x(1-x^2)=x(1-x)(1+x)=(1+\sqrt{3})x.(2+\sqrt{3})(1-x)(1+x).\frac{1}{(\sqrt{3}+1)(\sqrt{3}+2)}$

$=(x+x\sqrt{3})[2+\sqrt{3}-(2+\sqrt{3})x](1+x).\frac{1}{(\sqrt{3}+1)(\sqrt{3}+2)}$

\(\leq \left[\frac{x+x\sqrt{3}+2+\sqrt{3}-(2+\sqrt{3})x+1+x}{3}\right]^3.\frac{1}{(1+\sqrt{3})(\sqrt{3}+2)}\\ =\frac{(\sqrt{3}+1)^3}{3\sqrt{3}}.\frac{1}{(\sqrt{3}+1)(\sqrt{3}+2)}=\frac{2}{3\sqrt{3}}\)

Vậy $A_{\max}=\frac{2}{3\sqrt{3}}$. Giá trị này đạt tại $x=\frac{1}{\sqrt{3}}$

Bạn nên gõ đề bằng công thức toán (biểu tượng $\sum$ góc trái khung soạn thảo) để mọi người đọc hiểu đề và hỗ trợ bạn nhanh hơn nhé.

a, Với \(x\ge0;x\ne1\):

\(P=\left(\dfrac{\sqrt{x}-2}{x-1}-\dfrac{\sqrt{x}+2}{x+2\sqrt{x}+1}\right).\dfrac{\left(1-x\right)^2}{2}\)

\(=\left[\dfrac{\left(\sqrt{x}-2\right)\left(\sqrt{x}+1\right)}{\left(\sqrt{x}-1\right)\left(\sqrt{x}+1\right)^2}-\dfrac{\left(\sqrt{x}-1\right)\left(\sqrt{x}+2\right)}{\left(\sqrt{x}-1\right)\left(\sqrt{x}+1\right)^2}\right].\dfrac{\left(x-1\right)^2}{2}\)

\(=\dfrac{x-\sqrt{x}-2-\left(x+\sqrt{x}-2\right)}{\left(\sqrt{x}-1\right)\left(\sqrt{x}+1\right)^2}.\dfrac{\left(\sqrt{x}-1\right)^2\left(\sqrt{x}+1\right)^2}{2}\)

\(=\dfrac{-2\sqrt{x}\left(\sqrt{x}-1\right)}{2}=\sqrt{x}\left(1-\sqrt{x}\right)=\sqrt{x}-x\)

b, Thay \(x=7-4\sqrt{3}\) vào P, ta được:

\(P=\sqrt{7-4\sqrt{3}}-\left(7-4\sqrt{3}\right)\)

\(=\sqrt{\left(\sqrt{3}\right)^2-2.\sqrt{3}.2+2^2}+4\sqrt{3}-7\)

\(=\sqrt{\left(\sqrt{3}-2\right)^2}+4\sqrt{3}-7\)

\(=\left|\sqrt{3}-2\right|+4\sqrt{3}-7\)

\(=2-\sqrt{3}+4\sqrt{3}-7\) (vì \(\sqrt{3}< 2\))

\(=-5+3\sqrt{3}\)

$Toru$

a) \(P=\left(\dfrac{\sqrt{x}-2}{x-1}-\dfrac{\sqrt{x}+2}{x+2\sqrt{x}+1}\right).\dfrac{\left(1-x\right)^2}{2}\left(x\ge0,x\ne1\right)\\ =\left[\dfrac{\sqrt{x}-2}{\left(\sqrt{x}-1\right)\left(\sqrt{x}+1\right)}-\dfrac{\sqrt{x}+2}{\left(\sqrt{x}+1\right)^2}\right].\dfrac{\left(x-1\right)^2}{2}\\ =\dfrac{\left(\sqrt{x}-2\right)\left(\sqrt{x}+1\right)-\left(\sqrt{x}+2\right)\left(\sqrt{x}-1\right)}{\left(\sqrt{x}+1\right)^2\left(\sqrt{x}-1\right)}.\dfrac{\left(x-1\right)^2}{2}\\ \)

\(=\dfrac{x-2\sqrt{x}+\sqrt{x}-2-\left(x+2\sqrt{x}-\sqrt{x}-2\right)}{\left(x-1\right)\left(\sqrt{x}+1\right)}.\dfrac{\left(x-1\right)\left(\sqrt{x}-1\right)\left(\sqrt{x}+1\right)}{2}\\ =\left[x-\sqrt{x}-2-\left(x+\sqrt{x}-2\right)\right].\dfrac{\sqrt{x}-1}{2}\\ \)

\(=-2\sqrt{x}.\dfrac{\sqrt{x}-1}{2}\\ =-\sqrt{x}\left(\sqrt{x}-1\right)=-x+\sqrt{x}\)

b) \(x=7-4\sqrt{3}\left(TMDK\right)\)

\(\sqrt{x}=\sqrt{\left(2-\sqrt{3}\right)^2}=\left|2-\sqrt{3}\right|=2-\sqrt{3}\)

Thay vào biểu thức P, ta được:

\(P=-\left(7-4\sqrt{3}\right)+2-\sqrt{3}=-5+3\sqrt{3}\)

Phương trình không có dấu = thì sao giải đc bạn ? 

Đề sai kìa 

- Trích mẫu thử.

- Hòa tan từng mẫu thử vào nước.

+ Tan: Na₂O 

PT: \(Na_2O+H_2O\rightarrow2NaOH\)

+ Không tan: CuO, Al₂O₃, MgO. (1)

- Cho mẫu thử nhóm (1) pư với HCl rồi nhỏ NaOH thu được ở thí nghiệm trên vào.

+ Có tủa xanh: CuO

PT: \(CuO+2HCl\rightarrow CuCl_2+H_2O\)

\(CuCl_2+2NaOH\rightarrow2NaCl+Cu\left(OH\right)_{2\downarrow}\)

+ Có tủa keo trắng rồi tan trong NaOH dư: Al₂O₃

PT: \(Al_2O_3+6HCl\rightarrow2AlCl_3+3H_2O\)

\(2AlCl_3+6NaOH\rightarrow2Al\left(OH\right)_{3\downarrow}+6NaCl\)

\(Al\left(OH\right)_3+NaOH\rightarrow NaAlO_2+2H_2O\)

+ Có tủa trắng: MgO

PT: \(MgO+2HCl\rightarrow MgCl_2+H_2O\)

\(MgCl_2+2NaOH\rightarrow NaCl+Mg\left(OH\right)_{2\downarrow}\)

- Dán nhãn.