1: \(\frac{2x+6}{3x^2-x}:\frac{x^2+3x}{1-3x}\)

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

\(=\frac{2}{x}\cdot\frac{-\left(3x-1\right)}{x\left(3x-1\right)}=\frac{-2}{x^2}\)

2: \(\frac{x}{x-2y}+\frac{x}{x+2y}+\frac{4xy}{4y^2-x^2}\)

\(=\frac{x}{x-2y}+\frac{x}{x+2y}-\frac{4xy}{\left(x-2y\right)\left(x+2y\right)}\)

\(=\frac{x\left(x+2y\right)+x\left(x-2y\right)-4xy}{\left(x-2y\right)\left(x+2y\right)}=\frac{2x^2-4xy}{\left(x-2y\right)\left(x+2y\right)}\)

\(=\frac{2x\left(x-2y\right)}{\left(x-2y\right)\left(x+2y\right)}=\frac{2x}{x+2y}\)

3: \(\frac{1}{3x-2}-\frac{1}{3x+2}-\frac{3x-6}{4-9x^2}\)

\(=\frac{1}{3x-2}-\frac{1}{3x+2}+\frac{3x-6}{\left(3x-2\right)\left(3x+2\right)}\)

\(=\frac{3x+2-\left(3x-2\right)+3x-6}{\left(3x-2\right)\left(3x+2\right)}=\frac{3x+2-3x+2+3x-6}{\left(3x-2\right)\left(3x+2\right)}\)

\(=\frac{3x-2}{\left(3x-2\right)\left(3x+2\right)}=\frac{1}{3x+2}\)

4: \(\frac{x+3}{x+1}+\frac{2x-1}{x-1}+\frac{x+5}{x^2-1}\)

\(=\frac{x+3}{x+1}+\frac{2x-1}{x-1}+\frac{x+5}{\left(x-1\right)\left(x+1\right)}\)

\(=\frac{\left(x+3\right)\left(x-1\right)+\left(2x-1\right)\left(x+1\right)+x+5}{\left(x-1\right)\left(x+1\right)}\)

\(=\frac{x^2+2x-3+2x^2+2x-x-1+x+5}{\left(x-1\right)\left(x+1\right)}=\frac{3x^2+4x+1}{\left(x-1\right)\left(x+1\right)}\)

\(=\frac{\left(3x+1\right)\left(x+1\right)}{\left(x-1\right)\left(x+1\right)}=\frac{3x+1}{x-1}\)

\(\left\{{}\begin{matrix}x=\dfrac{5}{9}y\\x=\dfrac{10}{21}z\\2x=3y+z=50\end{matrix}\right.\)\(\Rightarrow2x-\dfrac{27}{5}+\dfrac{21}{10}x=50\)

\(\left\{{}\begin{matrix}x=\dfrac{500}{15}\\y=-\dfrac{900}{13}\\-\dfrac{1050}{13}\end{matrix}\right.\)

b: Ta có: \(\dfrac{x}{-3}=\dfrac{y}{7}\)

nên \(\dfrac{x}{6}=\dfrac{y}{-14}\left(1\right)\)

Ta có: \(\dfrac{y}{-2}=\dfrac{z}{5}\)

nên \(\dfrac{y}{-14}=\dfrac{z}{35}\left(2\right)\)

Từ \(\left(1\right),\left(2\right)\) suy ra \(\dfrac{x}{6}=\dfrac{y}{-14}=\dfrac{z}{35}\)

hay \(\dfrac{-2x}{12}=\dfrac{4y}{-56}=\dfrac{5z}{175}\)

Áp dụng tính chất của dãy tỉ số bằng nhau, ta được:

\(\dfrac{-2x}{12}=\dfrac{4y}{-56}=\dfrac{5z}{175}=\dfrac{-2x-4y+5z}{12+56+175}=\dfrac{146}{243}\)

Do đó: \(\left\{{}\begin{matrix}x=\dfrac{292}{81}\\y=-\dfrac{2044}{243}\\z=\dfrac{5110}{243}\end{matrix}\right.\)

\(\cos^225^0-\cos^235^0+\cos^245^0-\cos^255^0+\cos^265^0\)

\(=1-1+\dfrac{1}{2}=\dfrac{1}{2}\)

Câu 28:

Cho $25{,}6$ gam hỗn hợp gồm ancol etylic và phenol tác dụng với Na dư thu được $4{,}48$ lít khí $H_2$ (đktc).

Ta có:

$n_{H_2}=\dfrac{4{,}48}{22{,}4}=0{,}2$ mol.

Phản ứng: $2ROH + 2Na \rightarrow 2RONa + H_2$

=> $n_{ROH}=2n_{H_2}=0{,}4$ mol.

Gọi số mol ancol etylic là $x$, phenol là $y$.

Ta có: $x+y=0{,}4$.

Khối lượng hỗn hợp:

$46x+94y=25{,}6$.

Giải hệ: $x=0{,}3$, $y=0{,}1$.

Phần trăm khối lượng:

$\%m_{C_2H_5OH}=\dfrac{0{,}3\cdot46}{25{,}6}\cdot100\%=53{,}9\%$.

$\%m_{C_6H_5OH}=\dfrac{0{,}1\cdot94}{25{,}6}\cdot100\%=36{,}7\%$.

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