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11 c)
\(a^2+2\ge2\sqrt{a^2+1}\Leftrightarrow a^2+1-2\sqrt{a^2+1}+1\ge0\Leftrightarrow\left(\sqrt{a^2+1}-1\right)^2\ge0\) (luôn đúng)
12 a) Có a+b+c=1\(\Rightarrow\) (1-a)(1-b)(1-c)= (b+c)(a+c)(a+b) (*)
áp dụng BĐT cô-si: \(\left(b+c\right)\left(a+c\right)\left(a+b\right)\ge2\sqrt{bc}2\sqrt{ac}2\sqrt{ab}=8\sqrt{\left(abc\right)2}=8abc\) ( luôn đúng với mọi a,b,c ko âm )
b) áp dụng BĐT cô-si: \(c\left(a+b\right)\le\dfrac{\left(a+b+c\right)^2}{4}=\dfrac{1}{4}\)
Tương tự: \(a\left(b+c\right)\le\dfrac{1}{4};b\left(c+a\right)\le\dfrac{1}{4}\)
\(\Rightarrow abc\left(a+b\right)\left(b+c\right)\left(c+a\right)\le\dfrac{1}{4}\dfrac{1}{4}\dfrac{1}{4}=\dfrac{1}{64}\)
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12.1
Giả sử \(G=\left(m;2m-2\right)\left(m\in R\right)\)
\(\Rightarrow\left\{{}\begin{matrix}x_H=2x_E-x_G=6-m\\y_H=2y_E-y_G=2-2m\end{matrix}\right.\)
\(\Rightarrow H=\left(6-m;2-2m\right)\)
Mà \(H\in d_2\Rightarrow6-m+2-2m+3=0\Leftrightarrow m=\dfrac{11}{3}\)
\(\Rightarrow G=\left(\dfrac{11}{3};\dfrac{16}{3}\right)\)
\(\Rightarrow\Delta:8x-y-24=0\)
12.2
Giả sử \(A=\left(m;-m-1\right)\left(m\in R\right)\)
Ta có: \(\vec{AM}=\dfrac{1}{3}\vec{MB}\)
\(\Rightarrow\left\{{}\begin{matrix}x_M-x_A=\dfrac{1}{3}\left(x_B-x_M\right)\\y_M-y_A=\dfrac{1}{3}\left(y_B-y_M\right)\end{matrix}\right.\)
\(\Leftrightarrow\left\{{}\begin{matrix}1-m=\dfrac{1}{3}\left(x_B-1\right)\\m+1=\dfrac{1}{3}.y_B\end{matrix}\right.\)
\(\Leftrightarrow\left\{{}\begin{matrix}x_B=4-3m\\y_B=3m+3\end{matrix}\right.\)
\(\Rightarrow B=\left(4-3m;3m+3\right)\)
Mà \(B\in d_2\Rightarrow4-3m-2\left(3m+3\right)+2=0\Leftrightarrow m=0\)
\(\Rightarrow A=\left(0;-1\right)\)
\(\Rightarrow d:x-y-1=0\)
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`sin3x sinx+sin(x-π/3) cos (x-π/6)=0`
`<=> 1/2 (cos2x - cos4x) + 1/2(-sin π/6 + sin (2x-π/2)=0`
`<=> cos2x-cos4x-1/2+ sin(2x-π/2)=0`
`<=>cos2x-cos4x-1/2+ sin2x .cos π/2 - cos2x. sinπ/2=0`
`<=> cos2x - cos4x - cos2x = 1/2`
`<=> cos4x = cos(2π)/3`
`<=>` \(\left[{}\begin{matrix}4x=\dfrac{2\text{π}}{3}+k2\text{π}\\4x=\dfrac{-2\text{π}}{3}+k2\text{π}\end{matrix}\right.\)
`<=>` \(\left[{}\begin{matrix}x=\dfrac{\text{π}}{6}+k\dfrac{\text{π}}{2}\\x=-\dfrac{\text{π}}{6}+k\dfrac{\text{π}}{2}\end{matrix}\right.\)
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5a.
Pt có 2 nghiệm pb lhi:
\(\Delta=9+4m>0\Leftrightarrow m>-\dfrac{9}{4}\)
b. Phương trình có 2 nghiệm khi:
\(\Delta=1+4\left(-2m+1\right)\ge0\Rightarrow m\le\dfrac{5}{8}\)
6.
a. Pt có 2 nghiệm khi:
\(\Delta'=1-\left(m+2\right)\ge0\Leftrightarrow m\le-1\)
6b
Khi \(m\le-1\), theo hệ thức Viet ta có: \(\left\{{}\begin{matrix}x_1+x_2=2\\x_1x_2=m+2\end{matrix}\right.\)
\(x^2_1+x^2_2=10\Leftrightarrow\left(x_1+x_2\right)^2-2x_1x_2=10\)
\(\Leftrightarrow4-2\left(m+2\right)=10\)
\(\Leftrightarrow m=-5\)
B.
\(x^2_1+x_2^2+4x_1x_2=0\)
\(\Leftrightarrow\left(x_1+x_2\right)^2+2x_1x_2=0\)
\(\Leftrightarrow4+2\left(m+2\right)=0\)
\(\Leftrightarrow m=-4\)