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3.
\(y=\dfrac{1-sin^24x}{5}=\dfrac{cos^24x}{5}\)
\(cos4x\in\left[-1;1\right]\Rightarrow cos^24x\in\left[0;1\right]\Rightarrow y\in\left[0;\dfrac{1}{5}\right]\Rightarrow\left\{{}\begin{matrix}y_{min}=0\\y_{max}=\dfrac{1}{5}\end{matrix}\right.\)
6.
\(y=sinx+cosx+2=\sqrt{2}sin\left(x+\dfrac{\pi}{4}\right)+2\)
\(sin\left(x+\dfrac{\pi}{4}\right)\in\left[-1;1\right]\Rightarrow y=\sqrt{2}sin\left(x+\dfrac{\pi}{4}\right)+2\in\left[-\sqrt{2}+2;\sqrt{2}+2\right]\)
\(\Rightarrow y_{min}=-\sqrt{2}+2\)
\(y_{max}=\sqrt{2}+2\)
12.
\(y=\sqrt{2}sin\left(2x+\dfrac{\pi}{4}\right)\le\sqrt[]{2}\)
\(\Rightarrow M=\sqrt{2}\)
13.
Pt có nghiệm khi:
\(5^2+m^2\ge\left(m+1\right)^2\)
\(\Leftrightarrow2m\le24\)
\(\Rightarrow m\le12\)
14.
\(\Leftrightarrow\left[{}\begin{matrix}cosx=1\\cosx=-\dfrac{5}{3}\left(loại\right)\end{matrix}\right.\)
\(\Leftrightarrow x=k2\pi\)
15.
\(\Leftrightarrow\left[{}\begin{matrix}tanx=-1\\tanx=3\end{matrix}\right.\)
\(\Rightarrow\left[{}\begin{matrix}x=-\dfrac{\pi}{4}+k\pi\\x=arctan\left(3\right)+k\pi\end{matrix}\right.\)
Đáp án A
16.
\(\dfrac{\sqrt{3}}{2}sinx-\dfrac{1}{2}cosx=\dfrac{1}{2}\)
\(\Leftrightarrow sin\left(x-\dfrac{\pi}{6}\right)=\dfrac{1}{2}\)
\(\Leftrightarrow\left[{}\begin{matrix}x-\dfrac{\pi}{6}=\dfrac{\pi}{6}+k2\pi\\x-\dfrac{\pi}{6}=\dfrac{5\pi}{6}+k2\pi\end{matrix}\right.\)
\(\Leftrightarrow\left[{}\begin{matrix}x=\dfrac{\pi}{3}+k2\pi\\x=\pi+k2\pi\end{matrix}\right.\)
\(\left[{}\begin{matrix}2\pi\le\dfrac{\pi}{3}+k2\pi\le2018\pi\\2\pi\le\pi+k2\pi\le2018\pi\end{matrix}\right.\) \(\Rightarrow\left[{}\begin{matrix}1\le k\le1008\\1\le k\le1008\end{matrix}\right.\)
Có \(1008+1008=2016\) nghiệm
1.
\(sin^2x-4sinx.cosx+3cos^2x=0\)
\(\Rightarrow\dfrac{sin^2x}{cos^2x}-\dfrac{4sinx}{cosx}+\dfrac{3cos^2x}{cos^2x}=0\)
\(\Rightarrow tan^2x-4tanx+3=0\)
2.
\(\Leftrightarrow\dfrac{1}{2}cos2x+\dfrac{\sqrt{3}}{2}sin2x=\dfrac{1}{2}\)
\(\Leftrightarrow cos\left(2x-\dfrac{\pi}{3}\right)=\dfrac{1}{2}\)
3.
\(\Leftrightarrow2^2+m^2\ge1\)
\(\Leftrightarrow m^2\ge-3\) (luôn đúng)
Pt có nghiệm với mọi m (đề bài sai)
4.
\(\Leftrightarrow\dfrac{1}{2}sinx-\dfrac{\sqrt{3}}{2}cosx=1\)
\(\Leftrightarrow sin\left(x-\dfrac{\pi}{3}\right)=1\)
\(\Leftrightarrow x-\dfrac{\pi}{3}=\dfrac{\pi}{2}+k2\pi\)
\(\Leftrightarrow x=\dfrac{5\pi}{6}+k2\pi\)
6.
ĐKXĐ: \(cosx\ne0\)
Nhân 2 vế với \(cos^2x\)
\(sin^2x-4cosx+5cos^2x=0\)
\(\Leftrightarrow1-cos^2x-4cosx+5cos^2x=0\)
\(\Leftrightarrow\left(2cosx-1\right)^2=0\)
\(\Leftrightarrow cosx=\dfrac{1}{2}\Rightarrow x=\pm\dfrac{\pi}{3}+k2\pi\)
6.
\(cos^2x+\sqrt{3}sinx.cosx-1=0\)
\(\Leftrightarrow-sin^2x+\sqrt{3}sinx.cosx=0\)
\(\Leftrightarrow sinx\left(sinx-\sqrt{3}cosx\right)=0\)
\(\Leftrightarrow sinx\left(\dfrac{1}{2}sinx-\dfrac{\sqrt{3}}{2}cosx\right)=0\)
\(\Leftrightarrow sinx.sin\left(x-\dfrac{\pi}{3}\right)=0\)
\(\Leftrightarrow\left[{}\begin{matrix}sinx=0\\sin\left(x-\dfrac{\pi}{3}\right)=0\end{matrix}\right.\)
\(\Leftrightarrow\left[{}\begin{matrix}x=k\pi\\x=\dfrac{\pi}{3}+k\pi\end{matrix}\right.\)