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

Câu nào bạn, nếu mà cả thì đăng tách ra đi :)

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.

\(cos^2x+\sqrt{3}sinx.cosx-1=0\)

\(\Leftrightarrow-sin^2x+\sqrt{3}sinx.cosx=0\)

\(\Leftrightarrow sinx\left(sinx-\sqrt{3}cosx\right)=0\)

\(\Leftrightarrow sinx\left(\dfrac{1}{2}sinx-\dfrac{\sqrt{3}}{2}cosx\right)=0\)

\(\Leftrightarrow sinx.sin\left(x-\dfrac{\pi}{3}\right)=0\)

\(\Leftrightarrow\left[{}\begin{matrix}sinx=0\\sin\left(x-\dfrac{\pi}{3}\right)=0\end{matrix}\right.\)

\(\Leftrightarrow\left[{}\begin{matrix}x=k\pi\\x=\dfrac{\pi}{3}+k\pi\end{matrix}\right.\)

7.

\(\sqrt{3}sinx-cosx=2\)

\(\Leftrightarrow\dfrac{\sqrt{3}}{2}sinx-\dfrac{1}{2}cosx=1\)

\(\Leftrightarrow sin\left(x-\dfrac{\pi}{3}\right)=1\)

\(\Leftrightarrow x-\dfrac{\pi}{3}=\dfrac{\pi}{2}+k2\pi\)

\(\Leftrightarrow x=\dfrac{5\pi}{6}+k2\pi\)

8.

\(sin4x+\sqrt{3}cos4x=2sinx\)

\(\Leftrightarrow\dfrac{1}{2}sin4x+\dfrac{\sqrt{3}}{2}cos4x=sinx\)

\(\Leftrightarrow sin\left(4x+\dfrac{\pi}{3}\right)=sinx\)

\(\Leftrightarrow\left[{}\begin{matrix}4x+\dfrac{\pi}{3}=x+k2\pi\\4x+\dfrac{\pi}{3}=\pi-x+k2\pi\end{matrix}\right.\)

\(\Leftrightarrow\left[{}\begin{matrix}x=-\dfrac{\pi}{9}+\dfrac{k2\pi}{3}\\x=\dfrac{2\pi}{15}+\dfrac{k2\pi}{5}\end{matrix}\right.\)

11.

\(sin^2x-4sinx.cosx+3cos^2x=0\)

\(\Leftrightarrow\left(sinx-cosx\right)\left(sinx-3cosx\right)=0\)

\(\Leftrightarrow\left[{}\begin{matrix}sinx-cosx=0\\sinx-3cosx=0\end{matrix}\right.\)

\(\Leftrightarrow\left[{}\begin{matrix}sinx=cosx\\sinx=3cosx\end{matrix}\right.\)

Với \(cosx=0\Rightarrow\) pt vô nghiệm

Với \(cosx\ne0\)

\(pt\Leftrightarrow\left[{}\begin{matrix}tanx=0\\tanx=3\end{matrix}\right.\)

\(\Leftrightarrow\left[{}\begin{matrix}x=\dfrac{\pi}{4}+k\pi\\x=arctan3+k\pi\end{matrix}\right.\)

12.

\(pt\Leftrightarrow\sqrt{3}tanx+1=0\)

\(\Leftrightarrow tanx=-\dfrac{\sqrt{3}}{3}\)

\(\Leftrightarrow x=-\dfrac{\pi}{6}+k\pi\)

17.

\(2tan^2x+5tanx+3=0\Leftrightarrow\left[{}\begin{matrix}tanx=-1\\tanx=-\dfrac{3}{2}\end{matrix}\right.\)

\(\Rightarrow\left[{}\begin{matrix}x=-\dfrac{\pi}{4}+k\pi\\x=-arctan\left(\dfrac{3}{2}\right)+k\pi\end{matrix}\right.\)

Nghiệm âm lớn nhất là \(x=-\dfrac{\pi}{4}\)

18.

Pt vô nghiệm khi:

\(m^2+4^2< 6^2\)

\(\Leftrightarrow m^2< 20\)

\(\Rightarrow-2\sqrt{5}< m< 2\sqrt{5}\)

\(ab=20\)

19.

Pt có nghiệm khi:

\(m^2+4\ge\left(2m-1\right)^2\)

\(\Leftrightarrow3m^2-4m-3\le0\)

Theo Viet: \(\left\{{}\begin{matrix}a+b=\dfrac{4}{3}\\ab=-1\end{matrix}\right.\)

\(\Rightarrow a^2+b^2=\left(a+b\right)^2-2ab=\dfrac{34}{9}\)

20.

\(cos\left(2x-60^0\right)=sin\left(x+60^0\right)=cos\left(30^0-x\right)\)

\(\Leftrightarrow\left[{}\begin{matrix}2x-60^0=30^0-x+k360^0\\2x-60^0=x-30^0+k360^0\end{matrix}\right.\)

\(\Leftrightarrow\left[{}\begin{matrix}x=30^0+k120^0\\x=30^0+k360^0\end{matrix}\right.\) \(\Leftrightarrow x=30^0+k120^0\)

21.

Đáp án C, kiểm tra điều kiện có nghiệm \(a^2+b^2\ge c^2\) thì chỉ có \(1^2+\left(-1\right)^2>1^2\) thỏa mãn

22.

\(cos^2x-cosx=0\Rightarrow\left[{}\begin{matrix}cosx=0\\cosx=1\end{matrix}\right.\)

\(\Leftrightarrow\left[{}\begin{matrix}x=\dfrac{\pi}{2}+k\pi\\x=k2\pi\end{matrix}\right.\)

\(\Rightarrow x=\dfrac{\pi}{2}\)

23.

\(2sin^2x+5sinx-3=0\Rightarrow\left[{}\begin{matrix}sinx=\dfrac{1}{2}\\sinx=-3\left(loại\right)\end{matrix}\right.\)

\(\Rightarrow\left[{}\begin{matrix}x=\dfrac{\pi}{6}+k2\pi\\x=\dfrac{5\pi}{5}+k2\pi\end{matrix}\right.\)

Nghiệm dương bé nhất là \(x=\dfrac{\pi}{6}\)

24.

\(1-cos^2x-3cosx-4=0\)

\(\Leftrightarrow cos^2x+3cosx+3=0\)

Pt bậc 2 nói trên vô nghiệm nên pt đã cho vô nghiệm

25.

\(\Leftrightarrow\left(tanx+1\right)^2=0\)

\(\Leftrightarrow tanx=-1\)

\(\Rightarrow x=-\dfrac{\pi}{4}+k\pi\)

26.

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

\(\Leftrightarrow sin\left(x+\dfrac{\pi}{3}\right)=1\)

\(\Leftrightarrow x+\dfrac{\pi}{3}=\dfrac{\pi}{2}+k2\pi\)

\(\Leftrightarrow x=\dfrac{\pi}{6}+k2\pi\)

28.

\(\Leftrightarrow sinx-\left(1-2sin^2x\right)-2=0\)

\(\Leftrightarrow2sin^2x+sinx-3=0\)

\(\Leftrightarrow\left[{}\begin{matrix}sinx=1\\sinx=-\dfrac{3}{2}\left(loại\right)\end{matrix}\right.\)

\(\Leftrightarrow x=\dfrac{\pi}{2}+k2\pi\)

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