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Câu 39 trang 46 SGK Đại số và Giải tích 11 Nâng cao

Chứng minh rằng các phương trình sau đây vô nghiệm :
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Lựa chọn câu để xem lời giải nhanh hơn
Chứng minh rằng các phương trình sau đây vô nghiệm :

LG a

\(\sin x – 2\cos x = 3\)

Lời giải chi tiết:

\(\sin x - 2\cos x = 3 \) \(\Leftrightarrow {1 \over {\sqrt 5 }}\sin x - {2 \over {\sqrt 5 }}\cos x = {3 \over {\sqrt 5 }}\) \( \Leftrightarrow \sin \left( {x - \alpha } \right) = {3 \over {\sqrt 5 }}\) trong đó \(α\) là số thỏa mãn \(\cos \alpha = {1 \over {\sqrt 5 }}\,\text{ và }\,\sin \alpha = {2 \over {\sqrt 5 }}.\) Phương trình cuối cùng vô nghiệm do \({3 \over {\sqrt 5 }} > 1,\) nên phương trình đã cho vô nghiệm.

LG b

\(5\sin2x + \sin x + \cos x + 6 = 0\)

Phương pháp giải:

Đặt \(\sin x + \cos x = t\)

Lời giải chi tiết:

Đặt \(t = \sin x + \cos x\) ta có:

\(\begin{array}{l}
{t^2} = {\left( {\sin x + \cos x} \right)^2}\\
= {\sin ^2}x + 2\sin x\cos x + {\cos ^2}x\\
= 1 + \sin 2x\\
\Rightarrow \sin 2x = {t^2} - 1 
\end{array}\)

Lại có: \({t^2} = 1 + \sin 2x \le 2\)\( \Rightarrow  - \sqrt 2  \le t \le \sqrt 2 \) Thay vào pt đã cho được: \(5.\left( {{t^2} - 1} \right) + t + 6 = 0\) \( \Leftrightarrow 5{t^2} + t + 1 = 0\) Phương trình này vô nghiệm nên phương trình đã cho vô nghiệm.

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