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Tuesday, May 20, 2014

Maths Exercise #39: VNMO 30-4-2011 For Grade 11, P1

ដោះស្រាយសមីការខាងក្រោមលើសំនុំចំនួនពិតៈ
\sqrt{1+\sqrt{1-x^2}}[\sqrt{(1+x)^3}-\sqrt{(1-x)^3}]=2+\sqrt{1-x^2}

ដំណោះស្រាយ
លក្ខខណ្ឌៈ  -1\le x\le 1
តាង x=\cos t, 0\le t\le \pi
យើងបាន សមីការៈ \sqrt{1+\sin t}[\sqrt{(1+\cos t)^3}-\sqrt{(1-cos t)^3}]=2+\sin t
\Leftrightarrow \sqrt{(cos\frac{t}{2}+\sin\frac{t}{2})^2}[\cos^3\frac{t}{2}-\sin^3\frac{t}{2}]2\sqrt{2}=2+\sin t
\Leftrightarrow (\cos^2\frac{t}{2}-\sin^2\frac{t}{2})(1+\cos\frac{t}{2}\sin\frac{t}{2})2\sqrt{2}=2+\sin t
\Leftrightarrow \cos t(2+\sin t)\sqrt{2}=2+\sin t \Leftrightarrow (2+\sin t)(\sqrt{2}\cos t-1)=0
\Leftrightarrow \sqrt{2}\cos t-1=0\Leftrightarrow \cos t=\frac{1}{\sqrt{2}}\Leftrightarrow x=\frac{1}{\sqrt{2}}

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