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Class-8…অনুশীলনী-৪.১(৮ নং)

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                                  \displaystyle a-\frac{1}{a} =m হলে দেখাও যে  \displaystyle a^{4} +\frac{1}{a^{4}} =m^{4} +4m^{2} +2
দেওয়া আছে,,

 \displaystyle a-\frac{1}{a} =m 
এখন,
a\displaystyle ^{4}+\displaystyle \frac{1}{a^{4}}
=\displaystyle ( a^{2})\displaystyle ^{2}+\displaystyle \left(\frac{1}{a^{2}}\right)^{2}
=\displaystyle \left( a^{2} +\frac{1}{a^{2}}\right)^{2} -2.a^{2} .\frac{1}{a^{2}}
=\displaystyle \left\{\left( a-\frac{1}{a}\right)^{2} +2.a.\frac{1}{a}\right\}^{2} -2
=\displaystyle \left\{( m)^{2} +2\right\}^{2} -2
=\displaystyle \left( m^{2} +2\right)^{2}-2
=\displaystyle \left( m^{2}\right)^{2} +2.m^{2} .2+( 2)^{2} -2
=m\displaystyle ^{4} +4m^{2}+4-2
=m\displaystyle ^{4} +4m^{2}+2     (  প্রমানিত  )
                       
class-eight_math

দেওয়া আছে,,
\displaystyle a-\frac{1}{a} =m
দেখাতে হবে যে,, a\displaystyle ^{4} +\frac{1}{a^{4}} =m^{4} +4m^{2} +2
এখন,,
\displaystyle m^{4} +4m^{2} +2
=\displaystyle m^{4} +4m^{2} +2+2-2 [এখানে ২ যোগ করে আবার ২ বাদ দেওয়া হয়েছে। a\displaystyle ^{2}+2ab+\displaystyle b^{2} এরআকার ধারন করার জন্য ]
=\displaystyle m^{4} +4m^{2} +4-2
=\displaystyle \left( m^{2}\right)^{2} +2.m^{2} .2+( 2)^{2} -2 আমরা জানি (a+b)\displaystyle ^{2}=\displaystyle a^{2} +2ab+b^{2}
=\displaystyle \left( m^{2} +2\right)^{2} -2
=\displaystyle \left{\left( a-\frac{1}{a}\right)^{2} +2\right}^{2} -2 [ মান বসিয়ে পাই ]
=\displaystyle \left{a^{2} -2.a.\frac{1}{a} +\left(\frac{1}{a}\right)^{2} +2\right}^{2} -2
=\displaystyle \left( a^{2} -2+\frac{1}{a^{2}} +2\right)^{2} -2
=\displaystyle \left( a^{2} +\frac{1}{a^{2}}\right)^{2} -2
=\displaystyle \left( a^{2}\right)^{2} +2.a^{2} .\frac{1}{a^{2}} +\left(\frac{1}{a^{2}}\right)^{2} -2
=\displaystyle a^{4} +2+\frac{1}{a^{4}} -2
=\displaystyle a^{4} +\frac{1}{a^{4}}
সুতরাং .. a\displaystyle ^{4} +\frac{1}{a^{4}} =m^{4} +4m^{2} +2

Post Author: showrob

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