SATHEE: Quantitative Aptitude Ques 1591

Quantitative Aptitude Ques 1591

Question: If $ a+\frac{1}{a}=\sqrt{3}, $ then the value of $ a^{6}-\frac{1}{a^{6}}+2 $ will be

Options:

A) $ 3\sqrt{3} $

B) $ 5 $

C) $ 1 $

D) $ 2 $

Show Answer

Answer:

Correct Answer: D

Solution:

$ a+\frac{1}{a}=\sqrt{3} $ (i) On squaring both sides, we get

$ \Rightarrow $ $ a^{2}+\frac{1}{a^{2}}+2=3 $

$ \Rightarrow $ $ a^{2}+\frac{1}{a^{2}}=1 $ (ii) Now, multiplying Eqs. (i) and (Ii), we get $ ( a+\frac{1}{a} )( a^{2}+\frac{1}{a^{2}} )=\sqrt{3} $

$ \Rightarrow $ $ a^{3}+\frac{a}{a^{2}}+\frac{a^{2}}{a}+\frac{1}{a^{3}}=\sqrt{3} $

$ \Rightarrow $ $ a^{3}+\frac{1}{a^{3}}+( \frac{1}{a}+a )=\sqrt{3} $

$ \Rightarrow $ $ a^{3}+\frac{1}{a^{3}}+\sqrt{3}=\sqrt{3} $ [from Eq. (i)]

$ \Rightarrow $ $ a^{3}+\frac{1}{a^{3}}=0 $
$ \Rightarrow $ $ a^{6}=-1 $

$ \therefore $ $ a^{6}=\frac{1}{a^{6}}+2={{(-1)}^{6}}-\frac{1}{{{(-1)}^{6}}}+2 $ $ =1-1+2=2 $

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