A) \[(12,{{(20)}^{4}})\]
B) \[(8,5{{(10)}^{4}})\]
C) \[(24,{{(10)}^{8}})\]
D) \[(12,8{{(10)}^{4}})\]
Correct Answer: D
Solution :
\[{{\left[ \frac{1}{\sqrt{5{{x}^{3}}+1}-\sqrt{5{{x}^{3}}-1}} \right]}^{8}}+{{\left[ \frac{1}{\sqrt{5{{x}^{3}}+1}+\sqrt{5{{x}^{3}}-1}} \right]}^{8}}\]Rationalise the polynomial, \[{{\left[ \frac{1}{\sqrt{5{{x}^{3}}+1}-\sqrt{5{{x}^{3}}-1}}\times \frac{\sqrt{5{{x}^{3}}+1}+\sqrt{5{{x}^{3}}-1}}{\sqrt{5{{x}^{3}}+1}+\sqrt{5{{x}^{3}}-1}} \right]}^{8}}+\] \[{{\left[ \frac{1}{\sqrt{5{{x}^{3}}+1}-\sqrt{5{{x}^{3}}-1}}\times \frac{\sqrt{5{{x}^{3}}+1}-\sqrt{5{{x}^{3}}-1}}{\sqrt{5{{x}^{3}}+1}-\sqrt{5{{x}^{3}}-1}} \right]}^{8}}\] \[={{\left[ \frac{\sqrt{5{{x}^{3}}+1}+\sqrt{5{{x}^{3}}-1}}{(5{{x}^{3}}+1)-(5{{x}^{3}}-1)} \right]}^{8}}+{{\left[ \frac{\sqrt{5{{x}^{3}}+1-\sqrt{5{{x}^{3}}-1}}}{(5{{x}^{3}}+1)-(5{{x}^{3}}-1)} \right]}^{8}}\] \[=\frac{1}{{{2}^{8}}}\left[ {{[\sqrt{5{{x}^{3}}+1}+\sqrt{5{{x}^{3}}-1}]}^{8}}+{{(\sqrt{5{{x}^{3}}+1}-\sqrt{5{{x}^{3}}-1})}^{8}} \right]\]\[=\frac{1}{{{2}^{8}}}\left[ {{(a+b)}^{8}}+{{(a-b)}^{8}} \right]\] we know, \[{{(a+b)}^{8}}{{=}^{8}}{{C}_{0}}{{a}^{8}}{{b}^{0}}{{+}^{8}}{{C}_{1}}{{a}^{7}}{{b}^{1}}+......{{+}^{8}}{{C}_{8}}{{a}^{0}}{{b}^{8}}\] \[{{(a-b)}^{8}}{{=}^{8}}{{C}_{0}}{{a}^{8}}{{b}^{0}}{{-}^{8}}{{C}_{1}}{{a}^{7}}{{b}^{1}}+......{{+}^{8}}{{C}_{8}}{{a}^{0}}{{b}^{8}}\] \[{{(a+b)}^{8}}+{{(a-b)}^{8}}=8{{C}_{0}}{{a}^{8}}{{b}^{0}}{{+}^{8}}{{C}_{2}}{{a}^{6}}{{b}^{2}}{{+}^{8}}{{C}_{4}}{{a}^{4}}{{b}^{4}}{{+}^{8}}{{C}_{6}}{{a}^{2}}{{b}^{6}}{{+}^{8}}{{C}_{8}}{{a}^{0}}{{b}^{8}}\] Thus, our expression becomes, \[=\frac{1}{{{2}^{8}}}\left[ \begin{align} & ^{8}{{C}_{0}}{{(\sqrt{5{{x}^{3}}+1})}^{8}}{{+}^{8}}{{C}_{2}}{{(\sqrt{5{{x}^{3}}+1})}^{6}}{{(\sqrt{5{{x}^{3}}-1})}^{2}}+ \\ & ^{8}{{C}_{4}}{{(\sqrt{5{{x}^{3}}+1})}^{4}}{{(\sqrt{5{{x}^{3}}-1})}^{4}}{{+}^{8}}{{C}_{6}}{{(\sqrt{5{{x}^{3}}+1})}^{2}}{{(\sqrt{5{{x}^{3}}-1})}^{6}}+ \\ & ^{8}{{C}_{8}}{{(\sqrt{5{{x}^{3}}-1})}^{8}} \\ \end{align} \right]\] \[=\frac{1}{{{2}^{8}}}\left[ \begin{align} & ^{8}{{C}_{0}}{{(5{{x}^{3}}+1)}^{4}}{{+}^{8}}{{C}_{2}}{{(5{{x}^{3}}+1)}^{3}}{{(5{{x}^{3}}-1)}^{2}}+ \\ & ^{8}{{C}_{4}}{{(5{{x}^{3}}+1)}^{2}}{{(5{{x}^{3}}-1)}^{2}}{{+}^{8}}{{C}_{6}}(5{{x}^{3}}+1){{(5{{x}^{3}}-1)}^{3}}+ \\ & ^{8}{{C}_{8}}{{(5{{x}^{3}}-1)}^{4}} \\ \end{align} \right]\] From this, we can clearly see that the degree of polynomial is 12, Hence \[h=12\] which means option (2) & (3) are incorrect. now, for \[m\], let collect the coefficients of \[{{x}^{12}}\]from each term. coefficient of \[{{x}^{12}}=\left[ ^{8}{{C}_{0}}{{5}^{4}}{{+}^{8}}{{C}_{2}}{{5}^{4}}{{+}^{8}}{{C}_{4}}{{5}^{4}}{{+}^{8}}{{C}_{6}}{{5}^{4}}{{+}^{8}}{{C}_{8}}{{5}^{4}} \right]\] \[=\left[ {{5}^{4}}\times \frac{{{2}^{8}}}{2} \right]\] \[={{5}^{4}}\times {{2}^{4}}\times \frac{{{2}^{4}}}{2}\] \[={{10}^{4}}\times {{2}^{3}}\] \[=8({{10}^{4}})\] Hence, option D is correct.
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