| Statement-1: If \[{{x}^{2}}+x+1=0\] then the value of \[{{\left( x+\frac{1}{x} \right)}^{2}}+{{\left( {{x}^{2}}+\frac{1}{{{x}^{2}}} \right)}^{2}}+....+{{\left( {{x}^{27}}+\frac{1}{{{x}^{27}}} \right)}^{2}}\]is 54. |
| Statement-2: \[\omega ,{{\omega }^{2}}\] are the roots of equation\[{{x}^{2}}+x+1=0\] |
A) Statement-1 and 2 are true and Statement-2 is correct explanation of Statement-1.
B) Statement-1 and 2 are true and Statement-2 is not correct explanation of Statement-1.
C) Statement-1 is true, statement-2 is false
D) Statement-1 is false, Statement-2 is true.
Correct Answer: A
Solution :
\[x+\frac{1}{x}=-1,{{x}^{2}}+\frac{1}{{{x}^{2}}}=-1,{{x}^{3}}+\frac{1}{{{x}^{3}}}=2,{{x}^{4}}+\frac{1}{{{x}^{4}}}\] \[=x+\frac{1}{x},{{x}^{5}}+\frac{1}{{{x}^{5}}}=-1,{{x}^{6}}+\frac{1}{{{x}^{6}}}=2\,\text{etc}\text{.}\] \[\Rightarrow \]\[{{\left( x+\frac{1}{x} \right)}^{2}}+{{\left( {{x}^{2}}+\frac{1}{{{x}^{2}}} \right)}^{3}}+{{\left( {{x}^{3}}+\frac{1}{{{x}^{3}}} \right)}^{2}}+{{\left( {{x}^{4}}+\frac{1}{{{x}^{4}}} \right)}^{2}}\] \[+{{\left( {{x}^{5}}+\frac{1}{{{x}^{5}}} \right)}^{2}}+{{\left( {{x}^{6}}+\frac{1}{{{x}^{6}}} \right)}^{2}}+{{\left( {{x}^{7}}+\frac{1}{{{x}^{7}}} \right)}^{2}}\] \[+.......+{{\left( {{({{x}^{3}})}^{9}}+\frac{1}{{{({{x}^{3}})}^{9}}} \right)}^{2}}\] \[=(1+1+4)+(1+1+4)+(1+1+4)\]+??9 times\[=6\times 9=54.\]
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