• # question_answer DIRECTION (Qs. 84): Each of these questions contains two statements: Statement-1 (Assertion) and Statement-2 (Reason). Choose the correct answer (ONLY ONE option is correct) from the following- Statement-1 : If ${{x}^{2}}+x+1=0$ then the value of${{\left( x+\frac{1}{x} \right)}^{2}}+{{\left( {{x}^{2}}+\frac{1}{{{x}^{2}}} \right)}^{2}}+...+\left( {{x}^{27}}+\frac{1}{{{x}^{27}}} \right)\,\,\text{is}\,\,54.$ Statement-2: $\omega ,\,\,\,{{\omega }^{2}}$ are the roots of given equation and$x+\frac{1}{x}=-1,\,\,{{x}^{2}}+\frac{1}{{{x}^{2}}}=-1,\,\,{{x}^{3}}+\frac{1}{{{x}^{3}}}=2$ A)  Statement-1 is false, Statement-2 is true.B)  Statement-1 is true, Statement-2 is true; Statement-2 is a correct explanation for Statement-1.C)  Statement-1 is true, Statement-2 is true; Statement-2 is not a correct explanation for Statement-1.D)  Statement-1 is true, Statement-2 is false.

$x+\frac{1}{x}=-1,\,\,{{x}^{2}}+\frac{1}{{{x}^{2}}}=-1$,                 ${{x}^{3}}+\frac{1}{{{x}^{3}}}=2,\,\,{{x}^{4}}+\frac{1}{{{x}^{4}}}=x+\frac{1}{x},$                 ${{x}^{5}}+\frac{1}{{{x}^{5}}}=-1,\,\,{{x}^{6}}+\frac{1}{{{x}^{6}}}=2,\,\,\,etc.$ $\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