Gujarat BoardEnglish MediumSTD 9MathsPolynomials1 Mark
MCQ
If $\text{x}+\frac{1}{\text{x}}=3,$ then $\text{x}^6+\frac{1}{\text{x}^6}=$
A
927
B
414
C
364
D
322
✓
Answer
322 Solution: $\Big(\text{x}+\frac{1}{\text{x}}\Big)^2=\text{x}^2+\frac{1}{\text{x}^2}+2$ $\text{x}+\frac{1}{\text{x}}=3$ (given) $\Rightarrow\text{x}^2+\frac{1}{\text{x}^2}=(3)^2-2$ $\Rightarrow\text{x}^2+\frac{1}{\text{x}^2}=7\ ...(1)$ Cubing both side of equation (1). we have $\Big(\text{x}^2+\frac{1}{\text{x}^2}\Big)^3=(7)^3$ $\Rightarrow(\text{x}^2)^3+\Big(\frac{1}{\text{x}^2}\Big)^3+3(\text{x}^2)\frac{1}{\text{x}^2}\Big(\text{x}^2+\frac{1}{\text{x}^2}\Big)=7^3$ $\Rightarrow\text{x}^6+\frac{1}{\text{x}^6}+3(7)=7^3$ $\Rightarrow\text{x}^6+\frac{1}{\text{x}^6}=343-21$ $\Rightarrow\text{x}^6+\frac{1}{\text{x}^6}=322$ Hence, correct option is (d).
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