If \(x+\displaystyle\frac{1}{x}=7\), consider the following statements:
I. \(x^2+\displaystyle\frac{1}{x^2}=47\)
II. \(x^3+\displaystyle\frac{1}{x^3}=336\)
Which of the above statements is/are correct?
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → Only I
Use the identities \(x^2+\frac{1}{x^2}=\left(x+\frac{1}{x}\right)^2-2\) and \(x^3+\frac{1}{x^3}=\left(x+\frac{1}{x}\right)^3-3\left(x+\frac{1}{x}\right)\), and test each statement.
Statement I — Correct.
\(x^2+\frac{1}{x^2}=7^2-2=49-2=47\)
Statement II — Incorrect.
\(x^3+\frac{1}{x^3}=7^3-3\times7=343-21=322\), not 336
Only statement I is correct.