If \(x-\displaystyle\frac{1}{x}=4\), consider the following statements:
I. \(x^2+\displaystyle\frac{1}{x^2}=16\)
II. \(x^3-\displaystyle\frac{1}{x^3}=52\)
Which of the above statements is/are correct?
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → Neither I nor II
For a difference, the signs change: \(x^2+\frac{1}{x^2}=\left(x-\frac{1}{x}\right)^2+2\) and \(x^3-\frac{1}{x^3}=\left(x-\frac{1}{x}\right)^3+3\left(x-\frac{1}{x}\right)\).
Statement I — Incorrect.
\(x^2+\frac{1}{x^2}=4^2+2=16+2=18\), not 16
Statement II — Incorrect.
\(x^3-\frac{1}{x^3}=4^3+3\times4=64+12=76\), not 52
Neither statement is correct.