If the sum of a real number and its reciprocal is equal to 5, then what is the sum of the squares of the number and its reciprocal ?
25
24
23
22
Answer is Wrong!
Answer is Right!
This question was previously asked in
UPSC CISF-AC-EXE – 2021
We are given that the sum of the number and its reciprocal is 5:
$x + \frac{1}{x} = 5$.
We need to find the sum of the squares of the number and its reciprocal, which is $x^2 + \frac{1}{x^2}$.
Square both sides of the given equation:
$(x + \frac{1}{x})^2 = 5^2$
Using the identity $(a+b)^2 = a^2 + 2ab + b^2$, where $a=x$ and $b=1/x$:
$x^2 + 2 \cdot x \cdot \frac{1}{x} + (\frac{1}{x})^2 = 25$
$x^2 + 2 \cdot 1 + \frac{1}{x^2} = 25$
$x^2 + 2 + \frac{1}{x^2} = 25$
Subtract 2 from both sides:
$x^2 + \frac{1}{x^2} = 25 – 2$
$x^2 + \frac{1}{x^2} = 23$.