Question

Column A:\((\frac{1}{x})^2\)
Column B: x²

• A. The quantity in Column A is greater.
• B. The quantity in Column B is greater.
• C. The two quantities are equal.
• D. The relationship cannot be determined from the information given.

Hint:

When a quantitative comparison problem involves variables with no constraints, immediately think about testing different categories of numbers: positive integers, negative integers, positive fractions (between 0 and 1), and negative fractions. If you find two different relationships, the answer is (D).

Answer 1

The Correct Option is D

Step 1: Understanding the Concept:
We are asked to compare the square of a reciprocal with the square of the number itself, with no constraints on the value of \(x\) (other than \( x \neq 0 \)).
Step 2: Key Formula or Approach:
Simplify the expression in Column A. Then, test different types of numbers for \(x\) (e.g., integers greater than 1, fractions between -1 and 1) to see if the relationship is constant.
Step 3: Detailed Explanation:
First, simplify the expression in Column A: \[ \left(\frac{1}{x}\right)^2 = \frac{1^2}{x^2} = \frac{1}{x^2} \] So, we are comparing \( \frac{1}{x^2} \) (Column A) with \( x^2 \) (Column B).
Let's test different values for \(x\).
Case 1: Let \(x\) be an integer greater than 1, for example, \(x = 2\).

Column A: \( \frac{1}{2^2} = \frac{1}{4} \)
Column B: \( 2^2 = 4 \)
In this case, Column B>Column A.
Case 2: Let \(x\) be a fraction between 0 and 1, for example, \(x = \frac{1}{2}\).

Column A: \( \frac{1}{(1/2)^2} = \frac{1}{1/4} = 4 \)
Column B: \( \left(\frac{1}{2}\right)^2 = \frac{1}{4} \)
In this case, Column A>Column B.
Since we have found one case where Column B is greater and another where Column A is greater, the relationship depends on the value of \(x\).
Step 4: Final Answer:
The relationship cannot be determined from the information given.