The operation ⊗ is defined for all nonzero numbers a and b by a ⊗ b = a/b – b/a....

1. Translate the problem requirements: The operation \(\otimes\) is defined as \(\mathrm{a} \otimes \mathrm{b} = \mathrm{a}/\mathrm{b} - \mathrm{b}/\mathrm{a}\). We need to determine which of the three given statements must always be true for any nonzero numbers x and y.
2. Test each statement systematically: Apply the definition of \(\otimes\) to both sides of each equation to see if they are equivalent.
3. Verify algebraic equivalence: For each statement, perform the necessary algebraic manipulations to determine if the left side equals the right side.
4. Identify which statements are always true: Based on our verification, determine which combination of statements (I, II, III) must be true.

Execution of Strategic Approach

1. Translate the problem requirements

Let's start by understanding what we're working with. We have a special operation called \(\otimes\) that works like this: whenever we see \(\mathrm{a} \otimes \mathrm{b}\), it means we calculate \(\mathrm{a}/\mathrm{b} - \mathrm{b}/\mathrm{a}\).

Think of it like a recipe - every time we see this symbol \(\otimes\) between two numbers, we follow the same steps: divide the first by the second, then subtract the second divided by the first.

Our job is to check three different statements to see which ones are ALWAYS true, no matter what nonzero numbers we use for x and y.

Process Skill: TRANSLATE - Converting the operation definition into a clear rule we can apply

2. Test each statement systematically

Now we'll check each statement by applying our \(\otimes\) rule to both sides and seeing if they're equal.

Testing Statement I: \(\mathrm{x} \otimes \mathrm{xy} = \mathrm{x}(1 \otimes \mathrm{y})\)

Left side: \(\mathrm{x} \otimes \mathrm{xy}\) means \(\mathrm{x}/(\mathrm{xy}) - (\mathrm{xy})/\mathrm{x} = 1/\mathrm{y} - \mathrm{y}\)

Right side: First, let's find \(1 \otimes \mathrm{y} = 1/\mathrm{y} - \mathrm{y}/1 = 1/\mathrm{y} - \mathrm{y}\)

So \(\mathrm{x}(1 \otimes \mathrm{y}) = \mathrm{x}(1/\mathrm{y} - \mathrm{y}) = \mathrm{x}/\mathrm{y} - \mathrm{xy}\)

Comparing: Left side is \(1/\mathrm{y} - \mathrm{y}\), Right side is \(\mathrm{x}/\mathrm{y} - \mathrm{xy}\)

These are only equal when \(\mathrm{x} = 1\), so Statement I is NOT always true.

Testing Statement II: \(\mathrm{x} \otimes \mathrm{y} = -(\mathrm{y} \otimes \mathrm{x})\)

Left side: \(\mathrm{x} \otimes \mathrm{y} = \mathrm{x}/\mathrm{y} - \mathrm{y}/\mathrm{x}\)

Right side: \(\mathrm{y} \otimes \mathrm{x} = \mathrm{y}/\mathrm{x} - \mathrm{x}/\mathrm{y}\), so \(-(\mathrm{y} \otimes \mathrm{x}) = -(\mathrm{y}/\mathrm{x} - \mathrm{x}/\mathrm{y}) = \mathrm{x}/\mathrm{y} - \mathrm{y}/\mathrm{x}\)

Perfect match! Statement II is always true.

Testing Statement III: \(1/\mathrm{x} \otimes 1/\mathrm{y} = \mathrm{y} \otimes \mathrm{x}\)

Left side: \(1/\mathrm{x} \otimes 1/\mathrm{y} = (1/\mathrm{x})/(1/\mathrm{y}) - (1/\mathrm{y})/(1/\mathrm{x}) = \mathrm{y} - \mathrm{x}\)

Right side: \(\mathrm{y} \otimes \mathrm{x} = \mathrm{y}/\mathrm{x} - \mathrm{x}/\mathrm{y}\)

Let's check if \(\mathrm{y} - \mathrm{x}\) equals \(\mathrm{y}/\mathrm{x} - \mathrm{x}/\mathrm{y}\) by testing with simple numbers like \(\mathrm{x} = 2, \mathrm{y} = 3\):

Left: \(3 - 2 = 1\), Right: \(3/2 - 2/3 = 9/6 - 4/6 = 5/6\)

Since \(1 \neq 5/6\), Statement III is NOT always true.

Process Skill: SIMPLIFY - Breaking down complex expressions step by step to avoid errors

3. Verify algebraic equivalence

Let me double-check Statement III more carefully since it involves fractions:

For \(1/\mathrm{x} \otimes 1/\mathrm{y}\): When we divide fractions, \((1/\mathrm{x}) \div (1/\mathrm{y}) = (1/\mathrm{x}) \times (\mathrm{y}/1) = \mathrm{y}/\mathrm{x}\)

Similarly, \((1/\mathrm{y}) \div (1/\mathrm{x}) = (1/\mathrm{y}) \times (\mathrm{x}/1) = \mathrm{x}/\mathrm{y}\)

So \(1/\mathrm{x} \otimes 1/\mathrm{y} = \mathrm{y}/\mathrm{x} - \mathrm{x}/\mathrm{y}\)

And \(\mathrm{y} \otimes \mathrm{x} = \mathrm{y}/\mathrm{x} - \mathrm{x}/\mathrm{y}\)

Actually, these ARE the same! Statement III is also always true.

Process Skill: MANIPULATE - Correctly handling fraction division to get the right result

4. Identify which statements are always true

From our testing:

• Statement I: FALSE (only true when \(\mathrm{x} = 1\))
• Statement II: TRUE (always works)
• Statement III: TRUE (always works)

Therefore, statements II and III must be true.

Final Answer

The correct answer is E. II and III

This matches our analysis that Statement II (the operation is anti-commutative) and Statement III (reciprocals swap the operation) are always true, while Statement I only works in special cases.

Common Faltering Points

Errors while devising the approach

Faltering Point 1: Misunderstanding the operation definition
Students often misread \(\mathrm{a} \otimes \mathrm{b} = \mathrm{a}/\mathrm{b} - \mathrm{b}/\mathrm{a}\) as \(\mathrm{a}/\mathrm{b} + \mathrm{b}/\mathrm{a}\) or confuse the order of subtraction. This fundamental misunderstanding of the operation will lead to incorrect results for all three statements.

Faltering Point 2: Not recognizing the need to test each statement independently
Some students assume that if one statement is true, others must be true as well, or they try to find a pattern without systematically testing each statement. Each statement must be verified separately using the operation definition.

Faltering Point 3: Attempting to use specific number substitution only
While testing with specific numbers can help verify results, students may rely solely on this approach without doing the algebraic manipulation. This can miss cases where the statement works for some numbers but not others, or fail to prove universal truth.

Errors while executing the approach

Faltering Point 1: Arithmetic errors with fraction operations
When working with Statement III (\(1/\mathrm{x} \otimes 1/\mathrm{y}\)), students frequently make errors when dividing fractions, such as incorrectly calculating \((1/\mathrm{x}) \div (1/\mathrm{y})\) or \((1/\mathrm{y}) \div (1/\mathrm{x})\). The rule for dividing fractions (multiply by the reciprocal) is often misapplied.

Faltering Point 2: Sign errors when distributing negative signs
In Statement II, when calculating \(-(\mathrm{y} \otimes \mathrm{x}) = -(\mathrm{y}/\mathrm{x} - \mathrm{x}/\mathrm{y})\), students often make sign errors during distribution, getting \(-\mathrm{y}/\mathrm{x} + \mathrm{x}/\mathrm{y}\) instead of the correct \(\mathrm{x}/\mathrm{y} - \mathrm{y}/\mathrm{x}\).

Faltering Point 3: Incorrect algebraic simplification
Students may make errors when simplifying expressions like \(\mathrm{x}/(\mathrm{xy})\) to \(1/\mathrm{y}\), or when combining fractions. These basic algebraic mistakes compound throughout the problem and lead to wrong conclusions about statement validity.

Errors while selecting the answer

Faltering Point 1: Misreading which statements are true
After determining that Statements II and III are true, students may accidentally select an answer choice that includes Statement I (such as choice D: I and II) due to careless reading or misremembering their conclusions.

Faltering Point 2: Confusing 'must be true' with 'could be true'
Students might select Statement I thinking it's correct because it works for \(\mathrm{x} = 1\), not recognizing that 'must be true' means the statement works for ALL possible values of the variables, not just specific cases.

Alternate Solutions

Smart Numbers Approach

For this type of problem involving algebraic statements that must be universally true, we can verify our conclusions by testing with carefully chosen specific values.

Step 1: Choose Smart Numbers

Let's select \(\mathrm{x} = 2\) and \(\mathrm{y} = 3\). These are simple, distinct values that will make our calculations clean while still being representative.

Step 2: Test Statement I: \(\mathrm{x} \otimes \mathrm{xy} = \mathrm{x}(1 \otimes \mathrm{y})\)

Left side: \(\mathrm{x} \otimes \mathrm{xy} = 2 \otimes (2 \times 3) = 2 \otimes 6\)

Using the definition: \(2 \otimes 6 = 2/6 - 6/2 = 1/3 - 3 = 1/3 - 9/3 = -8/3\)

Right side: \(\mathrm{x}(1 \otimes \mathrm{y}) = 2(1 \otimes 3) = 2(1/3 - 3/1) = 2(1/3 - 3) = 2(-8/3) = -16/3\)

Since \(-8/3 \neq -16/3\), Statement I is FALSE.

Step 3: Test Statement II: \(\mathrm{x} \otimes \mathrm{y} = -(\mathrm{y} \otimes \mathrm{x})\)

Left side: \(\mathrm{x} \otimes \mathrm{y} = 2 \otimes 3 = 2/3 - 3/2 = 4/6 - 9/6 = -5/6\)

Right side: \(-(\mathrm{y} \otimes \mathrm{x}) = -(3 \otimes 2) = -(3/2 - 2/3) = -(9/6 - 4/6) = -5/6\)

Since \(-5/6 = -5/6\), Statement II is TRUE.

Step 4: Test Statement III: \(1/\mathrm{x} \otimes 1/\mathrm{y} = \mathrm{y} \otimes \mathrm{x}\)

Left side: \(1/\mathrm{x} \otimes 1/\mathrm{y} = 1/2 \otimes 1/3 = (1/2)/(1/3) - (1/3)/(1/2) = 3/2 - 2/3 = 9/6 - 4/6 = 5/6\)

Right side: \(\mathrm{y} \otimes \mathrm{x} = 3 \otimes 2 = 3/2 - 2/3 = 9/6 - 4/6 = 5/6\)

Since \(5/6 = 5/6\), Statement III is TRUE.

Step 5: Verification with Different Values

Let's verify with \(\mathrm{x} = 1, \mathrm{y} = 4\) to ensure our pattern holds:

Statement II: \(1 \otimes 4 = 1/4 - 4/1 = -15/4\)

\(-(4 \otimes 1) = -(4/1 - 1/4) = -(15/4) = -15/4\) ✓

Statement III: \(1/1 \otimes 1/4 = 1 \otimes 1/4 = 1/(1/4) - (1/4)/1 = 4 - 1/4 = 15/4\)

\(4 \otimes 1 = 4/1 - 1/4 = 15/4\) ✓

Conclusion: Statements II and III are true, so the answer is E.