Analysing the Question: Is the Expression Positive?

The question asks whether the expression \((p + q)^2 - 4pq\) is positive, given that \(p\) and \(q\) are natural numbers. Natural numbers are positive integers (1, 2, 3, ...).

Simplifying the Given Expression

Let's simplify the expression \((p + q)^2 - 4pq\). We can expand the term \((p + q)^2\):

Using the algebraic identity \((a+b)^2 = a^2 + 2ab + b^2\), we have:

\((p + q)^2 = p^2 + 2pq + q^2\)

Now substitute this back into the original expression:

\((p + q)^2 - 4pq = (p^2 + 2pq + q^2) - 4pq\)

\(= p^2 + 2pq + q^2 - 4pq\)

Combine the \(pq\) terms:

\(= p^2 - 2pq + q^2\)

This new expression \(p^2 - 2pq + q^2\) is also an algebraic identity, \((a-b)^2 = a^2 - 2ab + b^2\). So, we can write:

\(p^2 - 2pq + q^2 = (p - q)^2\)

Thus, the question is equivalent to asking: Is \((p - q)^2\) positive, where \(p\) and \(q\) are natural numbers?

Condition for \((p - q)^2\) to be Positive

For any real numbers \(p\) and \(q\), the square of their difference, \((p - q)^2\), is always greater than or equal to zero. That is, \((p - q)^2 \ge 0\). The expression \((p - q)^2\) is positive if and only if \((p - q) \ne 0\). This condition \(p - q \ne 0\) is equivalent to \(p \ne q\).

Since \(p\) and \(q\) are natural numbers, \((p-q)^2\) will be positive if and only if \(p\) and \(q\) are different natural numbers. If \(p\) and \(q\) are the same natural number (i.e., \(p=q\)), then \(p-q = 0\), and \((p-q)^2 = 0^2 = 0\), which is not positive.

So, the question is essentially asking: Are \(p\) and \(q\) different natural numbers (i.e., is \(p \ne q\))?

Evaluating Statement I: p < q

Statement I says that \(p < q\). If \(p\) is less than \(q\), it means \(p\) and \(q\) are definitely not equal. For example, if \(p=3\) and \(q=5\), then \(p < q\). In this case, \(p \ne q\). The expression \((p-q)^2 = (3-5)^2 = (-2)^2 = 4\), which is positive.

Since \(p\) and \(q\) are natural numbers, if \(p < q\), they must be distinct numbers. This guarantees that \(p \ne q\), which in turn guarantees that \((p - q)^2\) is positive.

Therefore, Statement I alone is sufficient to answer the question (the answer is YES, the expression is positive).

Evaluating Statement II: p > q

Statement II says that \(p > q\). If \(p\) is greater than \(q\), it means \(p\) and \(q\) are definitely not equal. For example, if \(p=7\) and \(q=2\), then \(p > q\). In this case, \(p \ne q\). The expression \((p-q)^2 = (7-2)^2 = (5)^2 = 25\), which is positive.

Since \(p\) and \(q\) are natural numbers, if \(p > q\), they must be distinct numbers. This guarantees that \(p \ne q\), which in turn guarantees that \((p - q)^2\) is positive.

Therefore, Statement II alone is sufficient to answer the question (the answer is YES, the expression is positive).

Conclusion on Data Sufficiency

Both Statement I and Statement II, when considered alone, provide enough information to determine that \(p \ne q\). Since the expression \((p+q)^2 - 4pq\) is positive if and only if \(p \ne q\), both statements alone are sufficient to answer the question.

Since the question can be answered using Statement I alone, and also using Statement II alone, the correct option is the one stating that the question can be answered using either Statement alone.

Revision Table: Positive Expression Data Sufficiency

Let's quickly summarise the key points for this positive expression data sufficiency problem:

• The expression \((p+q)^2 - 4pq\) simplifies to \((p-q)^2\).
• For natural numbers \(p, q\), \((p-q)^2\) is positive if and only if \(p \ne q\).
• Statement I (\(p < q\)) implies \(p \ne q\).
• Statement II (\(p > q\)) implies \(p \ne q\).
• Both statements independently lead to the conclusion that the expression is positive.

Additional Information: Algebraic Identities and Natural Numbers

Understanding algebraic identities like \((a+b)^2 = a^2 + 2ab + b^2\) and \((a-b)^2 = a^2 - 2ab + b^2\) is crucial for simplifying expressions like the one in the question. These identities help transform complex expressions into simpler forms, making analysis easier.

The concept of natural numbers (positive integers: 1, 2, 3, ...) is also important. When \(p\) and \(q\) are natural numbers, \((p-q)^2\) can only be a non-negative integer. It is zero only when \(p=q\), and a positive integer when \(p \ne q\).

Data sufficiency questions often test your ability to determine if the given statements provide enough information to answer the question, not necessarily to find the specific numerical answer. Here, we just needed to know if the expression is positive, which is equivalent to determining if \(p \ne q\).