Data Sufficiency - Algebra

1. A. The Question can be answered by using one of the Statements alone, but cannot be answered using the other statement alone.
2. B. The Question can be answered by using either Statement alone.
3. C. The Question can be answered by using both the Statements together, but cannot be answered using either Statement alone.
4. D. The Question can be answered even without using any of the Statements.

Answer: B

Explanation

First, simplify the expression:
(p + q)² – 4pq = p² + 2pq + q² – 4pq = p² – 2pq + q² = (p – q)²
The question asks: Is (p – q)² positive?
A square of any real number is always non-negative (≥ 0). It is positive if and only if the base is not zero. So, (p – q)² is positive if p – q ≠ 0, which means p ≠ q.

Statement I: p < q.
If p q.
If p > q, then p ≠ q. Therefore, (p – q)² will be positive. This statement alone is sufficient.

Since either Statement I alone or Statement II alone is sufficient to answer the question, the correct option is (b). This question tests basic algebraic identities and properties of numbers, a common element in CSAT.

1. A. The Question can be answered by using one of the Statements alone, but cannot be answered using the other Statement alone
2. B. The Question can be answered by using either Statement alone
3. C. The Question can be answered by using both the Statements together, but cannot be answered using either Statement alone
4. D. The Question cannot be answered even by using both the Statements together

Answer: C

Explanation

We are given: p + q + r = ₹50 and q = ₹16. We also know that q is the least price, so p ≥ 16 and r ≥ 16.
Substituting q=16 into the total cost equation: p + 16 + r = 50 => p + r = 34.

**Statement I: The cost of p is not more than that of r.** This means p ≤ r.
Given p+r=34 and p,r ≥ 16:
– If p=16, then r=18. This satisfies p ≤ r (16 ≤ 18).
– If p=17, then r=17. This satisfies p ≤ r (17 ≤ 17).
Since p could be 16 or 17, Statement I alone is insufficient to find a unique value for p.

**Statement II: The cost of r is not more than that of p.** This means r ≤ p.
Given p+r=34 and p,r ≥ 16:
– If p=18, then r=16. This satisfies r ≤ p (16 ≤ 18).
– If p=17, then r=17. This satisfies r ≤ p (17 ≤ 17).
Since p could be 17 or 18, Statement II alone is insufficient to find a unique value for p.

**Combining Statement I and Statement II:**
From Statement I, p ≤ r.
From Statement II, r ≤ p.
The only way both conditions can be true simultaneously is if p = r.
Substitute p = r into the equation p + r = 34:
p + p = 34 => 2p = 34 => p = 17.
Since p=17, then r=17. Both are indeed greater than or equal to q=16. Thus, the price of article p is uniquely determined as ₹17.
Therefore, both statements together are sufficient. This question tests data sufficiency and basic algebraic problem-solving.