CSAT Solved Papers/ 2023/Q57

2023 CSAT — Q57

Quant Data sufficiency 2.5 marks Medium

Question: Is (p+q−r)(p + q - r) greater than (p−q+r)(p - q + r), where p,qp, q and rr are integers?

Statement-1: (p−q)(p - q) is positive.

Statement-2: (p−r)(p - r) is negative.

Which one of the following is correct in respect of the above Question and the Statements?

  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 Answer
  4. D The Question cannot be answered even by using both the Statements together

Worked rationale

Simplify the question first. (p+q−r)−(p−q+r)=2q−2r=2(q−r)(p+q-r) - (p-q+r) = 2q - 2r = 2(q - r). So the question “is (p+q−r)>(p−q+r)(p+q-r) > (p-q+r)?” is exactly “is q>rq > r?”

Statement-1 alone (p−q>0⇒p>qp - q > 0 \Rightarrow p > q): says nothing about rr. Insufficient.

Statement-2 alone (p−r<0⇒p<rp - r < 0 \Rightarrow p < r): says nothing about qq. Insufficient.

Both together: q<pq < p (St-1) and p<rp < r (St-2) chain to

q<p<r  ⇒  q<r.q < p < r \;\Rightarrow\; q < r.

So q>rq > r is definitely false — a definite answer (“no”). Both statements together answer the question; neither alone does.

Answer: (c) The Question can be answered by using both the Statements together, but cannot be answered using either Statement alone.

Why the other options miss

  • A
    over-trusts a single statement: thinks one inequality (involving only p,qp,q or only p,rp,r) settles the qq-vs-rr comparison, which needs both.
  • B
    claims each statement is enough on its own: assumes each independently pins the order of qq and rr, ignoring the missing variable in each.
  • D
    skips the simplification: fails to reduce the question to ”q>rq > r,” so misses that q<p<rq < p < r chains to a definite “no.”

Specialist insight

The whole item collapses once you reduce the question to its core: (p+q−r)−(p−q+r)=2(q−r)(p+q-r)-(p-q+r) = 2(q-r), so it is purely ”qq vs rr.” Then the two statements are a transitivity chain: q<pq < p and p<rp < r give q<rq < r — a definite answer (the answer is “no,” but DS only needs a definite answer). Skipping the algebraic simplification is what makes (d) tempting; doing it makes (c) obvious.

The trap, in one line

Question reduces to "q>rq > r?"; St-1 gives q<pq<p, St-2 gives p<rp<r, chaining to q<rq<r (definite no) — both needed == (c).

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