CDS 2026 (I) Elementary Mathematics, question 80

Topic: Number system

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Q80

A Question is given followed by two Statements I and II. Consider the Question and the Statements and mark the correct option.

Question : Let XYZ be a 3-digit number and the difference between XYZ and ZYX is equal to PQR. Is (P+R)(P+R) equal to QQ ?

Statement-I : P=3P=3

Statement-II : R=6R=6

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
  4. (d) The Question can be answered even without using both the StatementsCorrect answer

Solution.

Answer: (d) The Question can be answered even without using both the Statements

  1. Subtracting a three-digit number and its reversal cancels the middle digit: XYZ−ZYX=99(X−Z)XYZ - ZYX = 99(X - Z).
  2. For a positive three-digit difference, let k=X−Zk = X - Z, so 2≤k≤92 \leq k \leq 9.
  3. Now 99k=100(k−1)+90+(10−k)99k = 100(k - 1) + 90 + (10 - k).
  4. Thus P=k−1,Q=9P = k - 1,Q = 9 and R=10−kR = 10 - k, giving P+R=9=QP + R = 9 = Q.