CDS 2026 (I) Elementary Mathematics, question 75

Topic: Mensuration

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Q75

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

Question : Two circles with radii p,qp,q (p>q)(p>q) in cm touch externally, where p,qp,q are natural numbers and each greater than 11. What is the value of (p−q)(p-q) ?

Statement-I : The sum of their areas is 130π cm2130\pi\text{ cm}^{2}

Statement-II : The distance between the centres of the two circles is 1414 cm

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 aloneCorrect answer
  4. (d) The Question cannot be answered even by using both the Statements together

Solution.

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

  1. I gives p2+q2=130p^{2} + q^{2} = 130. Both (p,q)=(11,3)(p,q) = (11,3) and (9, 7) satisfy this, but their differences are 8 and 2.
  2. Thus I alone is insufficient. II gives p+q=14p + q = 14 because the circles touch externally. The pairs (12, 2) and (11, 3) satisfy this, but their differences are 10 and 8.
  3. Thus II alone is insufficient.
  4. Together, (p−q)2=2(p2+q2)−(p+q)2=260−196=64(p - q)^{2} = 2\left( p^{2} + q^{2} \right) - (p + q)^{2} = 260 - 196 = 64.
  5. Since p>q,p−q=8p > q,p - q = 8; the valid radii are 11 cm and 3 cm.