CDS 2026 (I) Elementary Mathematics, question 71

Topic: Geometry

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Q71

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

Question : A circle is inscribed in the equilateral triangle Δ\Delta. What is the radius of the circle ?

Statement-I : Area of Δ\Delta is equal to 163 cm216\sqrt{3}\text{ cm}^{2}.

Statement-II : Perimeter of Δ\Delta is 2424 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 aloneCorrect answer
  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

Solution.

Answer: (b) The Question can be answered by using either Statement alone

  1. I alone: For equilateral side a,3a2/4=163a,\sqrt{3}a^{2}/4 = 16\sqrt{3} gives a=8cma = 8cm, so the inradius r=a3/6=43/3cmr = a\sqrt{3}/6 = 4\sqrt{3}/3cm.
  2. Sufficient.
  3. II alone: The perimeter gives a=24/3=8cma = 24/3 = 8cm and therefore the same unique radius.
  4. Sufficient.