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XAT 2011 — QA & DI Question 28

Simple EquationsEasy
Passage / Data

Answer the following question based on the information given below.

From a group of 545 contenders, a party has to select a leader. Even after holding a series of meetings, the politicians and the general body failed to reach a consensus. It was then proposed that all 545 contenders be given a number from 1 to 545. Then they will be asked to stand on a podium in a circular arrangement, and counting would start from the contender numbered 1. The counting would be done in a clockwise fashion. The rule is that every alternate contender would be asked to step down as the counting continued, with the circle getting smaller and smaller, till only one person remains standing. Therefore the first person to be eliminated would be the contender numbered 2.

Consider a square ABCD of side 60 cm. lt contains arcs BD and AC drawn with centres at A and D respectively. A circle is drawn such that it is tangent to side AB, and the arcs BD and AC. What is the radius of the circle?

Answer & solution

Correct answer: 10 cm

  • A

    9 cm

  • 10 cm

  • C

    12 cm

  • D

    15 cm

  • E

    None of the above

Solution

​​​​​​​

Let O be the centre of the smaller circle.

Let the small circle touch AB at P. OQ is perpendicular to AD.

Now, AO = 60 – r and & OP = r

Now,

QO2 = AP2 = (60 – r)2 – r2              … (i)

Now, in ΔDQO,

QO2 = DO2 – DQ2

Now, DO = 60 + r and DQ = 60 – r

DQ = 60 – r

∴ OQ2 = (60 + r)2 – (60 – r)2         … (ii) 

∴ (60 – r)2 +3600 – 120r = (60 + r)2

From (i) and (ii),

∴ r = 10

Hence, option (b).

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XAT 2011 QA & DI Q28: Consider a square ABCD of side 60 cm. lt contains arcs BD and AC drawn with centres at A and D respectively. A — Solution | TheCATExam