JEE Advanced 2021 · Paper 2 · Mathematics Q13

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MathematicsSingle correct+3 / -1Question 51 of 57

Passage

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Let
$M = \{(x, y) \in \mathbb R \times \mathbb R : x^2 + y^2 \le r^2\},$
where $r > 0$ . Consider the geometric progression $a_n = \dfrac1{2^{n-1}}$ , $n = 1, 2, 3, \ldots$ . Let $S_0 = 0$ and, for $n \ge 1$ , let $S_n$ denote the sum of the first $n$ terms of this progression. For $n \ge 1$ , let $C_n$ denote the circle with center $(S_{n-1}, 0)$ and radius $a_n$ , and $D_n$ denote the circle with center $(S_{n-1}, S_{n-1})$ and radius $a_n$ .

Consider $M$ with $r = \dfrac{1025}{513}$ . Let $k$ be the number of all those circles $C_n$ that are inside $M$ . Let $l$ be the maximum possible number of circles among these $k$ circles such that no two circles intersect. Then

k + 2l = 22

2k + l = 26

2k + 3l = 34

3k + 2l = 40

Answer and worked solution

Official answer: D

Which $C_n$ lie inside M. $C_n$ is inside when $S_{n-1}+a_n\le r$ . Now $S_{n-1}+a_n=S_n=2-\frac1{2^{n-1}}$ , and $r=2-\frac1{513}$ . So we need $2^{n-1}\le513$ , i.e. $n\le10$ : $k=10$ .

Which circles intersect.

$C_n$ and $C_{n+1}$ : centres $a_n$ apart, radii summing to $1.5a_n$ . They overlap.
$C_n$ and $C_{n+2}$ : centres $1.5a_n$ apart, radii summing to $1.25a_n$ . They are disjoint.

So the most non-intersecting circles is every other one: $l=5$ .

Check the options. $3k+2l=30+10=40$ . (D)

Topics:Equation of a circle Geometric progression

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JEE Advanced 2021 · Paper 2, Mathematics question 13