KVPY SX SB Exam 20141 Question Paper

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Question : 1 of 120

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PART - I

MATHEMATICS

Let

C_0

be a circle of radius I. For n ≥ 1, let C

_n

be a circle whose area equals the area of a square inscribed in C

_{n-1}

.
Then

∑↖{∞}↙{i=0}\Area(C_i)

equals

$π^2$
${π-2}/π^2$
$1/ π^2$
$π^2/ {π-2}$

Solution:

∑↖{∞}↙{i=0}\Area(C_i)= πr_0^ 2 + πr_1^ 2 + πr_2 ^2 + πr_3^ 2

+ ..... ∞

Area of

C_n = πr_n^ 2 = (√2r_{n-1})^2

r_n ^2 =2/π{r^2}_{n-1}

so

r_1^2=2/πr_0^2,r_2^2=2/πr_1^2

=2/π(2/πr_0^2)

r^2_3=2/π(r_2^2)=2/π(2/π2/πr_0^2)

So

∑↖{∞}↙{i=0}\Area(C_i ) = π[r_0^2+2/πr_0^2+2/π.2/π r_0^2+...∞]

= {πr_0^2}/{1-2/π}={π^2r_0^2}/{π-2}∀ r_0 = 1

=π^2/{π-2}

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