KVPY SA Exam 2016 Question Paper

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Question : 10 of 80

Marks: +1, -0

Let a and b be two positive real numbers such that a + 2b ≤ 1. Let

A_1

and

A_2

be respectively, the areas of circles with radii

ab^3

and

b^2

. Then the maximum possible value of

A_1/A_2

is

$1/16$
$1/64$
$1/{16√2}$
$1/32$

Solution:

a + 2b ≤ 1

\A_1=πa^2b^6

\A_2=πb^4

⇒\A_1/\A_2=a^2b^2

&

1 ≥ a+2b ≥ 2√{2ab}

(AM ≥ GM)

⇒1 ≥ 2√{2ab}

⇒1 ≥ 4×2ab

⇒1/64 ≥ a^2b^2

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