KVPY SA Exam 05-Nov-2017 Question Paper

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Question : 3 of 80

Marks: +1, -0

Let

a_1 , a_2 , a_3 , a_4

be real numbers such that

a_1^2 +a^2_2 +a_3^2 + a_4^2 = 1

. Then the smallest possible value of the expression

(a_1 -a_2 )^ 2 + (a_2 -a_3 )^ 2 + (a_3 -a_4 )^ 2 + (a_4 -a_1 )^ 2

lies in the interval

(0, 1.5)
(1.5, 2.5)
(2.5, 3)
$±1/2$

Solution:

∵

a_1^2 +a^2_2 +a_3^2 + a_4^2 = 1

Smallest possible value of

(a_1 -a_2 )^ 2 + (a_2 -a_3 )^ 2 + (a_3 -a_4 )^ 2

+ (a_4 -a_1 )^ 2=0

if a_1 = a_2 = a_3 = a_4 =±1/2

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