KVPY 2019 SB SX Exam Previous Papers

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Question : 89 of 120

Marks: +1, -0

Let A be the set of vectors

a↖{→} (a_1 ,a_2 ,a_3 )

satisfying

(∑↖{3}↙{i=1}a_i/2^i)^2=∑↖{3}↙{i=1}a_i^2/2^i

. Then

A is empty
A contains exactly one element
A has 6 elements
A has infinitely many elements

Solution:

(a_1/2+a_2/4 +a_3/8 ) ^2=a_1^2/2+a_2^2/4 +a_3^3/8

16a_1^2 + 12a_2^2 + 7a_3^2-16a_1a_2

-4a_2a_3 -8a_1a_3 = 0

⇒

(2√2a_1-2√2a_2)^2+(2√2a_1-2√2a_3)^2

+(2a_2-a_3)^2+4a_3^2=0

⇒

a_1 = a_2 = a_3 = 0

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