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KVPY2019MathematicsSets and Relations

Let A be the set of vectors a = a 1 , a 2 , a 3 satisfying ∑ i = 1 3 a i 2 i 2 = ∑ i = 1 3 a i 2 2 i Then,

Options

  1. AA is empty
  2. BA contains exactly one element
  3. CA has 6 elements
  4. DA has infinitely many elements

Correct answer

B. A contains exactly one element

Step-by-step solution

Given, ∑ i = 1 3 a i 2 i 2 = ∑ i = 1 3 a i 2 2 i ⇒ a 1 2 + a 2 4 + a 3 8 2 = a 1 2 2 + a 2 2 4 + a 3 2 8 ⇒ a 1 2 4 + a 2 2 16 + a 3 2 64 + a 1 a 2 4 + a 2 a 3 16 + a 3 a 1 8 = a 1 2 2 + a 2 2 4 + a 3 2 8 ⇒ a 1 2 4 + 3 a 2 2 16 + 7 a 3 2 64 = a 1 a 2 4 + a 2 a 3 16 + a 3 a 1 8 ⇒ 16 a 1 2 + 12 a 2 2 + 7 a 3 2 = 16 a 1 a 2 + 4 a 2 a 3 + 8 a 3 a 1 ⇒ 8 a 1 2 + 8 a 2 2 - 16 a 1 a 2 + 4 a 2 2 + a 3 2 - 4 a 2 a 3 + 2 a 3 2 + 8 a 1 2 - 8 a 3 a 1 + 4 a 3 2 = 0 ⇒ 8 a 1 - a 2 2 +

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