If the vectors a = c( _7 x) i + 2 j + 3 k and b = ( _7 x) i + 3c( _7 x) j - 4 k make obtuse angle for any x > 0, then c belongs to

MHT CET 2025 Mathematics question

If the vectors and make obtuse angle for any , then c belongs to

(A)
(B)
(C)Correct
(D)

Answer: (C)

Solution

Let ; as ranges over , takes every real value. The angle is obtuse iff . \bar a\cdot\bar b=ct\cdot t+2\cdot3ct+3(-4)=ct^2+6ct-12. We need for all real . This needs the leading coefficient and negative discriminant:

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Question ID VK-102731