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What are the values of x for which the angle between the vectors 2x2 + 3x +  and  −2 + x2 is obtuse ?

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  1. 0 < x < 2
  2. x < 0
  3. x > 2
  4. 0 ≤ x ≤ 2

Answer (Detailed Solution Below)

Option 1 : 0 < x < 2
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Detailed Solution

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Concept:

  • The angle between two vectors   and  is given by, ​ 
  • If  = a1 + a2 + a3 and   = b1 + b2 + b3, then  

Calculation:

Given: The angle between the vectors 2x2 + 3x +  and  −2 + x2 is obtuse

The angle between the vectors 2x2 + 3x +  and  −2 + x2 is given by

⇒ 

⇒ 

Since θ is obtuse, 

⇒ cos θ < 0

⇒ 3x2 - 6x < 0

⇒ x(x - 2) < 0

⇒ 0 < x < 2

∴ The correct option is (1).

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