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JEE Main202228 Jul 2022Evening ShiftMathematicsVector AlgebraActual

Let S be the set of all a ∈ R for which the angle between the vectors u → = a log e b i ^ - 6 j ^ + 3 k ^ and v → = log e b i ^ + 2 j ^ + 2 a log e b k ^ , b > 1 is acute. Then S is equal to

Options

  1. A- ∞ , - 4 3
  2. BΦ
  3. C- 4 3 , 0
  4. D12 7 , ∞

Correct answer

B. Φ

Step-by-step solution

Given, u → = a log e b i ^ - 6 j ^ + 3 k ^ and v → = log e b i ^ + 2 j ^ + 2 a log e b k ^ For angle to be acute u → · v → > 0 ⇒ a log e b i ^ - 6 j ^ + 3 k ^ . log e b i ^ + 2 j ^ + 2 a log e b k ^ > 0 ⇒ a log e b 2 - 12 + 6 a log e b > 0 ∀ b > 1 Let y = a log e b 2 - 12 + 6 a log e b Now let log e b = t ⇒ t > 0 as b > 1 Now y = a t 2 + 6 a t - 12 > 0 , ∀   t > 0 Since y > 0 , so a > 0   &   D &#60

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