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JEE Main202427 Jan 2024Morning ShiftMathematicsVector AlgebraActual

The least positive integral value of α , for which the angle between the vectors α i ^ - 2 j ^ + 2 k ^ and α i ^ + 2 α j ^ - 2 k ^ is acute, is _____.

Correct answer

0

Step-by-step solution

We know that, a → . b → = a → b → cos θ ⇒ cos θ = ( α i ^ - 2 j ^ + 2 k ^ ) · ( α i ^ + 2 α j ^ - 2 k ^ ) α 2 + 4 + 4 α 2 + 4 α 2 + 4 ⇒ cos θ = α 2 - 4 α - 4 α 2 + 8 5 α 2 + 4 Since θ is acute, ⇒ α 2 - 4 α - 4 > 0 ⇒ α 2 - 4 α + 4 > 8 ⇒ ( α - 2 ) 2 > 8 ⇒ α - 2 > 2 2 or α - 2 < - 2 2 α > 2 + 2 2 or α < 2 - 2 2 So, the least positive integral value of α is 5 .

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