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Question 9 The value of 𝛼 if the angle between 𝑝 βƒ—=2𝛼^2 Δ± Λ†βˆ’3𝛼ȷ Λ†+π‘˜ Λ† and π‘ž βƒ—=Δ± Λ†+Θ· Λ†+π›Όπ‘˜ Λ† is obtuse, is (A) π‘…βˆ’[0,1] (B) (0,1) (C) [0,∞) (D) [1,∞) Angle between two vectors is found using dot product We know that 𝑝 βƒ— . π‘ž βƒ— = "|" 𝑝 βƒ—"|" "|" π‘ž βƒ—"|" cos ΞΈ cos ΞΈ = (𝒑 βƒ—" . " 𝒒 βƒ—)/(|𝒑 βƒ— |" " |𝒒 βƒ—|) where ΞΈ is the angle between 𝑝 βƒ— and π‘ž βƒ— Since angle between 𝑝 βƒ— and π‘ž βƒ— is obtuse cos ΞΈ < 0 Putting values (𝑝 βƒ—" . " π‘ž βƒ—)/(|𝑝 βƒ— |" " |π‘ž βƒ—|)<0 Since "|" 𝑝 βƒ—"|" "|" π‘ž βƒ—"|" is a number, we can multiply it to right side 𝑝 βƒ—" . " π‘ž βƒ—<0 Γ—|𝑝 βƒ— |" "|π‘ž βƒ—| 𝒑 βƒ—" . " 𝒒 βƒ—<𝟎 (2𝛼^2 Δ± Λ†βˆ’3𝛼ȷ Λ†+π‘˜ Λ† ).(Δ± Λ†+Θ· Λ†+π›Όπ‘˜ Λ† )<0 2𝛼^2 Γ— 1βˆ’3𝛼 Γ— 1+1 Γ— 𝛼<0 2𝛼^2 βˆ’3𝛼+ 𝛼<0 2𝛼^2 βˆ’2𝛼<0 2𝛼(π›Όβˆ’1)<0 𝜢(πœΆβˆ’πŸ)<𝟎 So, the correct answer is (B) 𝛼(π›Όβˆ’1)<0 So, 𝜢∈(𝟎, 𝟏) So, the correct answer is (B)

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Davneet Singh

Davneet Singh has done his B.Tech from Indian Institute of Technology, Kanpur. He has been teaching from the past 14 years. He provides courses for Maths, Science and Computer Science at Teachoo