Let a and b be two unit vectors such that the angle between them is (π)/(3) . If λ a +2 b and 3 a -λ b are perpendicular to each other, then the number of values of λ in [-1,3] is :

Correct answer: C. 0 aligned & a · b =(1)/(2) & Now (λ a +2 b ) ·(3 a -λ b )=0 & 3 λ a · a -λ^2 a · b +6 a · b -2 λ b · b =0 & 3 λ-(λ^2)/(2)+3-2 λ=0 & λ^2-2 λ-6=0 & λ=1 ± 7 & number of values =0 aligned

JEE Main 202522 JanuaryEvening Shift

Exam: JEE Main
Year: 2025
Date: 22 January
Shift: Evening Shift
Subject: Mathematics
Chapter: Vector Algebra

JEE Main 2025 · 22 January · Evening Shift · Mathematics · Vector Algebra

Let a and b be two unit vectors such that the angle between them is...

Let

\vec{a}

and

\vec{b}

be two unit vectors such that the angle between them is

\frac{\pi}{3}

. If

\lambda \vec{a}+2 \vec{b}

and

3 \vec{a}-\lambda \vec{b}

are perpendicular to each other, then the number of values of

\lambda

[-1,3]

is :

Options

Correct answer

C. 0

Step-by-step solution

\begin{aligned} & \hat{a} \cdot \hat{b}=\frac{1}{2} \\ & \text{ Now }(\lambda \hat{a}+2 \hat{b}) \cdot(3 \hat{a}-\lambda \hat{b})=0 \\ & 3 \lambda \hat{a} \cdot \hat{a}-\lambda^2 \hat{a} \cdot \hat{b}+6 \hat{a} \cdot \hat{b}-2 \lambda \hat{b} \cdot \hat{b}=0 \\ & 3 \lambda-\frac{\lambda^2}{2}+3-2 \lambda=0 \\ & \lambda^2-2 \lambda-6=0 \\ & \lambda=1 \pm \sqrt{7} \\ & \Rightarrow \text{ number of values }=0\end{aligned}

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