JEE Advanced 2024 Paper 1 Mathematics Q12

Mathematics

Problem

Let O

\vec{P}

= ((α − 1)/α)î + ĵ +

\hat{k}

, O

\vec{Q}

= î + ((β − 1)/β)ĵ +

\hat{k}

and O

\vec{R}

= î + ĵ + (1/2)

\hat{k}

be three vectors, where α, β ∈ ℝ − {0} and O denotes the origin. If (O

\vec{P}

× O

\vec{Q}

) · O

\vec{R}

= 0 and the point (α, β, 2) lies on the plane 3x + 3y − z + l = 0, then the value of l is ________.