JEE Main 2023MathematicsThree Dimensional GeometryMediumMCQ

JEE Main 2023 — Three Dimensional Geometry Question with Solution

JEE Main 2023 (30 Jan Shift 1)

Question

Let a unit vector OP→ make angle α,β,γ with the positive directions of the co-ordinate axes OX,OY, OZ respectively, where β∈0,π2. OP→ is perpendicular to the plane through points 1,2,3, 2,3,4 and 1,5,7, then which one of the following is true ?

Choose an option

Show full solutionCorrect option: A
Correct answer
Aα∈π2,π and γ∈π2,π

Step-by-step explanation

Equation of plane passing through the points 1,2,3, 2,3,4 and 1,5,7 is

x-1y-2z-3111034=0

⇒x-1-4y-2+3z-3=0

⇒x-4y+3z=2

Direction ratios of normal of the plane is <1,-4,3>

Direction cosines of normal of the plane is 

±126,∓426,±326

Hence,

cosβ=426>0⇒β∈0,π2,

cosα=-126⇒α∈π2,π ,

cosγ=-326⇒γ∈π2,π    

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About this question

This is a previous-year question from JEE Main 2023, covering the Three Dimensional Geometry chapter of Mathematics. PrepSharp catalogues every PYQ from JEE Main with a verified answer key and step-by-step solution prepared by IIT alumni — so you can search by chapter, topic or year and revise efficiently.