JEE Advanced 2024 Paper 1
Exam JEE Advanced
Year 2024
Paper Paper 1
Subject Mathematics
Chapter Vector Algebra JEE Advanced 2024 · Paper 1 · Mathematics · Vector Algebra
Let O P =(α-1)/(α) i + j + k , O Q = i +(β-1)/(β) j + k and O R = i...
Let
O P → = α − 1 α i ^ + j ^ + k ^ , O Q → = i ^ + β − 1 β j ^ + k ^ \overrightarrow{O P}=\frac{\alpha-1}{\alpha} \hat{i}+\hat{j}+\hat{k}, \overrightarrow{O Q}=\hat{i}+\frac{\beta-1}{\beta} \hat{j}+\hat{k} OP = α α − 1 i ^ + j ^ + k ^ , OQ = i ^ + β β − 1 j ^ + k ^ and
O R → = i ^ + j ^ + 1 2 k ^ \overrightarrow{O R}=\hat{i}+\hat{j}+\frac{1}{2} \hat{k} OR = i ^ + j ^ + 2 1 k ^ be three vectors, where
α , β ∈ R − { 0 } \alpha, \beta \in \mathbb{R}-\{0\} α , β ∈ R − { 0 } and
O O O denotes the origin. If
( O P → × O Q → ) ⋅ O R → = 0 (\overrightarrow{O P} \times \overrightarrow{O Q}) \cdot \overrightarrow{O R}=0 ( OP × OQ ) ⋅ OR = 0 and the point
( α , β , 2 ) (\alpha, \beta, 2) ( α , β , 2 ) lies on the plane
3 x + 3 y − z + l = 0 3 x+3 y-z+l=0 3 x + 3 y − z + l = 0 , then the value of
l l l is ____
Step-by-step solution
(\overrightarrow{\mathrm{OP}} \times \overrightarrow{\mathrm{OQ}}) \cdot \overrightarrow{\mathrm{OR}}=0 \left| α − 1 α 1 1 1 β − 1 β 1 1 1 1 2 \begin{array}{ccc}\frac{\alpha-1}{\alpha} & 1 & 1 \\ 1 & \frac{\beta-1}{\beta} & 1 \\ 1 & 1 & \frac{1}{2}\end{array} α α − 1 1 1 1 β β − 1 1 1 1 2 1 \right|=0 \alpha+\beta+1=0...(1)
Also
( α , β , 2 ) \quad(\alpha, \beta, 2) ( α , β , 2 ) lies on
3 x + 3 y − z + l = 0 ⇒ 3 α + 3 β − 2 + l = 0 ⇒ l = 2 − 3 ( α + β ) 3 \mathrm{x}+3 \mathrm{y}-\mathrm{z}+l=0 \Rightarrow \quad 3 \alpha+3 \beta-2+l=0 \quad \Rightarrow \quad l=2-3(\alpha+\beta) 3 x + 3 y − z + l = 0 ⇒ 3 α + 3 β − 2 + l = 0 ⇒ l = 2 − 3 ( α + β ) ...(2)
use (1) in (2)
⇒ l = 5 \Rightarrow l=5 ⇒ l = 5
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