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JEE Advanced2025MathematicsVector AlgebraActual

Let w = i + j -2 k , and u and v be two vectors such that u v = w and v w = u . Let , , and t be real numbers such that u = i + j + k ,-t + + =0, -t + =0 , and + -t =0 . Match each entry in List-I to the correct entry in List-II and choose the correct option. List-I List-II (P) | v |^2 is equal to (1) 0 (Q) If = 3 ,then ^2 is equal to (2) 1 (R) If = 3 ,then ( + )^2 is equal to (3) 2 (S) If = 2 ,then t+3 is equal to (

Options

  1. A( P ) (2),( Q ) (1),( R ) (4),( S ) (5)
  2. B( P ) (2),( Q ) (4),( R ) (3),( S ) (5)
  3. C( P ) (2),( Q ) (1),( R ) (4),( S ) (3)
  4. D( P ) (5),( Q ) (4),( R ) (1),( S ) (3)

Correct answer

A. ( P ) (2),( Q ) (1),( R ) (4),( S ) (5)

Step-by-step solution

array lll u v = w & u v & u w v w = u & v w & array aligned & w = i + j -2 k & u = i + j + k aligned | array ccc - t & 1 & 1 1 & - t & 1 1 & 1 & - t array |=0 array ll u . w =0= + -2 & t=-1 or 2 ( v . w ) v = u v & t=-1 w | v |^2-0= w & + + =0 | v |=1(P) & t=2 & = = array aligned & Also, u v = w & | u || v |=| w | & | u |=| w |= 6 aligned Case-1: aligned & t =2 & = = & ^2+ ^2+ ^2=6 & = 2 ,- 2 & (S) t +3=5 aligned Case-2: aligned & t =-1 & + + =0 & + -2 =0 & =0, =- & ( Q ) = 3 , ^2=0, =- 3 & ( R ) = 3 ( + )^2= ^2=3

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