The number of distinct real values of \lambda for which the vectors \lambda^2 \hat{i}+\hat{j}+\hat{k}, \hat{i}+\lambda^2 \hat{j}+\hat{k} and \hat{i}+\hat{j}+\lambda^2 \hat{k} are coplanar is:

NIMCET 2024 Paper

Section: Mathematics

Question : 5 of 120

Marks: +1, -0

The number of distinct real values of

λ

for which the vectors

λ^2 {i}↖{∧}+{j}↖{∧}+{k}↖{∧}, \;\; {i}↖{∧}+λ^2 {j}↖{∧}+{k}↖{∧}

and

{i}↖{∧}+{j}↖{∧}+λ^2 {k}↖{∧}

are coplanar is:

Solution:

For coplanar vectors, the scalar triple product is zero. Hence

{|\table λ^2 , 1 , 1 ; 1 , λ^2 , 1 ; 1 , 1 , λ^2 |}=0

Apply

C_1=C_1+C_2+C_3

(λ^2+2) {|\table 1 , 1 , 1 ; 1 , λ^2 , 1 ; 1 , 1 , λ^2 |}=0

Apply

R_1=R_1-R_2

(λ^2+2)

1

1

⇒ (λ^2+2) (1-λ^2) (1-λ^2) =0

Hence

λ= ± 1

. There are two possible values.

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