The number of distinct real values of λ, for which the vectors -λ2 hat i + hat j + hat k , hat i -λ2 hat j + hat k and hat i + hat j -λ2 hat k are coplanar, is

Question

The number of distinct real values of λ, for which the vectors -λ2 hat i + hat j + hat k , hat i -λ2 hat j + hat k and hat i + hat j -λ2 hat k are coplanar, is (A) 0 (B) 1 (C) 2 (D) 3

Tardigrade Admin · Accepted Answer

Since, given vectors are coplanar ∴ |-λ2 1 1 1 -λ2 1 1 1 -λ2|=0 ⇒ λ6-3 λ2-2=0 ⇒(1+λ2)2(λ2-2)=0 ⇒ λ=± √2

Q. The number of distinct real values of $λ$ , for which the vectors $- λ^{2} \hat{i} + \hat{j} + \hat{k}, \hat{i} - λ^{2} \hat{j} + \hat{k}$ and $\hat{i} + \hat{j} - λ^{2} \hat{k}$ are coplanar, is

0

1

2

3

Since, given vectors are coplanar
$∴ ∣ ∣ - λ^{2} 11 1 - λ^{2} 1 11 - λ^{2} ∣ ∣ = 0$
$\Rightarrow λ^{6} - 3 λ^{2} - 2 = 0$
$\Rightarrow (1 + λ^{2})^{2} (λ^{2} - 2) = 0$
$\Rightarrow λ = \pm 2$

Q. The number of distinct real values of λ, for which the vectors −λ2i^+j^​+k^,i^−λ2j^​+k^ and i^+j^​−λ2k^ are coplanar, is

0

1

2

3

Since, given vectors are coplanar ∴∣∣​−λ211​1−λ21​11−λ2​∣∣​=0 ⇒λ6−3λ2−2=0 ⇒(1+λ2)2(λ2−2)=0 ⇒λ=±2​

Q. The number of distinct real values of $λ$ , for which the vectors $- λ^{2} \hat{i} + \hat{j} + \hat{k}, \hat{i} - λ^{2} \hat{j} + \hat{k}$ and $\hat{i} + \hat{j} - λ^{2} \hat{k}$ are coplanar, is

Since, given vectors are coplanar
$∴ ∣ ∣ - λ^{2} 11 1 - λ^{2} 1 11 - λ^{2} ∣ ∣ = 0$
$\Rightarrow λ^{6} - 3 λ^{2} - 2 = 0$
$\Rightarrow (1 + λ^{2})^{2} (λ^{2} - 2) = 0$
$\Rightarrow λ = \pm 2$