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Vectors — Maths STD 12 Science — Question

Maharashtra BoardEnglish MediumSTD 12 ScienceMathsVectors2 Marks
MCQ
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
  • A
    Zero
  • B
    One
  • ✓
    Two
  • D
    Three
✓

Answer

Correct option: C.
Two
(C) Let $\overline{ a }, \overline{ b }$ and $\overline{ c }$ be the given vectors. The vectors are coplanar.
$\therefore\left|\begin{array}{ccc}-\lambda^2 & 1 & 1 \\ 1 & -\lambda^2 & 1 \\ 1 & 1 & -\lambda^2\end{array}\right|=0$
$\Rightarrow-\lambda^2\left(\lambda^4-1\right)-1\left(-\lambda^2-1\right)+1\left(1+\lambda^2\right)=0 \\ \Rightarrow \lambda^6-3 \lambda^2-2=0 \\ \Rightarrow\left(1+\lambda^2\right)^2\left(\lambda^2-2\right)=0 \\ \Rightarrow \lambda= \pm \sqrt{2}$

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