Q71 / 150 · MHT CET 2021 (24 Sep Shift 1) · Mathematics

If $\bar{a}=2 \hat{i}-\hat{j}+\hat{k}, \bar{b}=\hat{i}+2 \hat{j}-3 \hat{k}$ and $\bar{c}=3 \hat{i}+\lambda \hat{j}+5 \hat{k}$ are coplanar, then $\lambda$ is the root of the equation

  • A
    $x^2+3 x=6$
  • B
    $x^2+2 x=4$
  • Correct answer
    C
    $x^2+3 x=4$
  • D
    $x^2+2 x=6$

Solution

Since given vectors are coplanar, we write $$ \begin{aligned} & \left|\begin{array}{ccc} 2 & -1 & 1 \\ 1 & 2 & -3 \\ 3 & \lambda & 5 \end{array}\right|=0 \\ & \therefore 2(10+3 \lambda)+1(5+9)+1(\lambda-6)=0 \\ & \therefore 20+6 \lambda+14+\lambda-6=0 \Rightarrow 7 \lambda+28=0 \Rightarrow \lambda=-4 \end{aligned} $$ Which is one of the root of equation in option $C$

Hence, the correct answer is option: 'C'.

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