Q47 / 160 · AP EAMCET 2020 (18 Sep Shift 2) · Mathematics

Find ' \(\lambda\) ' if \(\mathbf{a}, \mathbf{b}, \mathbf{c}\) are three non-coplanar vectors such that \(\left[\begin{array}{ccc} 4 a+3 b-c & 4 a+3 b+2 c \quad a-4 b-c \end{array}\right]\) \(=\left(\lambda^2+\lambda+1\right)\left[\begin{array}{lll}a & b & c\end{array}\right]\)

  • A
    \(-7,8\)
  • B
    \(-7,-6\)
  • Correct answer
    C
    \(7,-8\)
  • D
    \(-7,-8\)

Solution

\(\begin{array}{rlrl} \left|\begin{array}{ccc} 4 & 3 & -1 \\ 4 & 3 & 2 \\ 1 & -4 & -1 \end{array}\right|[\bar{a} \bar{b} \bar{c}] & =\left(\lambda^2+\lambda+1\right)[\bar{a} \bar{b} \bar{c}] \\ 4(5)-3(-6)-1(-19) & =\lambda^2+\lambda+1 \\ 57 & =\lambda^2+\lambda+1 \\ \Rightarrow & & \lambda^2+\lambda-56 & =0 \\ \Rightarrow & & (\lambda-7)(\lambda+8) & =0 \\ \Rightarrow & & \lambda & =7 \text { (or) }-8 \end{array}\) Hence, option (c) is correct.

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

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