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NDA 2021
Exam
NDA
Year
2021
Subject
Mathematics
Chapter
Determinants

NDA 2021 · Mathematics · Determinants

If | array ccc a & -b & a-b-c -a & b & -a+b-c -a & -b & -a-b+c...

If \left|a−ba−b−c−ab−a+b−c−a−b−a−b+c\begin{array}{ccc}a & -b & a-b-c \\ -a & b & -a+b-c \\ -a & -b & -a-b+c\end{array}$\right|-k a b c=0 \quad(a \neq 0, b \neq 0, c \neq 0) then what is the value of kk ?$

Options

  1. A-4
  2. B-2
  3. C2
  4. D4

Correct answer

A. -4

Step-by-step solution

 Applying c3→c3−c1−c2∣a−b−c−ab−c−a−bc∣−kabc=0⇒abc−kabc=0Applying R2→R2+R1 and  R3→R3+R1abc∣1−1−1−11−1−1−11∣−kabc=0⇒−4abc−kabc=0⇒k=−4\begin{aligned} & \text{ Applying } c_3 \rightarrow c_3-c_1-c_2 \\ & \left|\begin{array}{ccc}a & -b & -c \\ -a & b & -c \\ -a & -b & c\end{array}\right|-k a b c=0 \\ & \Rightarrow a b c-k a b c=0 \\ & \text{Applying } R_2 \rightarrow R_2+R_1 \text{\text{ and } } R_3 \rightarrow R_3+R_1 \\ & \qquad a b c\left|\begin{array}{ccc}1 & -1 & -1 \\ -1 & 1 & -1 \\ -1 & -1 & 1\end{array}\right|-k a b c=0 \\ & \Rightarrow-4 a b c-k a b c=0 \\ & \Rightarrow k=-4\end{aligned}
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