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NDA 2022
Exam
NDA
Year
2022
Subject
Mathematics
Chapter
Determinants

NDA 2022 · Mathematics · Determinants

Under which of the following conditions does the determinant |...

Under which of the following conditions does the determinant \left|abcbcacab\begin{array}{lll}a & b & c \\ b & c & a \\ c & a & b\end{array}$\right| vanish? 1. a+b+c=0a+b+c=0 2. a3+b3+c3=3abca^3+b^3+c^3=3 a b c 3. a2+b2+c2−ab−bc−ca=0a^2+b^2+c^2-a b-b c-c a=0 Select the correct answer using the code given below :$

Options

  1. A1 and 2 only
  2. B2 and 3 only
  3. C1 and 3 only
  4. D1, 2 and 3

Correct answer

D. 1, 2 and 3

Step-by-step solution

Let \Delta=\left|abcbcacab\begin{array}{lll}a & b & c \\ b & c & a \\ c & a & b\end{array}\right|=0 Applying \quad R_1 \rightarrow R_1+R_2+R_3=\left|a+b+ca+b+ca+b+cbcacab\begin{array}{ccc}a+b+c & a+b+c & a+b+c \\ b & c & a \\ c & a & b\end{array}\right|=0=(a+b+c)\left|111bcacab\begin{array}{ccc}1 & 1 & 1 \\ b & c & a \\ c & a & b\end{array}\right|=0 Applying c2→c2−c1\quad c_2 \rightarrow c_2-c_1 and c_3 \rightarrow c_3-c_1=(a+b+c)\left|100bc−ba−bca−cb−c\begin{array}{ccc}1 & 0 & 0 \\ b & c-b & a-b \\ c & a-c & b-c\end{array}\right|=0=(a+b+c)[−a2−b2−c2+ab+bc+ca]=0=−(a+b+1)[a2+b2+c2−ab−bc−ca]=0=−(a3+b3+c3−3abc)=0⇒a+b+c=0 or a2+b2+c2−ab−bc−ca=0or a3+b3+c3=3abc\begin{aligned} & =(a+b+c)\left[-a^2-b^2-c^2+a b+b c+c a\right]=0 \\ & =-(a+b+1)\left[a^2+b^2+c^2-a b-b c-c a\right]=0 \\ & =-\left(a^3+b^3+c^3-3 a b c\right)=0 \\ & \Rightarrow a+b+\mathrm{c}=0 \text{ or } a^2+b^2+c^2-a b-b c-c a=0 \\ & \text{or } a^3+b^3+c^3=3 a b c\end{aligned}$$ Hence all statement 1,2 and 3 are correct.
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