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NDA 2018
Exam
NDA
Year
2018
Subject
Mathematics
Chapter
Determinants

NDA 2018 · Mathematics · Determinants

If a+b+c=0 , then one of the solution of | array ccc a-x & c & b c...

If a+b+c=0a+b+c=0, then one of the solution of \left|a−xcbcb−xabac−x\begin{array}{ccc} a-x & c & b \\ c & b-x & a \\ b & a & c-x \end{array}\right|=0 \text{\text{ is } }

Options

  1. Ax=a\mathrm{x}=\mathrm{a}
  2. B\mathrm{x}=\sqrt{\frac{3\left(\mathrm{a}^2+\mathrm{b}^2+\mathrm{c}^2\right)}{2}}
  3. Cx=\sqrt{\frac{2\left(\mathrm{a}^2+\mathrm{b}^2+\mathrm{c}^2\right)}{3}}
  4. Dx=0\mathrm{x}=0

Correct answer

D. x=0\mathrm{x}=0

Step-by-step solution

∣a−xcbcb−xabac−x∣=0−∣x−(a+b+c)cbx−(a+b+c)b−xax−(a+b+c)ac−x∣=0⇒∣xcbxb−xaxac−x∣=0{ Applying C1→C1+C2+C3 } ∣xcb0c+x−bb−a0c−ab+x−c∣=0 {Applying R2→R1−R2 and R3→R1−R3 } x{(c+x−b)(b+x−c)−(b−a)(c−a)}=0x{(x2−(b−c)2−bc+ac+ab−a2}=0x(x2−a2−b2−c2+ab+bc+ca)=0x{x2−(a−b)2−(b−c)2−(c−a)2}=0∴x=0\begin{aligned} \left|\begin{array}{ccc} a-x & c & b \\ c & b-x & a \\ b & a & c-x \end{array}\right|=0 \\ -\left|\begin{array}{ccc} x-(a+b+c) & c & b \\ x-(a+b+c) & b-x & a \\ x-(a+b+c) & a & c-x \end{array}\right|=0 \\ \Rightarrow\left|\begin{array}{ccc} x & c & b \\ x & b-x & a \\ x & a & c-x \end{array}\right|=0 \\ \left\{\text{ Applying } \mathrm{C}_1 \rightarrow \mathrm{C}_1+\mathrm{C}_2+\mathrm{C}_3\right. \text{ \} } \\ \left|\begin{array}{ccc} x & c & b \\ 0 & c+x-b & b-a \\ 0 & c-a & b+x-c \end{array}\right|=0 \\ \text{ \{Applying } \mathrm{R}_2 \rightarrow \mathrm{R}_1-\mathrm{R}_2 \text{ and } \mathrm{R}_3 \rightarrow \mathrm{R}_1-\mathrm{R}_3 \text{ \} } \\ x\{(c+x-b)(b+x-c)-(b-a)(c-a)\}=0 \\ x\left\{\left(x^2-(b-c)^2-b c+a c+a b-a^2\right\}=0\right. \\ x\left(x^2-a^2-b^2-c^2+a b+b c+c a\right)=0 \\ x\left\{x^2-(a-b)^2-(b-c)^2-(c-a)^2\right\}=0 \\ \begin{array}{ll} \therefore & x=0 \\ \end{array} \end{aligned}
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