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NDA 2021
Exam
NDA
Year
2021
Subject
Mathematics
Chapter
Determinants

NDA 2021 · Mathematics · Determinants

If a+b+c=4 and a b+b c+c a=0 , then what is the value of the...

If a+b+c=4andab+bc+ca=0a+b+c=4 and a b+b c+c a=0, then what is the value of the following determinant? \left|abcbcacab\begin{array}{lll}a & b & c \\ b & c & a \\ c & a & b\end{array}$$\right|

Options

  1. A3232
  2. B−64-64
  3. C−128-128
  4. D6464

Correct answer

B. −64-64

Step-by-step solution

(a+b+c)=4(Given)Now,(Given) Now,\left|abcbcacab\begin{array}{lll}a & b & c \\ b & c & a \\ c & a & b\end{array}\right|=a \left(b c-a^2\right)+b \left(a c-b^2\right)+c(ab−c2)+c \left(a b-c^2\right)=3 a b c-a^3-b^3-c^3=−(a3+b3+c3−3abc)=−(a+b+c)3=−(4)3=−64.\begin{aligned} & =-\left(a^3+b^3+c^3-3 a b c\right)=-(a+b+c)^3 \\ & =-(4)^3=-64 .\end{aligned}
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