Q188 / 270 · NDA 2024 (Phase 1) · Mathematics

If in a triangle $A B C, \sin ^3 A+\sin ^3 B+\sin ^3 C$ $=3 \sin \mathrm{~A} \sin \mathrm{~B} \sin \mathrm{C}$, then what is the value of the determinant $\left|\begin{array}{lll}a & b & c \\ b & c & a \\ c & a & b\end{array}\right|$; where $a, b, c$ are sides of the triangle?

  • A
    $a+b+c$
  • B
    $a b+b c+c a$
  • C
    $(a+b)(b+c)(c+a)$
  • Correct answer
    D
    0

Solution

Given that, $\begin{aligned} & \sin ^3 A+\sin ^3 B+\sin ^3 C \\ & =3 \sin \mathrm{~A} \cdot \sin \mathrm{~B} \cdot \sin \mathrm{C} \\ & \Rightarrow \sin \mathrm{~A}+\sin \mathrm{B}+\sin \mathrm{C}=0 \\ & \Rightarrow \quad a k+k b+k c=0 \\ & \Rightarrow \quad a+b+c=0 \\ & \left|\begin{array}{lll} a & b & c \\ b & c & a \\ c & a & b \end{array}\right|=\left|\begin{array}{lll} a+b+c & b & c \\ b+c+a & c & a \\ c+a+b & a & b \end{array}\right| \\ & \text { (Applying } \mathrm{C}_1 \rightarrow \mathrm{C}_1+\mathrm{C}_2+\mathrm{C}_3 \text { ) } \\ & =\left|\begin{array}{lll} 0 & b & c \\ 0 & c & a \\ 0 & a & b \end{array}\right|=0 . \end{aligned}$

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

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