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NDA 2022
Exam
NDA
Year
2022
Subject
Mathematics
Chapter
Determinants

NDA 2022 · Mathematics · Determinants

If (a-b)(b-c)(c-a)=2 and a b c=6 , then what is the value of |...

If (a−b)(b−c)(c−a)=2andabc=6(a-b)(b-c)(c-a)=2 and a b c=6, then what is the value of \left|abca2b2c2a3b3c3\begin{array}{ccc}a & b & c \\ a^2 & b^2 & c^2 \\ a^3 & b^3 & c^3\end{array}\right| ?

Options

  1. A3
  2. B12
  3. C14
  4. D15

Correct answer

B. 12

Step-by-step solution

Let \Delta=\left|abca2b2c2a3b3c3\begin{array}{ccc}a & b & c \\ a^2 & b^2 & c^2 \\ a^3 & b^3 & c^3\end{array}\right|=abc∣111abca2b2c2∣=abc∣c1→c1−c2 and c2→c2−c3a−b01(a−b)(a+b)(b−c)(b+c)c2∣=abc(a−b)(b−c)∣00111ca+b(b+c)c2∣\begin{aligned} & \quad=a b c\left|\begin{array}{ccc}1 & 1 & 1 \\ a & b & c \\ a^2 & b^2 & c^2\end{array}\right| \\ & =a b c\left|\begin{array}{ccc} \\ c_1 \rightarrow c_1-c_2 & \text{ and } c_2 \rightarrow c_2-c_3 \\ a-b & 0 & 1 \\ (a-b)(a+b) & (b-c)(b+c) & c^2\end{array}\right| \\ & =a b c(a-b)(b-c)\left|\begin{array}{ccc}0 & 0 & 1 \\ 1 & 1 & c \\ a+b & (b+c) & c^2\end{array}\right|\end{aligned}$\begin{aligned} & =a b c(a-b)(b-c)(c-a) \\ & =6 \times 2=12\end{aligned}$
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