If - a a2 1-a3 b b2 1-b3 c c2 1-c3- -0 and the vectors A-1-a-a2 - B-1-b-b2- C-1-c-c2are noncoplanar- the value of abc-

The correct option is B 1 0 = | a amp;a^2 amp;1+a^3 b amp;b^2 amp;1+b^3 c amp;c^2 amp;1+c^3| nbsp; =| a amp;a^2 amp;1 ...

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Condition for Coplanarity of Four Points

If | a a2 ...

Question

If $∣ ∣ ∣ ∣ ∣ \begin{matrix} a & a^{2} & 1 + a^{3} b & b^{2} & 1 + b^{3} c & c^{2} & 1 + c^{3} \end{matrix} ∣ ∣ ∣ ∣ ∣ = 0$ and the vectors
$\to A = (1, a, a^{2}), \to B = (1, b, b^{2}), \to C = (1, c, c^{2})$ are noncoplanar, the value of $a b c =$

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∣ ∣ ∣ ∣ ∣ \begin{matrix} a & a^{2} & 1 + a^{3} b & b^{2} & 1 + b^{3} c & c^{2} & 1 + c^{3} \end{matrix} ∣ ∣ ∣ ∣ ∣ = 0

A = (1, a, a^{2}), B = (1, b, b^{2}), C = (1, c, c^{2})

a b c

∣ ∣ ∣ ∣ ∣ \begin{matrix} a & a^{2} & 1 + a^{3} b & b^{2} & 1 + b^{3} c & c^{2} & 1 + c^{3} \end{matrix} ∣ ∣ ∣ ∣ ∣ = 0

\to A = 1, a, a^{2}, \to B = 1, b, b^{2}, \to C = 1, c, c^{2}

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∣ ∣ ∣ ∣ ∣ \begin{matrix} a & a^{2} & 1 + a^{3} b & b^{2} & 1 + b^{3} c & c^{2} & 1 + c^{3} \end{matrix} ∣ ∣ ∣ ∣ ∣ = 0

(1, a, a^{2}), (1, b, b^{2})

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∣ ∣ ∣ ∣ ∣ \begin{matrix} a & a^{2} & 1 + a^{3} b & b^{2} & 1 + b^{3} c & c^{2} & 1 + c^{3} \end{matrix} ∣ ∣ ∣ ∣ ∣ = 0

¯ A = (1, a, a^{2})

¯ B = (1, b, b^{2})

¯ C = (1, c, c^{2})

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∣ ∣ ∣ ∣ ∣ \begin{matrix} a & a^{2} & 1 + a^{3} b & b^{2} & 1 + b^{3} c & c^{2} & 1 + c^{3} \end{matrix} ∣ ∣ ∣ ∣ ∣ = 0

- a b c =

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