Question

If a b c belongs to R such that $a^{3} + b^{3} + c^{3} = 2$ $and \left[\begin{array}{ccc}a&b&c\\b&c&a\\c&a&b\end{array}\right]$ = 0 then find a b c

Answer 1

Answer:

Given Data:

a, b, c ∈ R

a³ + b³ + c³ = 2

and

$\left[\begin{array}{ccc}a&b&c\\b&c&a\\c&a&b\end{array}\right] = 0$

To find: the value of a, b, c

Solution:

So if,

$\left[\begin{array}{ccc}a&b&c\\b&c&a\\c&a&b\end{array}\right] = 0\\\\then,$

=> 3abc - a³ -b³ -c³ = 0

we know that,  a³ + b³  + c³ = 2

So,

=> 3abc = a³ + b³ + c³

=> 3abc = 2

=> abc = 2/3

The value of abc is $\frac{2}{3}$ .

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Answer 2

GIVEN:-

• a b c belongs to R such that $a^{3} + b^{3} + c^{3} = 2$ $and \left[\begin{array}{ccc}a&b&c\\b&c&a\\c&a&b\end{array}\right]$ = 0

TO FIND :-

• Value of a b c

SOLUTION :-

$\rightarrow \: \left[ \begin{array}{ccc}a b \: c \: \\ b \: c \: a \\ c \: a \: b \end{array}\right ]$

$\implies \: a(bc - a {}^{2} ) - b(b {}^{2} - ac) + c(ab - c {}^{2} ) \\ \\ \implies \: abc - a {}^{3} - b {}^{3} + abc - c {}^{3} + abc = 0 \\ \\ \implies \: 3abc - (a {}^{3} + b {}^{3} + c {}^{3} ) = 0$

Given that ,

$\rightarrow \: a {}^{3} + b {}^{3} + c {}^{3} = 2 \\ \\ so \: \: \: 3abc - 2=0 \\ \\ \implies \: 3abc = 2 \\ \\ \implies \: abc = \frac{2}{3}$

Hence ,Answer is ,

$\rightarrow \: \boxed{abc = \frac{2}{3} }$

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