What must be added to $1 - x + x^{2} - 2 x^{3}$ to get $x^{3}$ ?

$x^{3} - x^{2} + x - 1$

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$- 1 + x + x^{2} - 3 x^{3}$

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$3 x^{3} - x^{2} + x - 1$

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None of these

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Solution

The correct option is B: $3 x^{3} - x^{2} + x - 1$
Let the term to be added be $y$ .
$\Rightarrow (1 - x + x^{2} - 2 x^{3}) + y = x^{3}$
$\Rightarrow y = x^{3} - (1 - x + x^{2} - 2 x^{3})$
$\Rightarrow$ $y = x^{3} - 1 + x - x^{2} + 2 x^{3}$
$∴$ $y = 3 x^{3} - x^{2} + x - 1$
so, $3 x^{3} - x^{2} + x - 1$ must be added to $(1 - x + x^{2} - 2 x^{3})$ to get $x^{3}$ .

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