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Many-One onto Function

If a and ...

Question

If $a$ and $b$ are the roots of the quadratic equation $x^{2} - 4 x + 3 = 0$ , then $(1 + a + a^{2} + a^{3} . . .) (1 + b + b^{2} + b^{3} + . . . .)$ equal to

\infty

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\frac{1}{4}

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\frac{- 1}{6}

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none of these

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Solution

The correct option is A $\infty$
$x^{2} - 4 x + 3 = 0 (x - 1) (x - 3) = 0 x = 1 o r 3 ∴ a = 3, b = 1 a + b = 4, a b = 3 1 + a + a^{2} + a^{3} + . . . . . . . . + \infty = \frac{1}{1 - a} (s u m o f i n f i n i t e G . P .) ∴ (1 + a + a^{2} + a^{3} + . . . . . . . . + \infty) (1 + b + b^{2} + b^{3} + . . . . . . . + \infty) = (\frac{1}{1 - a}) (\frac{1}{1 - b}) = \frac{1}{1 - (a + b) + a b} = \frac{1}{1 - 4 + 3} = \frac{1}{0} = \infty$

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