The smaller value of n for which $x^{2} - 2 x - 3$ and $x^{3} - 2 x^{2} - n x - 3$ have an H.C.F. involving $x$ is

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Solution

The correct option is B 2
$L e t f (x) = x^{2} - 2 x - 3 a n d g (x) = x^{3} - 2 x^{2} - n x - 3. f (x) a n d g (x) w i l l h a v e a n H C F i f a n y o f t h e f a c t o r s o f f (x) d i v i d e s g (x) c o m p l e t e l y . N o w f (x) = x^{2} - 2 x - 3 = (x - 3) (x + 1) \Rightarrow x = 3, - 1. ∴ E i t h e r (x - 3) o r (x + 1) o r b o t h w i l l b e a f a c t o r o f g (x) . ∴ g (3) a n d g (- 1) s h o u l d b e z e r o . S o g (3) = 27 - 18 - 3 n - 3 = 0 \Rightarrow n = 2 a n d g (- 1) = - 1 - 2 + n - 3 = 0 \Rightarrow n = 6 ∴ s m a l l e r v a l u e o f n i s 2 (A n s)$

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