Let $p (x) = x^{2} - 5 x + 1$ and $q (x) = x^{2} - 3 x + b$ . where $a$ and $b$ are positive integers. Suppose HCF $(p (x), q (x)) = x - 1$ and $k (x) =$ LCM $(p (x), q (x))$ . If the coefficient of the highest degree term of $k (x)$ is $1$ , the sum of the roots of $(x - 1) + k (x)$ is $?$

4

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7

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Solution

The correct option is D $7$
$p (x) = x^{2} - 5 x + a$
$q (x) = x^{2} - 3 x + b$

Given that $H C F (p (x), q (x)) = x - 1$ , the other factors of $p (x)$ and $q (x)$ become $(x - 4)$ and $(x - 2)$ respectively.

$∴ p (x) = (x - 1) (x - 4)$

$q (x) = (x - 1) (x - 2)$

$\Rightarrow k (x) = L C M (p (x), q (x)) = (x - 1) (x - 2) (x - 4)$

$\Rightarrow (x - 1) + k (x) = (x - 1) + (x - 1) (x - 2) (x - 4)$

$= (x - 1) (x^{2} - 6 x + 8 + 1)$

$= (x - 1) (x^{2} - 6 x + 9)$

$= x^{3} - 7 x^{2} + 15 x - 9$

So, by applying the relation between the coefficients and the roots , we get sum of roots $= 7$ .

Sum of roots is thus $7$ ,

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Let $p (x) = x^{2} - 5 x + a$ and $q (x) = x^{2} - 3 x + b$ , where a and b are positive integers. Suppose hof(p(x),q(x)) = x – 1 and k(x) = 1cm (p(x), q(x)). If the coefficient of the highest degree term of k(x) is 1, the sum of the roots of (x – 1) + k(x) is.