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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 $\operatorname{hof}(p(x), q(x)$ $=x-1$ and $k(x)=1 c m(p(x), q(x))$. If the coefficient of the highest degree term of $k(x)$ is 1, the sum of the roots of $(x-1)+k(x)$ is.
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The correct answer is:
7
$\quad \therefore$ HCF $=x-1$
$\quad \Rightarrow p(x)=x^{2}-5 x+a$
$=x^{2}-5 x+4$
$=(x-1)(x-4) \quad ......\text {(1)}$
and $\quad q(x) x^{2}-3 x+b=x^{2}-3 x+2$
$=(x-1)(x-2)$
$\Rightarrow k(x)=(x-1)(x-2)(x-4) \quad ......\text {(2)}$
Hence
$(x-1)+R(x)=(x-1)+(x-1)(x-2)(x-2)(x-4)$
$=(x-1)(x-3)^{2}$
Hence sum of roots $=7$
$\quad \Rightarrow p(x)=x^{2}-5 x+a$
$=x^{2}-5 x+4$
$=(x-1)(x-4) \quad ......\text {(1)}$
and $\quad q(x) x^{2}-3 x+b=x^{2}-3 x+2$
$=(x-1)(x-2)$
$\Rightarrow k(x)=(x-1)(x-2)(x-4) \quad ......\text {(2)}$
Hence
$(x-1)+R(x)=(x-1)+(x-1)(x-2)(x-2)(x-4)$
$=(x-1)(x-3)^{2}$
Hence sum of roots $=7$
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