The number of all common roots of the equatio | vedclass

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(D) First,factorize the given equation $x^4-10x^3+37x^2-60x+36=0$. By testing values,we find $x=2$ and $x=3$ are roots. Dividing by $(x-2)^2(x-3)^2$,w

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$2$

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(D) First,factorize the given equation $x^4-10x^3+37x^2-60x+36=0$. By testing values,we find $x=2$ and $x=3$ are roots. Dividing by $(x-2)^2(x-3)^2$,we get $(x-2)^2(x-3)^2=0$. Thus,the roots are $2, 2, 3, 3$. Let the roots be $r_1=2, r_2=2, r_3=3, r_4=3$. We increase two distinct roots by $1$. The distinct roots are $2$ and $3$. Increasing these by $1$ gives new roots $3$ and $4$. The other two roots remain fixed as $2$ and $3$. So the new roots are $2, 3, 3, 4$. The original roots are ${2, 2, 3, 3}$ and the new roots are ${2, 3, 3, 4}$. The common roots are $2, 3, 3$. However,the question asks for the number of common roots. Comparing the sets,the common values are $2$ and $3$. Thus,there are $2$ distinct common roots.

The number of all common roots of the equation $x^4-10x^3+37x^2-60x+36=0$ and the transformed equation obtained by increasing any two distinct roots of it by $1$,keeping the other two roots fixed,is

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