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Question

Number of common roots of the equation 
$$\displaystyle x^{3}-2x^{2}-x+2=0$$ and $$\displaystyle x^{3}+6x^{2}+11x+6=0$$ is 


A
zero
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B
one
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C
two
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D
three
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Solution

The correct option is A one
let ,
$$x^{3}-2x^{2}-x+2=0$$ ___ (1)
$$x^{3}+6x^{2}+11x+6=0$$___(2)
now put x=1 in eq (1)
$$=1^{3}-2\times 1^{2}-1+2$$
$$=1-2-1+2\Rightarrow 0$$
put x=1 in eq (2)
$$=1^{3}+6\times 1^{2}+11\times 1+6$$
$$=1+6+11+6\Rightarrow 24$$
so x=1 is not a common root of both equations.
now put x=-1 in eq (1)
$$=-1^{3}-2\times -1^{2}+1+2$$
$$=-1-2+1+2\Rightarrow 0$$
put x=-1 in eq (2)
$$=-1^{3}+6\times -1^{2}+11\times -1+6$$
$$=-1+6-11+6\Rightarrow 0$$
With similar $$x=\pm 2$$ puts in both equations has no common roots finds.
$$\therefore x=-1$$ is the common roots of the equations.

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