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Question

If $$b>a$$, then the equation $$(x - a) (x - b)-1=0$$ has


A
both roots in [a,b]
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B
both roots in (,a)
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C
both roots in (b,)
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D
one root in (,a) and other in (b,)
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Solution

The correct option is D one root in $$ \left( -\infty ,a \right) $$ and other in $$ \left( b,\infty \right) $$
Let $$f(x)=(x-a)(x-b)-1$$
$$f(a)=-1$$    which is negative.
$$f(b)=-1$$   this is also negative.
Therefore, roots do not lie between $$a$$ and $$b$$.
Thus, one root in $$(-\infty ,a)$$ and other root lies in $$(b,\infty )$$.
Hence, option 'D' is correct.

Mathematics

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