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$f\left ( x \right )= \left ( x-a \right )\left ( x-b \right )-1=0$    $\left ( Where\; a< b \right )$ has

Option 1)
Both roots in   $\left ( -\infty,a \right )$

Option 2)
Both roots in $\left ( a,b \right )$

Option 3)
Both roots in $\left ( b,\infty \right )$

Option 4)
One in $\left ( -\infty,a \right )$ and one in   $\left ( b,\infty \right )$

Answers (1)

$f\left ( -\infty \right )=+\infty$

$f\left ( a \right )=-1$

$f\left ( b \right )=-1$

$f\left ( \infty \right )=+\infty$

$\because$ $f\left (- \infty \right )$ and $f\left (a \right )$ are of opposite sign so atleast one root lies in $\left ( -\infty ,a \right )$

Similarly atleast one root lies in $\left ( b,\infty \right )$

But, since it is a quadratic equation so it can't have more than two roots so exactly one lies in $\left ( -\infty ,a \right )$ and exactly one lies in $\left ( b,\infty \right )$

$\therefore$ Option (D)

Number of roots of polynomial equation -

For a polynomial equation $P\left ( x \right )= 0$ if $P\left ( a \right )$ and $P\left ( b \right )$ are of opposite sign then odd number of roots lie between $a$ and $b$ , if they are of same sign then either no root or even number of roots lie between them.

-

Option 1)

Both roots in $\left ( -\infty,a \right )$

This is incorrect

Option 2)

Both roots in $\left ( a,b \right )$

This is incorrect

Option 3)

Both roots in $\left ( b,\infty \right )$

This is incorrect

Option 4)

One in $\left ( -\infty,a \right )$ and one in $\left ( b,\infty \right )$

This is correct

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