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# Test of Mathematics Solution Subjective 76 - Range of a Rational Polynomial  This is a Test of Mathematics Solution Subjective 76 (from ISI Entrance). The book, Test of Mathematics at 10+2 Level is Published by East West Press. This problem book is indispensable for the preparation of I.S.I. B.Stat and B.Math Entrance.

## Problem

Find the set of all values of such that can take all real values.

## Solution       Now takes all real values if discriminant is allways .
So now we have to find the all values of such that  for all  |R.    ...(i)
Now this is a equation of upside open parabola. If the discriminant is of equation (i) then will always positive.        Conclusion:

If , then can take all the values as varies over |R. This is a Test of Mathematics Solution Subjective 76 (from ISI Entrance). The book, Test of Mathematics at 10+2 Level is Published by East West Press. This problem book is indispensable for the preparation of I.S.I. B.Stat and B.Math Entrance.

## Problem

Find the set of all values of such that can take all real values.

## Solution       Now takes all real values if discriminant is allways .
So now we have to find the all values of such that  for all  |R.    ...(i)
Now this is a equation of upside open parabola. If the discriminant is of equation (i) then will always positive.        Conclusion:

If , then can take all the values as varies over |R.

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