How Cheenta works to ensure student success?
Explore the Back-Story

# Test of Mathematics Solution Subjective 87 - Complex Roots of a Real Polynomial  This is a Test of Mathematics Solution Subjective 87 (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.

Also visit: I.S.I. & C.M.I. Entrance Course of Cheenta

## Problem:

Let , where are complex numbers. If is real for all real , show that are real numbers. In addition to above, assume that is not real whenever is not real. Show that .

## Solution: As is real for all real , we have  is real. is real. is real. is real.

As is real is also real.

Similarly as is real is also real.

Implying is also real.

Thus all are real. Let us assume that .

Thus the equation can be written as Let be a root of the equation. If is imaginary that means is imaginary. But , thus is real. Similarly , the other root of the equation, is also real.

Therefore . Take Then  Using , we get,  Thus is real even when is imaginary. Thus our assumption that is wrong.

Hence Proved . This is a Test of Mathematics Solution Subjective 87 (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.

Also visit: I.S.I. & C.M.I. Entrance Course of Cheenta

## Problem:

Let , where are complex numbers. If is real for all real , show that are real numbers. In addition to above, assume that is not real whenever is not real. Show that .

## Solution: As is real for all real , we have  is real. is real. is real. is real.

As is real is also real.

Similarly as is real is also real.

Implying is also real.

Thus all are real. Let us assume that .

Thus the equation can be written as Let be a root of the equation. If is imaginary that means is imaginary. But , thus is real. Similarly , the other root of the equation, is also real.

Therefore . Take Then  Using , we get,  Thus is real even when is imaginary. Thus our assumption that is wrong.

Hence Proved .

### Knowledge Partner  