INTRODUCING 5 - days-a-week problem solving session for Math Olympiad and ISI Entrance. Learn More

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

Let \(P(z) = az^2+ bz+c\), where \(a,b,c\) are complex numbers.

\((a)\) If \(P(z)\) is real for all real \(z\), show that \(a,b,c\) are real numbers.

\((b)\) In addition to \((a)\) above, assume that \(P(z)\) is not real whenever \(z\) is not real. Show that \(a=0\).

\((a)\) As \(P(z)\) is real for all real \(z\), we have \(P(0)=c\) \(=> c\) is real.

\(P(1) = a+b+c\) is real.

\(P(-1) = a-b+c\) is real.

\(P(1) + P(-1) = 2a+2c\) is real.

As \(c\) is real \(=> a\) is also real.

Similarly as \((a+b+c)\) is real \(=> (a+b+c)-(a+c)\) is also real.

Implying \(b\) is also real.

Thus all \(a,b,c\) are real.

\((b)\)Let us assume that \(a\neq 0\).

Thus the equation can be written as \(P'(z)=z^2 + \frac{b}{a} z + \frac{c}{a} = 0\)

Let \(\alpha\) be a root of the equation. If \(\alpha\) is imaginary that means \(P'(\alpha)\) is imaginary. But \(P'(\alpha)=0\), thus \(\alpha\) is real. Similarly \(\beta\), the other root of the equation, is also real.

Therefore \(\alpha + \beta = -\frac{b}{a}\). \(\cdots (i)\)

Take \(x=\frac{\alpha + \beta}{2} + i\)

Then \(P'(x) = \frac{(\alpha + \beta)^2}{4} + (\alpha + \beta)i -1 + \frac{b}{a}(\frac{\alpha + \beta}{2} + i) + \frac{c}{a}\)

\(=> P'(x) = \frac{(\alpha + \beta)^2}{4} + (\alpha + \beta)i -1 + \frac{b}{a} \frac{\alpha + \beta}{2} +\frac{b}{a} i + \frac{c}{a}\)

Using \((i)\), we get,

\(=> P'(x) = \frac{(\alpha + \beta)^2}{4} - \frac{b}{a} i -1 + \frac{b}{a} \frac{\alpha + \beta}{2} +\frac{b}{a} i + \frac{c}{a}\)

\(=> P'(x) = \frac{(\alpha + \beta)^2}{4} -1 + \frac{b}{a} \frac{\alpha + \beta}{2} + \frac{c}{a}\)

Thus \(P'(x)\) is real even when \(x\) is imaginary. Thus our assumption that \( a \neq 0\) is wrong.

Hence Proved \(a=0\).

Advanced Mathematical Science. Taught by olympians, researchers and true masters of the subject.

JOIN TRIAL