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 , 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
.
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
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
.
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 .