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

May 16, 2011

I.S.I. 10+2 Subjectives Solution

P148. Show that there is no real constant c > 0 such that (\cos\sqrt{x+c}=\cos\sqrt{x}) for all real numbers (x\ge 0).
Solution:

If the given equation holds for some constant c>0 then,

f(x) = (\cos\sqrt{x}-\cos\sqrt{x+c}=0) for all (x\ge 0)
(\Rightarrow 2\sin\frac{\sqrt{x+c}+\sqrt{x}}{2}\sin\frac{\sqrt{x+c}-\sqrt{x}}{2}=0)
Putting x=0, we note
(\Rightarrow\sin^2\frac{\sqrt{c}}{2}=0)
As (c\not=0)
(\sqrt{c}=2n\pi)
(\Rightarrow c=4n^2\pi^2)
We put n=1 and x=(\frac{\pi}{2}) to note that f(x) is not zero.
Hence no c>0 allows f(x) =0 for all (x\ge 0). (proved)

Leave a Reply

This site uses Akismet to reduce spam. Learn how your comment data is processed.

Cheenta. Passion for Mathematics

Advanced Mathematical Science. Taught by olympians, researchers and true masters of the subject.
JOIN TRIAL
support@cheenta.com
enter