Trigonometric Identities Question 272
Question: The least positive non-integral solution of the equation $ \sin \pi (x^{2}+x)=\sin \pi x^{2} $ is
Options:
A) rational
B) irrational of the form $ \sqrt{p} $
C) irrational of the form $ \frac{\sqrt{p}-1}{4}, $ where p is an odd integer
D) irrational of the form $ \frac{\sqrt{p}+1}{4}, $ where p is an even integer
Show Answer
Answer:
Correct Answer: A
Solution:
We have, $ \sin \pi (x^{2}+x)=\sin \pi x^{2} $
$ \Rightarrow \pi (x^{2}+x)=n\pi +{{(-1)}^{n}}\pi x^{2} $
$ \therefore $ Either $ x^{2}+x=2m+x^{2}\Rightarrow x=2m\in I $ or $ x^{2}+x=k-x^{2}, $ where k is an odd integer
$ \Rightarrow 2x^{2}+x-k=0\Rightarrow x=\frac{-1\pm \sqrt{1+8k}}{4} $ For least positive non-integral solution is $ x=\frac{1}{2}, $ when $ k=1 $
BETA
