Try this problem from I.S.I. B.Stat Entrance Objective Problem based on Periodic Function.
If f(x) = \(a_0+a_1cosx+a_2cos2x+....+a_ncosnx\) where \(a_0,a_1,....,a_n\) are non zero real numbers and \(a_n > |a_0|+|a_1|+....+|a_{n-1}|\), then number of roots of f(x)=0 in \( 0 \leq x \leq 2\pi\), is
Periodic
Real Numbers
Inequality
Answer:at least 2n
B.Stat Objective Problem 710
Challenges and Thrills of Pre-College Mathematics by University Press
f is periodic with period \(2\pi\)
here \(0 <|\displaystyle\sum_{k=0}^{n-1}a_kcos(kx)| \leq \displaystyle\sum_{k=0}^{n-1}|a_k|<a_n, x\in\)set of reals
for points \(x_k=\frac{k\pi}{n}\) \(1 \leq k \leq 2n\)
\(f(x_k)=a_n[cos(k\pi)+\theta], |\theta|<1 for 1 \leq k \leq 2n\)
[ since cos \(\theta\) is periodic and f(x) is expressed for every point x=\(x_k\)]
f has at least 2n points in such a period interval where f has alternating sign.
I.S.I. & C.M.I. Entrance Program
Taught by students and alumni of I.S.I. & C.M.I.
In 2026, the following Cheenta students have been successful for Indian Statistical Institute's M.Stat Entrance. They ranked within the first 50 in the entire country in these entrances. I.S.I. M.Stat Entrance
In 2026, the following Cheenta students have been successful for Indian Statistical Institute's B.Stat Entrance and Chennai Mathematical Institute's B.Sc. Math Entrance. They ranked within the first 200 in the entire country in these entrances. Most of these students attended the problem solving workshops regularly, which happen 5 days every week. CMI B.Sc. Math Entrance […]
In 2025, 8 students from Cheenta Academy cracked the prestigious Regional Math Olympiad. In this post, we will share some of their success stories and learning strategies. The Regional Mathematics Olympiad (RMO) and the Indian National Mathematics Olympiad (INMO) are two most important mathematics contests in India.These two contests are for the students who are […]
Cheenta Academy proudly celebrates the success of 27 current and former students who qualified for the Indian Olympiad Qualifier in Mathematics (IOQM) 2025, advancing to the next stage — RMO. This accomplishment highlights their perseverance and Cheenta’s ongoing mission to nurture mathematical excellence and research-oriented learning.