Test of Mathematics Solution Subjective 66 - Range of a Polynomial
Join Trial or Access Free ResourcesJoin Trial or Access Free Resources
This is a Test of Mathematics Solution Subjective 66 (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
If c is a real number with 0 < c < 1, then show that the values taken by the function y = $ {frac {x^2+2x+c}{x^2+4x+3c}} $ , as x varies over real numbers, range over all real numbers.
y = $ {\frac {x^2+2x+c}{x^2+4x+3c}} $
or $ {yx^2} $ + 4yx + 3cy = $ {x^2} $ + 2x + c
or $ {x^2} $ (y-1) + (4y-2)x + 3cy - c = 0
Now if we can show that the discriminant $ {\ge{0}} $ for all y then for all real y there exist a real x.
Now, discriminant is $ {(4y-2)^2} $ - 4(y-1)(3cy-c)
We need to show $ {(4y-2)^2} $ - 4(y-1)(3cy-c) > 0 for any 0 < c < 1. (16-12c) $ {4^2} $ - (16-16c)y + (4-4c) > 0.
This is parabola opening upword.
Now if its discriminant < 0 then this equation is always > 0.
So this is equivalent to prove
$ {(16-16c)^2} $ - 4 (16-12c)(4-4c) < 0.
or $ {(2-2c)^2} $ - (4-3c)(1-c) < 0
or $ {c^2} $ -C < 0
Now given 1>c>0 so $ {c^2} $ <c
or $ {c^2} $ -c < 0
Conclusion: So for 1>c>0 y = $ {\frac{x^2+2x+c}{x^2+4x+3c}} $
Range over all real number when x varies over all real number.
Publishing original scientific research in a peer-reviewed journal is an achievement many scholars work toward for years. Achieving this feat early in one's academic journey requires exceptional intellectual rigor, creativity, and persistence. Aaditya Punatar and Muskan Sadana have reached this remarkable milestone, successfully publishing their collaborative research paper in the prestigious Journal of Emerging Investigators […]
Her project on quantum-accelerated Steiner trees for 15-minute cities earned a place among the finalists at the world's largest pre-college STEM competition. Prishaa Shrimali of Marymount School of New York, was selected as a Grand Finalist at the 2026 Regeneron International Science and Engineering Fair (ISEF) in Phoenix, Arizona. Her project was Optimizing Urban Accessibility: […]
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 […]