Complex Numbers And Quadratic Equations question 35
Question: If the equation $ a_{n}x^{n}+{a_{n-1}}{x^{n-1}}+….+a_1x=0 $ , $ a_1\ne 0 $ , $ n\ge 2 $ , has a positive root $ x=\alpha $ , then the equation $ na_{n}{x^{n-1}}+(n-1){a_{n-1}}{x^{n-2}}+….+a_1=0 $ has a positive root, which is [AIEEE 2005]
Options:
A) Greater than or equal to a
B) Equal to $ \alpha $
C) Greater than $ \alpha $
D) Smaller than $ \alpha $
Show Answer
Answer:
Correct Answer: D
Solution:
Let $ f(x)=a_{n}x^{n}+{a_{n-1}}{x^{n-1}}+….+a_1x $ ; $ f(0)=0; $ $ f(\alpha )=0 $
Þ $ {f}’(x)=0 $ , has atleast one root between $ (0,\alpha ) $ i.e., equation $ na_{n}{x^{n-1}}+(n-1){a_{n-1}}{x^{n-2}}+….+a_1=0 $ has a positive root smaller than .
BETA
