KVPY Sample Paper KVPY Stream-SX Model Paper-13

If equations \[{{x}^{n}}-n{{x}^{n\,-\,1}}+{{a}_{2}}{{x}^{n\,-\,2}}+{{a}_{3}}{{x}^{n\,-\,3}}+...\] has \[+{{a}_{n-1}}x+{{(-1)}^{n}}=0\] positive roots, then the least value of n for which \[{{a}_{2}}+{{a}_{3}}\]is negative is

A) 2

B) 6

C) 4

D) 1

Correct Answer: B

Solution :

We have, \[{{x}^{n}}-n{{x}^{n-1}}+{{a}_{2}}{{x}^{n-2}}+{{a}_{3}}{{x}^{n-3}}+...+\,{{a}_{n\,-\,1}}x+{{(-1)}^{n\,=\,0}}\]

has ra positive roots.

\[\therefore \]AM of all roots is equal to GM of roots,

so all roots are equal and equal to 1.

\[\therefore \]\[{{a}_{2}}={}^{n}{{C}_{2}}\]and \[{{a}_{3}}=-{}^{n}{{C}_{3}}\]

\[\Rightarrow \]\[{{a}_{2}}+{{a}_{3}}={}^{n}{{C}_{2}}-{}^{n}{{C}_{3}}<0\]

\[\Rightarrow \]\[n>2+3\]

\[\Rightarrow \]\[n>5\]