\frac{\sum_{r=0}^{k-1}{x^{2r}}}{\sum_{r=0}^{k-1}{x^r}} is a polynomial,p and q are any 2 values of k then the roots of the equation
3x^2 + px + 5q = 0 can not be
A)Real
B)Imaginary
C)Rational
D)Irrational
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3 Answers
21VIV C this : pretty much the same sum :
http://targetiit.com/iit_jee_forum/posts/quadratic_2024.html
62given expression equals...
\huge \frac{\frac{1-x^{2k}}{1-x^{2}}}{\frac{1-x^k}{1-x}}
\huge = \frac{(1+x^k)}{(1+x)} \\
so what we know is that k has to be odd.. otherwise it will not be a polynomial. (why?)
now find the discriminant..
p2-15q
4n+1-15(4m+1)
=4(n-15m)-14
which is of the form 4k+2 which cannot be a perfect square..
hence it will not have rational roots..