Q. If $\alpha,\ \beta\ and\ \gamma$ are the roots of the equation $x^{3} - 3x + 1 = 0,\ then\ \lbrack\alpha\rbrack + \lbrack\beta\rbrack + \lbrack\gamma\rbrack\ is\ (\lbrack \bullet \rbrack$ denotes the greatest integer function)
A) $- 3$
B) $- 2$
C) $- 1$
D) Does not exist
Explanation
(c)
Since, the value of function at different points are
$f( - 2) < 0,\ f( - 1) > 0,\ f(0) > 0,\ f(1) < 0,\ f(2) > 0$
Hence, one root lie in ($- 2,\ 1)$.
$\therefore\$2$nd$ root lie in (0, 1) and last root lie in (1, 2).
$\therefore\lbrack\alpha\rbrack = - 2,\ \lbrack\beta\rbrack = 0,\ \lbrack\gamma\rbrack = 1$
$\therefore\lbrack\alpha\rbrack + \lbrack\beta\rbrack + \lbrack\gamma\rbrack = - 1$
Reviewed by Dr. Abubacker
Physics Educator & NEET/JEE Specialist
Ph.D. in Physics with 19+ years of teaching experience.