Question bank/JEE Main/Mathematics

Number of positive integral values of x satisfying the
inequality
$\frac{(x - 4)^{2017} \cdot (x + 8)^{2016}(x + 1)}{x^{2016}(x - 2)^{3} \cdot (x + 3)^{5} \cdot (x - 6)(x + 9)^{2018}} \leq 0$
is

A0B1C2D3

Show answer and solution▾

$\frac{(x - 4)^{2017}(n + 8)^{2016}(n + 1)}{x^{2016}(x - 2)^{3}(x + 3)^{5}(n + 6)(x + 9)} \leq 0$
Critical points: $x = - 9,\ - 8,\ - 3,\ - 1,\ 0,\ 2,\ 4,\ 6$
Open interval at $x = - 9,\ - 3,\ 0,\ 2,\ 6$
At $x = 1$, the sign of expression is negative
For $x \in \lbrack - 1,\ 0);\ - ve$.
[IMAGE] $x = 0$ has even power; So sign of the expression remains same w.r.t
$x = 0 \Rightarrow - ve$ for
$x \in (0,\ \ 2)$
Similarly check w.r.t. each critical point.
$\Rightarrow x \in ( - \infty, - 9) \cup ( - 9, - 3) \cup \lbrack - 1,0) \cup (0,2) \cup (4,6\rbrack$
$\Rightarrow$ Positive integral values of x.
$\Rightarrow 1,\ 4,\ 5$

Source: Grass-G / M-05 Mathematics Paper 10 October.docx