Let $S=\{a\in N,a\le 100\}$ . If the equation $\left[{\tan }^{2}x\right]-\tan x-a=0$  has real roots then number of elements in S is [ where [.] G.I.F]

$a=\left[{\tan }^{2}x\right]-\tan x\Rightarrow \tan x$  is
$\Rightarrow \left[{\tan }^{2}x\right]={\tan }^{2}x\Rightarrow {\tan }^{2}x-\tan x-a=0\Rightarrow \tan x=\frac{1\pm \sqrt{1+4a}}{2\left(1\right)}$ 
Let  $\tan x=k\Rightarrow {(2k-1)}^2=1+4a$
 $\Rightarrow 4k^2+1-4k=1+4a\Rightarrow \omega =k(k-1)$
$1\le a\le 100$ 
 $k=2,3,4,5,6,7,8,9,10$
Value of a are
$k=2,3,4,5,6,7,8,9,10$