1. Tardigrade
  2. Question
  3. Mathematics
  4. If α β are roots of quadratic x 2+( k -5) x +3=0 and α<1<β and k ∈ I. If largest possible value of k is n then [ log (n2+2)(n2+n+1)] is equal to [Note: [ k ] denotes greatest integer function less than or equal to k.)

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Q. If $\alpha \& \beta$ are roots of quadratic $x ^2+( k -5) x +3=0$ and $\alpha<1<\beta$ and $k \in I$. If largest possible value of $k$ is $n$ then $\left[\log _{\left(n^2+2\right)}\left(n^2+n+1\right)\right]$ is equal to
[Note: $[ k ]$ denotes greatest integer function less than or equal to $k$.)

Complex Numbers and Quadratic Equations

Solution:

$ \Theta 1$ lies between roots
$\therefore f (1)<0 $
$ 1+ k -5+3< 0 $
$\Rightarrow k <1$
$\therefore n =0 $
$\therefore \log _{\left( n ^2+2\right)}\left( n ^2+ n +1\right)=0$