Question

Let $f: R \rightarrow\left(0, \frac{\pi}{4}\right]$ be defined as $f(x)=\cot ^{-1}\left(x^2+x+k\right)$, where $k \in R$. If $f(x)$ is surjective function then $1+\frac{1}{ k }+\frac{1}{ k ^2}+\frac{1}{ k ^3}+\ldots \ldots \ldots \infty$ terms is equal to

• A. 2
• B. 3
• C. 4
• D. 5

Answer 1

Answer: (D)

$ \Theta f ( x ) \text { is surjective } $
$\therefore 1 \leq x ^2+ x + k <\infty $
$\therefore \frac{- D }{4 a }=1 \Rightarrow \frac{-(1-4 k )}{4}=1 \Rightarrow k =\frac{5}{4}$
$\therefore \text { given sum }=\frac{1}{1-(1 / k )}=\frac{1}{1-(4 / 5)}=5$