Question

Let \( f:(-\infty,-1] \rightarrow\left(\frac{\pi}{2}, \pi\right] \) be defined as \( f(x)=\sec ^{-1}\left(-x^{2}+x+a\right) \). If \( f(x) \) is surjective, then the range of \( a \) is :

(a) (1) 

(b) \( \left\{\frac{-5}{4}\right\} \) 

(c) \( \left(-\infty, \frac{-5}{4}\right] \) 

(d) \( (-\infty, 1] \)

Answer 1

Correct option is (b) \(\left\{\frac{-5}4\right\}\)

\(-x^2 + x + a \le-1\)

\(a \le x^2 - x - 1\)

⇒ \(a \le \frac{-5}4\)

min\((x^2 - x - 1) = \frac{-5}4\)