Question

f : R →$\left( 0 , \frac { \pi } { 2 } \right]$where f(x) = cot^-1 (x² + x + a). Complete set of values of 'a' such that f(x) is onto is

• A. $\left[ \frac { 3 } { 4 } , \infty \right)$
• B. [1, ∞)
• C. $\left[ \frac { 1 } { 2 } , \infty \right)$
• D. $\left[ \frac { 1 } { 4 } , \infty \right)$

Answer 1

Answer: (D)

$\left[ \frac { 1 } { 4 } , \infty \right)$

Answer 2

We know that cot^-1(x) ∈(0, π/2] ∀ x > 0. Thus x² + x + a ≥ 0 ∀ x ∈ R

⇒ 1 - 4a ≤ 0 ⇒ $a \geq \frac { 1 } { 4 }$