JEE Main 2022 · 29 July · evening · Single correct · +4 / −1
The domain of the function f(x)=sin ⁻¹((x²-3 x+2)/(x²+2 x+7)) is :
- (a) [1, ∞)
- (b) [-1,2]
- (c) [-1, ∞)
- (d) (-∞, 2]
Answer and solution
Answer (c) [-1, ∞)
Solution
Maths sir · 4 steps
Trap the fraction between -1 and 1, and the positive denominator lets you cross-multiply safely.
- Condition for arcsin. sin⁻¹ t is defined only for -1 ≤ t ≤ 1. Also x²+2x+7 = (x+1)²+6 > 0 for every x, so you can multiply through without flipping any inequality.
-(x²+2x+7) ≤ x²-3x+2 ≤ x²+2x+7
- Upper bound. From x²-3x+2 ≤ x²+2x+7, the x² terms cancel to give -5x ≤ 5. Dividing by -5 flips the sign, so x ≥ -1.
- Lower bound. From x²-3x+2 ≥ -x²-2x-7 you get 2x² - x + 9 ≥ 0. Its discriminant is 1 - 72 = -71 < 0 and the leading coefficient is positive, so this holds for every real x.
- So. Only the upper bound restricts x, and the domain is [-1, ∞), option (c).
The idea to keep. For sin⁻¹ or cos⁻¹ of a fraction, check the sign of the denominator first. If it is always positive, cross-multiply both bounds and solve each one separately.
Answer from the NTA final answer key. Solution by PrepWiser.