JEE Main 2021 · 18 March · morning · Single correct · +4 / −1
The real valued function
f(x) = (cos ec⁻¹x)/√(x - [x]), where [x] denotes the greatest integer less than or equal to x, is defined for all x belonging to :
- (a) all real except integers
- (b) all non-integers except the interval [ -1, 1 ]
- (c) all integers except 0, -1, 1
- (d) all real except the interval [ -1, 1 ]
Answer and solution
Answer (b) all non-integers except the interval [ -1, 1 ]
Solution
Maths sir · 4 steps
Two separate conditions: the numerator needs |x| ≥ 1, the denominator needs x not an integer.
- Numerator. csc⁻¹ x is defined only when |x| ≥ 1, that is, x ≤ -1 or x ≥ 1. Any x strictly between -1 and 1 is ruled out.
- Denominator. x - [x] = {x} is the fractional part, which lies in [0, 1). Under a square root in the denominator it must be strictly positive, so {x} ≠ 0. That means x cannot be an integer.
- Combine. Take both conditions together. The integers -1 and 1 are removed by the second condition, so the domain is the non-integers in (-∞, -1) ∪ (1, ∞).
- So. f(x) is defined for all non-integers outside the interval [-1, 1], which is option (b).
The idea to keep. For a quotient, find the domain of each part separately and intersect them. Remember that x - [x] is zero at every integer, so it can never sit in a denominator there.
Answer from the NTA final answer key. Solution by PrepWiser.