If the domain of the function f(x) = (cos ⁻¹√(x² - x + 1))/√(sin ⁻¹( (2x - 1)/2 )) is the interval (α, β], then α + β is equal to :

Solution

Maths sir · 3 steps

Find where the top works and where the bottom works, then keep only the x values that pass both.

1. Numerator condition. cos⁻¹ t needs -1 ≤ t ≤ 1. Here t = √(x² - x + 1) is never negative, and x² - x + 1 = (x - 1/2)² + 3/4 > 0, so the root always exists. The only real condition is x² - x + 1 ≤ 1, that is x(x-1) ≤ 0, which gives 0 ≤ x ≤ 1.
2. Denominator condition. √sin⁻¹((2x-1)/2) sits in the denominator, so the quantity under the root must be strictly positive. sin⁻¹ u > 0 exactly when 0 < u ≤ 1. So 0 < (2x-1)/2 ≤ 1, which gives 0 < 2x - 1 ≤ 2, that is 1/2 < x ≤ 3/2.

   [0, 1] ∩ (1/2, 3/2] = (1/2, 1]

3. So. The domain is (1/2, 1], so α = 1/2 and β = 1. That gives α + β = 3/2, which is option (a).

The idea to keep. For a fraction, the numerator only has to be defined, but the denominator has to be defined and nonzero. A square root in the denominator means the inside must be strictly positive.

Answer from the NTA final answer key. Solution by PrepWiser.