Sets, Relations & Functions
Functions
nta_pyq_2025_jan
Grade 11
Question:
Let the range of the function f (x) = 6 + 16 cos x ⋅ cos( \pi - x) ⋅ cos( \pi + x) ⋅ sin 3x ⋅ cos 6x, x \in R be [\alpha, \beta]. 3 3 Then the distance of the point (\alpha, \beta) from the line 3x + 4y + 12 = 0 is :
Step-by-Step Solution
Key Concept: Apply the core result for domains, ranges and functional equations and simplify using the given constraints.
f (x) = 6 + 16 cos x ⋅ cos( \pi - x) 3 (1) \pi cos( + x) ⋅ sin 3x ⋅ cos 6x 3 f (x) = 6 + 4 cos 3x ⋅ sin 3x ⋅ cos 6x \therefore f (x) = 6 + sin 12x \therefore Range of f (x) = [5, 7] \therefore [\alpha, \beta] = [5, 7] \therefore Distance of point from 3x + 4y + 12 = 0 ∣ ∣ 3.5 + 4.7 + 12 ∣ ∣ = ∣ ∣ 2 2 ∣ \sqrt3 + 4 ∣ = 11 units 4 3 2
Correct Answer: 1