JEE Main 2023 · 15 April · morning · Single correct · +4 / −1

If the domain of the function
f(x)=log ₑ(4 x²+11 x+6)+sin ⁻¹(4 x+3)+cos ⁻¹((10 x+6)/3) is (α, β], then
36|α+β| is equal to :

  1. (a) 72
  2. (b) 54
  3. (c) 45
  4. (d) 63
Answer and solution

Answer (c) 45

Solution

Maths sir · 3 steps

Write the interval each of the three pieces allows, then keep only the overlap.

  1. Three conditions. The log needs a positive argument. Both sin⁻¹ and cos⁻¹ need their argument between -1 and 1. The domain of f is where all three hold at once.
    PieceConditionAllowed x
    logₑ(4x²+11x+6)(4x+3)(x+2) > 0x < -2 or x > -3/4
    sin⁻¹(4x+3)-1 ≤ 4x+3 ≤ 1-1 ≤ x ≤ -1/2
    cos⁻¹((10x+6)/3)-3 ≤ 10x+6 ≤ 3-9/10 ≤ x ≤ -3/10
  2. Find the overlap. The two inverse trig pieces together give [-9/10, -1/2]. The log condition removes everything up to and including -3/4, leaving (-3/4, -1/2].
  3. So. α = -3/4 and β = -1/2, so |α+β| = 5/4 and 36 × 5/4 = 45, which is option (c).

The idea to keep. For a sum of functions, find each piece's domain separately and intersect them. A strict inequality from a log becomes the open end of the final interval.

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

Study Functions (Advanced)

Asked about 70 times in JEE Main since 2020. This question is from 2023.