Exams › JEE Advanced › Maths

Solve the system of inequalities simultaneously: 4 (arctan x)² - 8 (arctan x) + 3 < 0 and 4 (arccot x) - (arccot x)² - 3 >= 0.

1. (tan(1/2), tan(3/2))
2. (cot 3, cot 1)
3. (tan(1/2), cot 1)
4. empty set

Correct answer: (tan(1/2), cot 1)

Solution

First inequality: with u = arctan x, 4u² - 8u + 3 < 0 factors as (2u - 1)(2u - 3) < 0, so 1/2 < u < 3/2, i.e. 1/2 < arctan x < 3/2. Since arctan x < pi/2 ~ 1.571, the upper bound 3/2 is allowed: x in (tan(1/2), tan(3/2)). Second: with v = arccot x, 4v - v² - 3 >= 0 => v² - 4v + 3 <= 0 => (v-1)(v-3) <= 0 => 1 <= v <= 3. Since arccot x in (0, pi), this gives 1 <= arccot x <= 3, i.e. x in [cot 3, cot 1] (note arccot decreasing). Intersecting the two solution sets in x yields x in (tan(1/2), cot 1).

Related JEE Advanced Maths questions

• Identify the periodic function among the following:
• Given that α, β, γ, and θ are the smallest positive angles in increasing order for which their sine values equal a positive constant k, what is the result of 4 sin(α/2) + 3 sin(β/2) + 2 sin(γ/2) + sin(θ/2)?
• Consider the function fn(θ) = tan(θ/2)(1 + sec²θ)(1 + sec⁴θ)... (1 + sec²ⁿθ). Which of the following is true?
• Given that (a−b)sin(θ+ϕ) = (a+b)sin(θ−ϕ) and a tan(θ/2) − b tan(ϕ/2) = c, which of the following holds true?
• In a right-angled triangle ABC where ∠B = 90° and the sum of the two legs a and b equals 4, the triangle's area is maximized when ∠C is:
• If the altitudes p₁, p₂, and p₃ of a triangle enclose a circle with a diameter of 4/3 units, what is the minimum possible value of p₁ + p₂ + p₃?

⚔️ Practice JEE Advanced Maths free + battle 1v1 →