Exams › JEE Advanced › Maths

Determine the range of each function: (i) f(x) = ln(3x² - 8x + 1); (ii) f(x) = log base 2 of ((3 cos x + 4 sin x + 10)/(3 cos x + 4 sin x + 15)).

1. (i) all real numbers R; (ii) [log2(1/2), log2(3/4)] = [-1, log2(3/4)]
2. (i) [0, infinity); (ii) [-1, 0]
3. (i) (-infinity, ln(13/3)]; (ii) [log2(2/3), 0]
4. (i) R; (ii) [0, 1]

Correct answer: (i) all real numbers R; (ii) [log2(1/2), log2(3/4)] = [-1, log2(3/4)]

Solution

The natural log of a quadratic that can take any value from just above 0 up to +infinity gives all reals (on the domain where the quadratic is positive). For (ii) the bounded sinusoid makes the argument range over a closed interval, and the log of that interval gives the range.

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 →