Exams › JEE Advanced › Maths

Let A be the area of the region enclosed between the two circles x² + y² = 1 and (x-1)² + y² = 1. Which of the following statements are true?

1. A > pi/2
2. A < pi/2
3. A is irrational
4. A is rational

Correct answer: A < pi/2

Solution

The circles intersect at x = 1/2, y = +-sqrt(3)/2. Each circular segment subtends angle 2*pi/3 at the center. Area of each segment = (1²/2)*(2*pi/3 - sin(2*pi/3)) = (1/2)*(2*pi/3 - sqrt(3)/2). Total A = 2*(1/2)*(2*pi/3 - sqrt(3)/2) = 2*pi/3 - sqrt(3)/2 ≈ 2.094 - 0.866 = 1.228. Since pi/2 ≈ 1.571, we have A < pi/2. Also A = 2*pi/3 - sqrt(3)/2 is irrational (contains pi and sqrt(3)).

Related JEE Advanced Maths questions

• What is the area enclosed by the curves y = 2 − |2 − x| and y = 3/|x|?
• If f(x) and g(x) are continuous functions over the interval a ≤ x ≤ b, and p(x) represents the greater of f(x) and g(x), while q(x) represents the lesser of f(x) and g(x), what is the expression for the area enclosed by the curves y = p(x), y = q(x), and the vertical lines x = a and x = b?
• The area enclosed by the curves y = sin x + cos x and y = |cos x − sin x| over the interval [0, π/2] is -
• Let f: [1/2, 1] → R (the set of all real numbers) be a positive, non-constant and differentiable function such that f'(x) < 2 f(x) and f(1/2) = 1. Then the value of ∫[1/2 to 1] f(x) dx lies in the interval -
• Let S = {(x, y) ∈ R × R: x ≥ 0, y ≥ 0, y² ≤ 4x, y² ≤ 12 - 2x and 3y + √8x ≤ 5√8}. If the area of the region S is α√2, then α is equal to -
• The area of the region enclosed by the curves y = |(x + 2)/(x - 2)| and y = |(x - 2)/(x + 2)| together with the x-axis equals 4 ln(a/e). Find the value of a.

⚔️ Practice JEE Advanced Maths free + battle 1v1 →