JEE Advanced 2013 · Paper 1

Let complex numbers α and 1/α̅ lie on circles (x − x₀)² + (y − y₀)² = r² and (x − x₀)² + (y − y₀)² = 4r², respectively. If z₀ = x₀ + iy₀ satisfies the equation 2|z₀|² = r² + 2, then |α| =

1. A1/√2
2. B1/2
3. C1/√7
4. D1/3

Four persons independently solve a certain problem correctly with probabilities 1/2, 3/4, 1/4, 1/8. Then the probability that the problem is solved correctly by at least one of them is

1. A235/256
2. B21/256
3. C3/256
4. D253/256

Let f : [1/2, 1] → ℝ (the set of all real numbers) be a positive, non-constant and differentiable function such that f′(x) < 2f(x) and f(1/2) = 1. Then the value of ∫₁/₂¹ f(x) dx lies in the interval

1. A(2e − 1, 2e)
2. B(e − 1, 2e − 1)
3. C((e − 1)/2, e − 1)
4. D(0, (e − 1)/2)

The number of points in (−∞, ∞), for which x² − x sin x − cos x = 0, is

1. A6
2. B4
3. C2
4. D0

The area enclosed by the curves y = sin x + cos x and y = |cos x − sin x| over the interval [0, π/2] is

1. A4(√2 − 1)
2. B2√2(√2 − 1)
3. C2(√2 + 1)
4. D2√2(√2 + 1)

A curve passes through the point (1, π/6). Let the slope of the curve at each point (x, y) be y/x + sec(y/x), x > 0. Then the equation of the curve is

1. Asin(y/x) = log x + 1/2
2. Bcosec(y/x) = log x + 2
3. Csec(2y/x) = log x + 2
4. Dcos(2y/x) = log x + 1/2

The value of cot(∑ₙ₌₁²³ cot⁻¹(1 + ∑ₖ₌₁ⁿ 2k)) is

1. A23/25
2. B25/23
3. C23/24
4. D24/23

For a > b > c > 0, the distance between (1, 1) and the point of intersection of the lines ax + by + c = 0 and bx + ay + c = 0 is less than 2√2. Then

1. Aa + b − c > 0
2. Ba − b + c < 0
3. Ca − b + c > 0
4. Da + b − c < 0