JEE Advanced 2008 · Paper 2

A particle P starts from the point z₀ = 1 + 2i, where i = √−1. It moves first horizontally away from origin by 5 units and then vertically away from origin by 3 units to reach a point z₁. From z₁ the particle moves √2 units in the direction of the vector î + ĵ and then it moves through an angle π/2 in anticlockwise direction on a circle with centre at origin, to reach a point z₂. The point z₂ is given by

1. A6 + 7i
2. B−7 + 6i
3. C7 + 6i
4. D−6 + 7i

Let the function g : (−∞, ∞) → (−π/2, π/2) be given by g(u) = 2 tan⁻¹(eᵘ) − π/2. Then, g is

1. Aeven and is strictly increasing in (0, ∞)
2. Bodd and is strictly decreasing in (−∞, ∞)
3. Codd and is strictly increasing in (−∞, ∞)
4. Dneither even nor odd, but is strictly increasing in (−∞, ∞)

Consider a branch of the hyperbola x² − 2y² − 2√2x − 4√2y − 6 = 0 with vertex at the point A. Let B be one of the end points of its latus rectum. If C is the focus of the hyperbola nearest to the point A, then the area of the triangle ABC is

1. A1 − √(2/3)
2. B√(3/2) − 1
3. C1 + √(2/3)
4. D√(3/2) + 1

The area of the region between the curves y = √((1 + sin x)/cos x) and y = √((1 − sin x)/cos x) bounded by the lines x = 0 and x = π/4 is

1. A∫₀^(√2−1) t/((1 + t²)√(1 − t²)) dt
2. B∫₀^(√2−1) 4t/((1 + t²)√(1 − t²)) dt
3. C∫₀^(√2+1) 4t/((1 + t²)√(1 − t²)) dt
4. D∫₀^(√2+1) t/((1 + t²)√(1 − t²)) dt

Consider three points P = (−sin(β − α), −cos β), Q = (cos(β − α), sin β) and R = (cos(β − α + θ), sin(β − θ)), where 0 < α, β, θ < π/4. Then,

1. AP lies on the line segment RQ
2. BQ lies on the line segment PR
3. CR lies on the line segment QP
4. DP, Q, R are non-collinear

An experiment has 10 equally likely outcomes. Let A and B be two non-empty events of the experiment. If A consists of 4 outcomes, the number of outcomes that B must have so that A and B are independent, is

1. A2, 4 or 8
2. B3, 6 or 9
3. C4 or 8
4. D5 or 10

Let two non-collinear unit vectors â and b̂ form an acute angle. A point P moves so that at any time t the position vector OP (where O is the origin) is given by â cos t + b̂ sin t. When P is farthest from origin O, let M be the length of OP and û be the unit vector along OP. Then,

1. Aû = (â + b̂)/|â + b̂| and M = (1 + â · b̂)¹ᐟ²
2. Bû = (â − b̂)/|â − b̂| and M = (1 + â · b̂)¹ᐟ²
3. Cû = (â + b̂)/|â + b̂| and M = (1 + 2â · b̂)¹ᐟ²
4. Dû = (â − b̂)/|â − b̂| and M = (1 + 2â · b̂)¹ᐟ²

Let I = ∫ eˣ/(e⁴ˣ + e²ˣ + 1) dx, J = ∫ e⁻ˣ/(e⁻⁴ˣ + e⁻²ˣ + 1) dx. Then, for an arbitrary constant C, the value of J − I equals

1. A½ log((e⁴ˣ − e²ˣ + 1)/(e⁴ˣ + e²ˣ + 1)) + C
2. B½ log((e²ˣ + eˣ + 1)/(e²ˣ − eˣ + 1)) + C
3. C½ log((e²ˣ − eˣ + 1)/(e²ˣ + eˣ + 1)) + C
4. D½ log((e⁴ˣ + e²ˣ + 1)/(e⁴ˣ − e²ˣ + 1)) + C