IIT-JEE-Mathematics-Paper2-2008

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1.     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 the particles moves √2 units in the direction of the unit vector i + j 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

        (A)    6 + 7i

        (B)    -7 + 6i

        (C)    7 + 6i

        (D)    -6 + 7i

2.     Let the function g : (-∞, ∞) -->( -Π/2 , Π/2) be given by g(u) = 2tan^-1(e^u) -Π/2. Then, g is

        (A)    even and is strictly increasing in (0, ∞)

        (B)    odd and is strictly decreasing in (-∞, ∞)

        (C)    odd and is strictly increasing in (-∞, ∞)

        (D)    neither even or odd, but is strictly increasing in (-∞, ∞)

3.     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

        (A)    1 - √(2/3)

        (B)    √(3/2) - 1

        (C)    1 + √(2/3)

        (D)    √(3/2) + 1

4.     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

5.     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,

        (A)    P lies on the line segment RQ

        (B)    Q lies on the line segment PR

        (C)    R lies on the line segment QP

        (D)    P, Q, R are non-collinear

6.     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

        (A)    2, 4 or 8

        (B)    3, 6 or 9

        (C)    4 or 8

        (D)    5 or 10

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

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