2008 (AIEEE / IIT-JEE, in the JEE Main bank) questions

Mathematics

1. Let a and b be non-zero and real numbers. Then, the equation (ax^2 + by^2 + c)(x^2 - 5xy + 6y^2) = 0 represents

   Straight Lines and Pair of Lines

2. A particle P starts from the point z_0 = 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 z_1. From z_1 the particle moves…

   Complex Numbers and Quadratic Equations

3. The edges of a parallelopiped are of unit length and are parallel to non-coplanar unit vector a, b, c such that a · b = b · c = c · a = 1/2. Then, the volume of the parallelopiped is

   Vector Algebra · with solution

4. If the function g: (-∞, ∞) arrow (-(π)/2, (π)/2 ) is given by g(u) = 2tan^-1(e^u) - (π)/2. Then, g is

   Application of Derivatives

5. 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, then the number of outcomes that B must have, so that A and B are independent, is

   Probability · with solution

6. Consider a branch of the hyperbola x^2 - 2y^2 - 2√(2)x - 4√(2)y - 6 = 0 with vertex at the point A. Let B be one of the end points of its latusrectum. If C is the focus of the hyperbola nearest to the point A, then the…

   Conic Sections

7. Let two non-collinear unit vectors a and b form an acute angle. A point P moves, so that at any time t the position vector overrightarrowOP (where, O is the origin) is given by acos t + bsin t. When P is farthest from…

   Vector Algebra

8. Let g(x) = ((x-1)^n)/(logcos^m(x-1)); 0 0 and let p be the left hand derivative of |x-1| at x = 1. If lim_x → 1^- g(x) = p, then

   Limits Continuity and Differentiability

9. If I = ∫ frace^xe^4x + e^2x + 1 dx, J = ∫ frace^-xe^-4x + e^-2x + 1 dx. Then, for an arbitrary constant c, the value of J - I equals

   Indefinite Integration · with solution

10. The total number of local maxima and local minima of the function f(x) = begincases (2+x)^3, & -3 < x ≤ -1 \ x^2/3, & -1 < x < 2 endcases is

    Application of Derivatives · with solution

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

    Definite Integrals and Areas · with solution