2006 (AIEEE / IIT-JEE, in the JEE Main bank) questions

Mathematics

1. The set of values of θ satisfying the inequation 2sin^2θ - 5sinθ + 2 > 0, where 0 < θ < 2π, is

   Trigonometry · with solution

2. In radius of a circle which is inscribed in a isosceles triangle one of whose angle is 2π/3, is √(3), then area of triangle (in sq units) is

   Trigonometry · with solution

3. Axis of a parabola is y = x and vertex and focus are at a distance √(2) and 2√(2) respectively from the origin. Then, equation of the parabola is

   Conic Sections

4. Let, a = i + 2j + k, b = i - j + k, c = i + j - k. A vector coplanar to a and b has a projection along c of magnitude dfrac1√(3), then the vector is

   Vector Algebra · with solution

5. For x > 0, lim_x → 0 [(sin x)^1/x + (1/x )^sin x ]

   Limits Continuity and Differentiability

6. If w = α + iβ, where β ≠ 0 and z ≠ 1, satisfies the condition that (dfracw - wz1-z ) is purely real, then the set of values of z is

   Complex Numbers and Quadratic Equations · with solution

7. If e_1 is the eccentricity of the ellipse (x^2)/16 + (y^2)/25 = 1 and e_2 is the eccentricity of the hyperbola passing through the foci of the ellipse and e_1 e_2 = 1, then equation of the hyperbola is

   Conic Sections · with solution

8. If r, s, t are prime numbers and p, q are the positive integers such that LCM of p, q is r^2 s^4 t^2, then the number of ordered pairs (p, q) is

   Permutations and Combinations · with solution

9. If f''(x) = -f(x), where f(x) is a continuous double differentiable function and g(x) = f'(x). If F(x) = \f (x/2 ) \^2 + \g (x/2 ) \^2 and F(5) = 5, then F(10) is

   Limits Continuity and Differentiability · with solution

10. The value of ∫ dfrac(x^2-1) dxx^3√(2x^4-2x^2+1) is \hfill

    Indefinite Integration · with solution

11. If a, b, c are the sides of a triangle ABC such that x^2 - 2(a+b+c)x + 3λ(ab+bc+ca) = 0 has real roots, then

    Complex Numbers and Quadratic Equations · with solution

12. A plane passes through the point (1, -2, 1) and is perpendicular to two planes 2x - 2y + z = 0 and x - y + 2z = 4, then the distance of the plane from the point (1, 2, 2) is

    Three Dimensional Geometry · with solution