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2005 (AIEEE / IIT-JEE, in the JEE Main bank) questions

24 questions, by subject. Each opens with its options and answer.

Mathematics

  1. If X and Y are two non-empty sets where f: X → Y, is function is defined such that f(C) = \f(x) : x ∈ C\ for C ⊂eq X and f^-1(D) = \x : f(x) ∈ D\ for D ⊂eq Y, for any A ⊂eq Y and B ⊂eq Y, then

    Sets Relations and Functions · with solution

  2. The area of the equilateral triangle, in which three coins of radius 1 cm are placed, as shown in the figure, is [figure]

    Definite Integrals and Areas · with solution

  3. The minimum value of |a + bω + cω^2|, where a, b and c are all not equal integers and ω (≠ 1) is a cube root of unity, is

    Complex Numbers and Quadratic Equations

  4. If int_sin x^1 t^2 f(t) dt = 1 - sin x, forall x ∈ (0, π/2), then f (frac1√(3) ) is

    Definite Integrals and Areas · with solution

  5. binom300binom3010 - binom301binom3011 + binom302binom3012 + ·s + binom3020binom3030 is equal to

    Binomial Theorem · with solution

  6. In a triangle ABC, among the following which one is true?

    Trigonometry · with solution

  7. The locus of the centre of circle which touches (y-1)^2 + x^2 = 1 externally and also touches X-axis, is

    Circles · with solution

  8. Let f(x) = ax^2 + bx + c, a ≠ 0 and Δ = b^2 - 4ac. If α + β, α^2 + β^2 and α^3 + β^3 are in GP, then

    Sequences and Series

  9. If P = beginbmatrix √(3)/2 & 1/2 \ -1/2 & √(3)/2 endbmatrix, A = beginbmatrix 1 & 1 \ 0 & 1 endbmatrix and Q = PAP^T, then P^TQ^2005P is

    Matrices and Determinants · with solution

  10. The tangent at (1, 7) to the curves x^2 = y - 6x touches the circle x^2 + y^2 + 16x + 12y + c = 0 at

    Circles · with solution

  11. The number of ordered pairs (α, β), where α, β ∈ (-π, π) satisfying cos(α - β) = 1 and cos(α + β) = 1/e is

    Trigonometry

  12. f(x) = begincases x, & if x is rational \ 0, & if x is irrational endcases, g(x) = begincases 0, & if x is rational \ x, & if x is irrational endcases Then, f - g is

    Sets Relations and Functions

  13. If a, b, c are three non-zero, non-coplanar vectors and b_1 = b - dfracb·a|a|^2a, b_2 = b + dfracb·a|a|^2a, c_1 = c - dfracc·a|a|^2a - dfracc·b|b|^2b, c_2 = c - dfracc·a|a|^2a - dfracc·b_1|b_1|^2b_1, c_3 = c -…

    Vector Algebra

  14. A fair die is rolled. The probability that the first time 1 occurs at the even throw, is

    Probability

  15. If A = beginbmatrix 1 & 0 & 0 \ 0 & 1 & 1 \ 0 & -2 & 4 endbmatrix, 6A^-1 = A^2 + cA + dI, then (c, d) is

    Matrices and Determinants · with solution

  16. If f is a differentiable function satisfying f (1/n ) = 0, forall n ≥ 1, n ∈ I, then

    Limits Continuity and Differentiability

  17. Let f be twice differentiable function satisfying f(1) = 1, f(2) = 4, f(3) = 9, then

    Limits Continuity and Differentiability

  18. Let f(x) = ||x| - 1|, then points where, f(x) is not differentiable is/are

    Limits Continuity and Differentiability

  19. If x dy = y (dx + y dy), y(1) = 1 and y(x) > 0. Then, y(-3) is equal to

    Differential Equations · with solution

  20. The value of int_-1^0 [x^3 + 3x^2 + 3x + 3 + (x+1)cos(x+1)] dx is

    Definite Integrals and Areas

  21. The area bounded by the curves y = (x-1)^2, y = (x+1)^2 and y = 1/4 is

    Definite Integrals and Areas · with solution

  22. A variable plane x/a + y/b + z/c = 1 at a unit distance from origin cuts the coordinate axes at A, B and C. Centroid (x, y, z) satisfies the equation 1/(x^2) + 1/(y^2) + 1/(z^2) = K. The value of K is

    Three Dimensional Geometry

Chemistry

  1. The number of radial nodes in 3s and 2p respectively are

    Atomic Structure · with solution

Physics

  1. Which of the following sets have different dimensions?

    Units and Measurements

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