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1999 (AIEEE / IIT-JEE, in the JEE Main bank) questions

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

Mathematics

  1. If x = 9 is the chord of contact of the hyperbola x^2 - y^2 = 9, then the equation of the corresponding pair of tangents is

    Conic Sections · with solution

  2. The number of real solutions of tan^-1√(x(x+1)) + sin^-1√(x^2+x+1) = (π)/2 is

    Inverse Trigonometric Functions · with solution

  3. The function f(x) = [x]^2 - [x^2] (where, [x] is the greatest integer less than or equal to x), is discontinuous at

    Limits Continuity and Differentiability

  4. Let a_1, a_2, …, a_10 be in AP and h_1, h_2, equal to …, h_10 be in HP. If a_1 = h_1 = 2 and a_10 = h_10 = 3, then a_4 h_7 is

    Sequences and Series · with solution

  5. Let a = 2i + j + k, b = i + 2j - k and a unit vector c be coplanar. If c is perpendicular to a, then c is equal to

    Vector Algebra · with solution

  6. If the function f:[1,∞) → [1,∞) is defined by f(x) = 2^x(x-1), then f^-1(x) is

    Sets Relations and Functions · with solution

  7. If the roots of the equation x^2 - 2ax + a^2 + a - 3 = 0 are real and less than 3, then

    Complex Numbers and Quadratic Equations · with solution

  8. If two distinct chords, drawn from the point (p, q) on the circle x^2 + y^2 = px + qy (where, pq ≠ 0) are bisected by the X-axis, then

    Circles · with solution

  9. Let a = 2i + j - 2k and b = i + j. If c is a vector such that a · c = |c|, |c - a| = 2√(2) and the angle between (a × b) and c is 30°, then |(a × b) × c| is equal to

    Vector Algebra

  10. A solution of the differential equation (dy/dx )^2 - xdy/dx + y = 0 is

    Differential Equations · with solution

  11. In a triangle PQR, ∠ R = (π)/2, if tan (P/2 ) and tan (Q/2 ) are the roots of the equation ax^2 + bx + c = 0 (a ≠ 0), then

    Trigonometry

  12. If i = √(-1), then 4 + 5 (-1/2 + dfraci√(3)2 )^334 + 3 (-1/2 + dfraci√(3)2 )^365 is equal to

    Complex Numbers and Quadratic Equations

  13. The curve described parametrically by x = t^2 + t + 1, y = t^2 - t + 1 represents

    Conic Sections · with solution

  14. The function f(x) = sin^4 x + cos^4 x increases, if

    Application of Derivatives

  15. lim_x → 0 (x tan 2x - 2x tan x)/((1 - cos 2x)^2) is

    Limits Continuity and Differentiability · with solution

  16. The harmonic mean of the roots of the equation (5 + √(2))x^2 - (4 + √(5))x + 8 + 2√(5) = 0 is

    Sequences and Series · with solution

  17. If f(x) = beginvmatrix 1 & x & x+1 \ 2x & x(x-1) & (x+1)x \ 3x(x-1) & x(x-1)(x-2) & (x+1)x(x-1) endvmatrix, then f(100) is equal to

    Matrices and Determinants

  18. The function f(x) = (x^2 - 1)|x^2 - 3x + 2| + cos(|x|) is not differentiable at

    Limits Continuity and Differentiability · with solution

  19. If in the expansion of (1 + x)^m (1 - x)^n, the coefficients of x and x^2 are 3 and -6 respectively, then m is equal to

    Binomial Theorem · with solution

  20. If for a real number y, [y] is the greatest integer less than or equal to y, then the value of the integral int_π/2^3π/2 [2sin x] dx is

    Definite Integrals and Areas

  21. int_π/4^3π/4 dx/(1 + cos x) is equal to

    Definite Integrals and Areas · with solution

Chemistry

  1. The oxidation number of sulphur in S_8, S_2F_2, H_2S respectively, are

    Redox and Electrochemistry

  2. The electrons, identified by quantum numbers n and l, (i) n = 4, l = 1, (ii) n = 4, l = 0, (iii) n = 3, l = 2, (iv) n = 3, l = 1 can be placed in order of increasing energy, from the lowest to highest, as

    Atomic Structure

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