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2012 (AIEEE / IIT-JEE, in the JEE Main bank) questions

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

Mathematics

  1. If a_1, a_2, a_3, … are in a harmonic progression with a_1 = 5 and a_20 = 25. Then, the least positive integer n for which a_n < 0, is

    Sequences and Series · with solution

  2. The locus of the mid-point of the chord of contact of tangents drawn from points lying on the straight line 4x - 5y = 20 to the circle x^2 + y^2 = 9 is

    Circles · with solution

  3. The total number of ways in which 5 balls of different colours can be distributed among 3 persons so that each person gets at least one ball, is

    Permutations and Combinations · with solution

  4. If the lines (x-2)/1 = (y-3)/1 = (z-4)/k and (x-1)/k = (y-4)/2 = (z-5)/1 are coplanar, then k can have

    Three Dimensional Geometry · with solution

  5. The ellipse E_1: (x^2)/9 + (y^2)/4 = 1 is inscribed in a rectangle R whose sides are parallel to the coordinate axes. Another ellipse E_2 passing through the point (0, 4) circumscribes the rectangle R. The eccentricity…

    Conic Sections · with solution

  6. If a and b are vectors such that |a + b| = √(29) and a × (2i + 3j + 4k) = (2i + 3j + 4k) × b, then a possible value of (a + b) · (-7i + 2j + 3k) is

    Vector Algebra · with solution

  7. The function f: [0, 3] → [1, 29], defined by f(x) = 2x^3 - 15x^2 + 36x + 1, is

    Sets Relations and Functions · with solution

  8. Four fair dice D_1, D_2, D_3 and D_4 each having six faces numbered 1, 2, 3, 4, 5 and 6 are rolled simultaneously. The probability that D_4 shows a number appearing on one of D_1, D_2 and D_3, is

    Probability

  9. If P is a 3 × 3 matrix such that P^T = 2P + I, where P^T is the transpose of P and I is the 3 × 3 identity matrix, then there exists a column matrix, X = beginbmatrix x \ y \ z endbmatrix ≠ beginbmatrix 0 \ 0 \ 0…

    Matrices and Determinants · with solution

  10. Let z be a complex number such that the imaginary part of z is non-zero and a = z^2 + z + 1 is real. Then, a cannot take the value

    Complex Numbers and Quadratic Equations · with solution

  11. Let α(a) and β(a) be the roots of the equation (3√(1+a)-1)x^2-(√(1+a)-1)x+(6√(1+a)-1)=0, where a>-1. Then lim_a→ 0^+α(a) and lim_a→ 0^+β(a) are

    Limits Continuity and Differentiability

  12. If lim_x → ∞ ( (x^2 + x + 1)/(x + 1) - ax - b ) = 4, then

    Limits Continuity and Differentiability · with solution

  13. Let P = [a_ij] be a 3 × 3 matrix and let Q = [b_ij], where b_ij = 2^i+j a_ij for 1 ≤ i, j ≤ 3. If the determinant of P is 2, then the determinant of the matrix Q is

    Matrices and Determinants · with solution

  14. The value of the integral int_-π/2^π/2 (x^2 + log(π - x)/(π + x) ) cos x dx is

    Definite Integrals and Areas · with solution

  15. The equation of a plane passing through the line of intersection of the planes x + 2y + 3z = 2 and x - y + z = 3 and at a distance 2√(3) from the point (3, 1, -1) is

    Three Dimensional Geometry

  16. The point P is the intersection of the straight line joining the points Q(2, 3, 5) and R(1, -1, 4) with the plane 5x - 4y - z = 1. If S is the foot of the perpendicular drawn from the point T(2, 1, 4) to QR, then the…

    Three Dimensional Geometry

Chemistry

  1. The kinetic energy of an electron in the second Bohr orbit of a hydrogen atom is [a_0 is Bohr radius]

    Atomic Structure · with solution

Physics

  1. In the determination of Young's modulus (Y = 4MLg/(π l d^2) ) by using Searle's method, a wire of length L = 2 m and diameter d = 0.5 mm is used. For a load M = 2.5 kg, an extension l = 0.25 mm in the length of the wire…

    Experimental Physics

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