Practice Questions

Q70.Let C be the circle with center at (1, 1) and radius = 1. If T is the circle centered at (0, y), passing through the origin and touching the circle C externally, then the radius of T is equal to (1) 1 (2) 1 2 4 (3) √3 (4) √3 √2 2

Q70.Let L1 be the length of the common chord of the curves x2 + y2 = 9 and y2 = 8x, and L2 be the length of the latus rectum of y2 = 8x, then: (1) L1 > L2 (2) L1 = L2 (3) L1 < L2 (4) L1L2 = √2

Q71.The tangent at an extremity (in the first quadrant) of the latus rectum of the hyperbola x24 −y25 = 1 , meets the x-axis and y-axis at A and B, respectively. Then OA2 −OB2 , where O is the origin, equals (1) −209 (2) 169 (3) 4 (4) −43

Q71.Let a and b be any two numbers satisfying 1 + 1 = 14 . Then, the foot of perpendicular from the origin on a2 b2 the variable line x a + yb = 1 lies on : (1) A circle of radius = 2 (2) A hyperbola with each semi-axis = √2 . (3) A hyperbola with each semi-axis = 2 (4) A circle of radius = √2

Q71.The locus of the foot of perpendicular drawn from the centre of the ellipse x2 + 3y2 = 6 on any tangent to it is (1) (x2 + y2) 2 = 6x2 + 2y2 (2) (x2 + y2) 2 = 6x2 −2y2 (3) (x2 −y2) 2 = 6x2 + 2y2 (4) (x2 −y2) 2 = 6x2 −2y2

Q71.A stair-case of length l rests against a vertical wall and a floor of a room. Let P be a point on the stair-case, nearer to its end on the wall, that divides its length in the ratio 1 : 2. If the staircase begins to slide on the floor, then the locus of P is: (1) an ellipse of eccentricity 1 (2) an ellipse of eccentricity √3 2 2 (3) a circle of radius 2 1 (4) a circle of radius √32 l

Q71.Two tangents are drawn from a point (−2, −1) to the curve, y2 = 4x. If α is the angle between them, then |tan α| is equal to: (1) 1 (2) 1 3 √3 (3) √3 (4) 3 y2

Q72.Let P(3 sec θ, 2 tan θ) and Q(3 sec ϕ, 2 tan ϕ) where θ + ϕ = π2 , be two distinct points on the hyperbola x2 . Then the ordinate of the point of intersection of the normals at P and Q is: 9 −y24 = 1 (1) 11 3 (2) −113 (3) 13 2 (4) −132 = 5, then k is equal to:

Q72. sin(πcos2x) lim is equal to x→0 x2 (1) −π (2) π (3) π (4) 1 2

Q72.If the point (1, 4) lies inside the circle x2 + y2 −6x + 10y + p = 0 and the circle does not touch or intersect the coordinate axes, then the set of all possible values of p is the interval (1) (25, 39) (2) (25, 29) (3) (0, 25) (4) (9, 25)

Q72.The contrapositive of the statement ''if I am not feeling well, then I will go to the doctor" is (1) if I will go to the doctor, then I am not feeling (2) if I am feeling well, then I will not go to the well. doctor. (3) if I will not go to the doctor, then I am feeling (4) if I will go to the doctor, then I am feeling well. well. ¯

Q72.The minimum area of a triangle formed by any tangent to the ellipse x2 = 1 and the co-ordinate axes is: 16 + 81 (1) 12 (2) 18 (3) 26 (4) 36

Q73.The statement ∼(p ↔~q) is (1) A tautology (2) A fallacy (3) Equivalent to p ↔q (4) Equivalent to ~p ↔q

Q73.Let p, q, r denote arbitrary statements. Then the logically equivalent of the statement p ⇒(q ∨r) is: (1) (p ∨q) ⇒r (2) (p ⇒q) ∨(p ⇒r) (3) (p ⇒∼q) ∧(p ⇒r) (4) (p ⇒q) ∧(p ⇒∼r)

Q73.Let x , M and σ2 be respectively the mean, mode and variance of n observations x1, x2, . ..., xn and di = −xi −a, i = 1, 2, . ..., n, where a is any number. Statement I: Variance of d1, d2, . . . , dn is σ2 . ¯Statement II: Mean and mode of d1, d2, . . . . , dn are −x −a and −M −a, respectively. (1) Statement I and Statement II are both true (2) Statement I and Statement II are both false (3) Statement I is true and Statement II is false (4) Statement I is false and Statement II is true

Q73.If limx→2 tan(x−2{x2+k+2x−2k}x2−4x+4 JEE Main 2014 (11 Apr Online) JEE Main Previous Year Paper (1) 0 (2) 1 (3) 2 (4) 3

Q73.If OB is the semi-minor axis of an ellipse, F1 and F2 are its focii and the angle between F1B and F2B is a right angle, then the square of the eccentricity of the ellipse is (1) 1 (2) 1 4 √2 (3) 1 (4) 1 2 2√2

Q74.Let A and B be any two 3 × 3 matrices. If A is symmetric and B is skew symmetric, then the matrix AB −BA is (1) skew symmetric (2) I or − I, where I is an identity matrix (3) symmetric (4) neither symmetric nor skew symmetric

Q74.Let ¯X and M.D. be the mean and the mean deviation about ¯X of n observations xi, i = 1, 2,n. If each of the observations is increased by 5 , then the new mean and the mean deviation about the new mean, respectively, are : – (1) ¯X , M.D. (2) X + 5, M.D. – (3) ¯X , M.D. +5 (4) X + 5, M.D. +5 JEE Main 2014 (12 Apr Online) JEE Main Previous Year Paper

Q74.If f(x) is continuous and f( 29 ) = 29 , then lim f( 1−cosx2 3x ) equals to x→0 (1) 8 (2) 0 9 (3) 2 (4) 9 9 2

Q74.The variance of the first 50 even natural numbers is : JEE Main 2014 (06 Apr) JEE Main Previous Year Paper (1) 437 (2) 4374 (3) 833 (4) 833 4

Q74.The proposition ∼(p∨∼q)∨∼(p ∨q) is logically equivalent to: (1) p (2) q (3) ∼p (4) ∼q

Q75.A relation on the set A = {x : |x| < 3, x ∈Z}, where Z is the set of integers is defined by R = {(x, y) : y = |x|, x ≠−1}. Then the number of elements in the power set of R is: (1) 32 (2) 16 (3) 8 (4) 64

Q75. r 2r −1 3r −2 If Δr = n2 n −1 a , then the value of ∑n−1r=1 Δr 2 1 n(n −1) (n −1)2 12 (n −1)(3n + 4) (1) Is independent of both a and n (2) Depends only on a (3) Depends only on n (4) Depends both on a and n

Q75.The contrapositive of the statement "I go to school if it does not rain" is (1) If it rains, I go to school. (2) If it rains, I do not go to school. (3) If I go to school, it rains. (4) If I do not go to school, it rains.