Practice Questions

Q76.Let A and B denote the statements A: cos α + cos β + cos γ = 0 B: sin α + sin β + sin γ = 0 If cos(β −γ) + cos(γ −α) + cos(α −β) = −32 , then (1) A is true and B is false (2) A is false and B is true (3) both A and B are true (4) both A and B are false

Q77.For real x, let f(x) = x3 + 5x + 1, then (1) f is one-one but not onto R (2) f is onto R but not one-one (3) f is one-one and onto R (4) f is neither one-one nor onto R

Q78.Let f(x) = (x + 1)2 −1, x ≥−1 Statement-1: The set {x : f(x) = f −1(x)} = {0, −1} Statement-2 : f is a bijection. (1) Statement-1 is true, Statement-2 is true; (2) Statement-1 is true, Statement-2 is true; Statement-2 is a correct explanation for Statement-2 is not a correct explanation for Statement-1 Statement-1 (3) Statement-1 is true, Statement-2 is false (4) Statement-1 is false, Statement-2 is true

Q79.Let f(x) = x|x| and g(x) = sin x. Statement-1 : gof is differentiable at x = 0 and its derivative is continuous at that point. Statement-2 : gof is twice differentiable at x = 0 . (1) Statement-1 is true, Statement-2 is true; (2) Statement-1 is true, Statement-2 is true; Statement-2 is a correct explanation for Statement-2 is not a correct explanation for Statement-1 Statement-1 (3) Statement-1 is true, Statement-2 is false (4) Statement-1 is false, Statement-2 is true

Q80.Let y be an implicit function of x defined by x2x −2xx cot y −1 = 0 . Then y′(1) equals (1) −1 (2) 1 (3) log 2 (4) −log 2

Q81.Given P(x) = x4 + ax3 + bx2 + cx + d such that x = 0 is the only real root of P ′(x) = 0 . If P(−1) < P(1), then in the interval [−1, 1] (1) P(−1) is the minimum and P(1) is the (2) P(−1) is not minimum but P(1) is the maximum maximum of P of P (3) P(−1) is the minimum and P(1) is not the (4) neither P(−1) is the minimum nor P(1) is the maximum of P maximum of P

Q82.The shortest distance between the line y −x = 1 and the curve x = y2 is (1) 3√2 (2) 2√3 8 8 (3) 3√2 (4) √3 5 4 Q83. ∫π0 [cot x]dx, [∙] denotes the greatest integer function, is equal to (1) π (2) 1 2 (3) −1 (4) −π2

Q84.The area of the region bounded by the parabola (y −2)2 = x −1, the tangent to the parabola at the point (2, 3) and the x-axis is (1) 3 (2) 6 (3) 9 (4) 12 JEE Main 2009 JEE Main Previous Year Paper

Q85.The differential equation which represents the family of curves y = c1ec2x , where c1 and c2 are arbitrary constants is (1) y′ = y2 (2) y′′ = y′y (3) yy′′ = y′ (4) yy′′ = (y′)2

Q86.If →u, →v, ¯w are non-coplanar vectors and p, q are real numbers, then the equality [ 3→u p→v p→w ] −[ p→v →w q→u ] −[ 2→w q→v q→u ] = 0 holds for (1) exactly one value of (p, q) (2) exactly two values of (p, q) (3) more than two but not all values of (p, q) (4) all values of (p, q)

Q87.Let the line x−2 3 = y−1−5 = z+22 lies in the plane x + 3y −αz + β = 0. Then (α, β) equals (1) (6, −17) (2) (−6, 7) (3) (5, −15) (4) (−5, 15)

Q88.The projections of a vector on the three coordinate axis are 6, −3, 2 respectively. The direction cosines of the vector are (1) 6, −3, 2 (2) 65 , −35 , 25 (3) 7 6 , −37 , 27 (4) −67 , −37 , 27

Q89.In a binomial distribution B (n, p = 41 ), if the probability of at least one success is greater than or equal to 109 , then n is greater than 1 1 (1) 3 (2) 3 log10 4+log10 log10 4−log10 (3) 9 (4) 4 log10 4−log10 3 log10 4−log10 3

Q90.One ticket is selected at random from 50 tickets numbered 00, 01, 02, … , 49. Then the probability that the sum of the digits on the selected ticket is 8 , given that the product of these digits is zero, equals (1) 1 (2) 1 14 7 (3) 5 (4) 1 14 50 JEE Main 2009 JEE Main Previous Year Paper

Q71.Statement - 1: For every natural number n ≥2, 1 + 1 + … + 1 > √n. Statement −2 : For every √1 √2 √n natural number n ≥2, √n(n + 1) < n + 1. (1) Statement −1 is false, Statement −2 is true (2) Statement −1 is true, Statement −2 is true, Statement −2 is a correct explanation for Statement −1 (3) Statement −1 is true, Statement −2 is true; (4) Statement −1 is true, Statement −2 is false. Statement −2 is not a correct explanation for Statement −1.

Q72.The quadratic equations x2 −6x + a = 0 and x2 −cx + 6 = 0 have one root in common. The other roots of the first and second equations are integers in the ratio 4 : 3. Then the common root is (1) 1 (2) 4 (3) 3 (4) 2

Q73.The conjugate of a complex number is 1 . Then the complex number is i−1 (1) −1 (2) 1 i−1 i+1 (3) −1 (4) 1 i+1 i−1

Q74.In a shop there are five types of ice-creams available. A child buys six ice-creams. Statement -1: The number of different ways the child can buy the six ice-creams is 10C5 . Statement −2 : The number of different ways the child can buy the six ice-creams is equal to the number of different ways of arranging 6 A's and 4 B's in a row. (1) Statement −1 is false, Statement −2 is true (2) Statement −1 is true, Statement −2 is true, Statement −2 is a correct explanation for Statement −1 (3) Statement −1 is true, Statement −2 is true; (4) Statement −1 is true, Statement −2 is false. Statement −2 is not a correct explanation for Statement −1.

Q75.How many different words can be formed by jumbling the letters in the word MISSISSIPPI in which no two S are adjacent? (1) 8 ⋅6C4 ⋅7C4 (2) 6.8 ⋅7C4 (3) 6 ⋅7 ⋅8C4 (4) 7 ⋅6C4 ⋅8C4

Q76.The first two terms of a geometric progression add up to 12. The sum of the third and the fourth terms is 48. If the terms of the geometric progression are alternately positive and negative, then the first term is (1) −4 (2) −12 (3) 12 (4) 4

Q77.Statement-1: ∑nr=0(r + 1)nCr = (n + 2)2n−1 Statement -2: ∑nr=0(r + 1)nCrxr = (1 + x)n + nx(1 + x)n−1 . JEE Main 2008 JEE Main Previous Year Paper (1) Statement −1 is false, Statement −2 is true (2) Statement −1 is true, Statement −2 is true, Statement −2 is a correct explanation for Statement −1 (3) Statement −1 is true, Statement −2 is true; (4) Statement −1 is true, Statement −2 is false. Statement −2 is not a correct explanation for Statement −1.

Q78.The perpendicular bisector of the line segment joining P(1, 4) and Q(k, 3) has y-intercept - 4. Then a possible value of k is (1) 1 (2) 2 (3) −2 (4) −4

Q79.The point diametrically opposite to the point P(1, 0) on the circle x2 + y2 + 2x + 4y −3 = 0 is (1) (3, −4) (2) (−3, 4) (3) (−3, −4) (4) (3, 4)

Q80.A parabola has the origin as its focus and the line x = 2 as the directrix. Then the vertex of the parabola is at (1) (0, 2) (2) (1, 0) (3) (0, 1) (4) (2, 0)

Q81.A focus of an ellipse is at the origin. The directrix is the line x = 4 and the eccentricity is 1/2. Then the length of the semi-major axis is (1) 8 (2) 2 3 3 (3) 4 (4) 5 3 3