Practice Questions

Q60.Let 50∪ = ∪n = T , where each Xi contains 10 elements and each Yi contains 5 elements. If each element i=1Xi i=1Yi of the set T is an element of exactly 20 of sets Xi 's and exactly 6 of sets Yi 's then n is equal to : (1) 15 (2) 50 (3) 45 (4) 30

Q61.If for some α and β in R , the intersection of the following three planes x + 4y −2z = 1 x + 7y −5z = β x + 5y + αz = 5 is a line in R3 , then α + β is equal to: (1) 0 (2) 10 (3) 2 (4) −10 Q62. ; x < 0 ⎧ sin(a+2)x+sinxx If f(x) = is continuous at x = 0 , then a + 2b is equal to: ⎨ b ; x = 0 ; x > 0 ⎩ (x+3x2)1/3−x1/3x1/3 (1) 1 (2) −1 (3) 0 (4) −2

Q62.Let [t] denote the greatest integer ≤t and x→0x[lim discontinuous, when x is equal to: (1) √A + 1 (2) √A + 5 (3) √A + 21 (4) √A

Q63.Let f be any function continuous on [a, b] and twice differentiable on (a, b) . If all x ∈(a, b), f '(x) > 0 and f ''(x) < 0 , then for any c ∈(a, b), f(c)−f(a)f(b)−f(c) (1) b+a (2) 1 b−a (3) b−c (4) c−a c−a b−c

Q63.The set of all real values λ for which the function f(x) = (1 −cos2 x). (λ + sin x), xε (−π2 2 ), has exactly one maxima and exactly one minima, is : (1) (−12 , 12 ) −{0} (2) (−32 , 32 ) (3) (−12 , 12 ) (4) (−32 , 32 ) −{0}

Q63.Suppose f(x) is a polynomial of degree four having critical points at −1, 0, 1. If T = {x ∈R |f(x) = f(0)}, then the sum of squares of all the elements of T is : (1) 4 (2) 6 (3) 2 (4) 8

Q64.Let f(x) be a polynomial of degree 5 such that x = ±1 are its critical points. If x→0(2lim + f(x)x3 ) = 4, then which one of the following is not true? (1) f is an odd function (2) f(1) −4f(−1) = 4 . x = 1 is a point of maximum and x = −1 (3) x = 1is a point of local minimum and x = −1 is (4) x = 1 is a point of local maxima of f a point of local maximum JEE Main 2020 (07 Jan Shift 2) JEE Main Previous Year Paper

Q64.If the tangent to the curve y = x + sin y at a point (a, b) is parallel to the line joining (0, 23 ) and ( 21 , 2) , then (1) b = a (2) |b −a| = 1 (3) |a + b| = 1 (4) b = π2 + a JEE Main 2020 (02 Sep Shift 1) JEE Main Previous Year Paper

Q65.If I1 = ∫10 (1 −x50)100dx and I2 = ∫10 (1 −x50)101dx such that I2 = αI1 then (1) 5049 (2) 5050 5050 5049 (3) 5050 (4) 5051 5051 5050 Q66. ∫(x−1)20 t cos t2dt lim (x−1) sin(x−1) x→1( ) (1) is equal to 1 . (2) is equal to 1. 2 (3) is equal to −12 . (4) is equal to 0.

Q65.If f(a + b + 1 −x) = f(x), for all x, where a and b are fixed positive real numbers, then b 1 ∫ x(f(x) + f(x + 1))dx is equal to a+b a (1) b−1 (2) b−1 ∫ f(x + 1)dx ∫ f(x)dx a−1 a−1 (3) b+1 (4) b+1 ∫ f(x)dx ∫ f(x + 1)dx a+1 a+1

Q65.The equation of the normal to the curve y = (1 + x)2y + cos2(sin−1 x) , at x = 0 is (1) y + 4x = 2 (2) y = 4x + 2 (3) x + 4y = 8 (4) 2y + x = 4

Q65.If the function f(x) = {k1(xk2−π)2cos x,−1, xx ≤π> π to: (1) ( 21 , 1) (2) (1, 0) (3) ( 21 , −1) (4) (1, 1) + c, where c is a constant of integration, then g(0) is

Q65.The integral ∫ 8dx 6 is equal to: (where C is a constant of integration) (x+4) 7 (x−3) 7 (1) x−3 71 (2) x−3 −17 ( x+4 ) + C ( x+4 ) + C (3) 1 x−3 73 (4) x−3 −137 2 ( x+4 ) + C −113 ( x+4 ) + C

Q65.Let f be a twice differentiable function on (1, 6), If f(2) = 8, f ′(2) = 5, f ′(x) ≥1 and f′′(x) ≥4, for all x ∈(1, 6), then : (1) f(5) + f ′(5) ≤26 (2) f(5) + f ′(5) ≥28 (3) f ′(5) + f′′(5) ≤20 (4) f(5) ≤10 is equal to, (where C is a constant of integration):

Q66.The area of the region (in sq. units), enclosed by the circle x2 + y2 = 2 which is not common to the region bounded by the parabola y2 = x and the straight line y = x , is (1) 1 6 (24π −1) (2) 13 (6π −1) (3) 1 3 (12π −1) (4) 16 (12π −1) JEE Main 2020 (07 Jan Shift 1) JEE Main Previous Year Paper = ex such that y(0) = 0, then y(1) is

Q66.If the value of the integral ∫ 01 3 dx is k6 , then k is equal to: (1−x2) 2 JEE Main 2020 (03 Sep Shift 2) JEE Main Previous Year Paper (1) 2√3 + π (2) 2√3 −π (3) 3√2 + π (4) 3√2 −π

Q66.Let f : (−1, ∞) →R be defined by f(0) = 1 and f(x) = x1 loge(1 + x), x ≠0 . Then the function f (1) Decreases in (−1, 0) and increases in (0, ∞) (2) Increases in (−1, ∞) (3) Increases in (−1, 0) and decreases in (0, ∞) (4) Decreases in (−1, ∞)

Q68.The area (in sq. units) of the region A = {(x, y) : (x −1)[x] ≤y ≤2√x, 0 ≤x ≤2}, where [t] denotes the greatest integer function, is : (1) 3 8 √2 −12 (2) 34 √2 + 1 (3) 8 3 √2 −1 (4) 43 √2 −12

Q68.Let a, b, c ∈R be such that a2 + b2 + c2 = 1. If a cos θ = b cos(θ + 2π3 ) = c cos(θ + 4π3 ),where θ = π9 , then the angle between the vectors aˆi + bˆj + cˆk and bˆi + cˆj + aˆk is: (1) 0 (2) 2π3 (3) π (4) π 2 9

Q69.The shortest distance between the lines x−1 0 = y+1−1 = 1z and x + y + z + 1 = 0, 2 x −y + z + 3 = 0 is JEE Main 2020 (06 Sep Shift 1) JEE Main Previous Year Paper (1) 1 (2) 1 √3 (3) 1 (4) 1 √2 2

Q69.If 10 different balls are to be placed in 4 distinct boxes at random, then the probability that two of these boxes contain exactly 2 and 3 balls is: (1) 965 (2) 965 211 210 (3) 945 (4) 945 210 211

Q70.The probability that a randomly chosen 5- digit number is made from exactly two digits is : (1) 135 (2) 150 104 104 (3) 134 (4) 121 104 104

Q70.Let x0 be the point of local maxima of f(x) =→a⋅(→ ×→c), →c= 7ˆi −2ˆj + xˆk. Then the value of →a⋅→b +→b ⋅→c+→c⋅→a at x = x0 is: (1) −4 (2) −30 (3) 14 (4) −22 is equal to ______

Q70.In a game two players A and B take turns in throwing a pair of fair dice starting with player A and total of scores on the two dice, in each throw is noted. A wins the game if he throws a total of 6 before B throws a total of 7 and B wins the game if he throws a total of 7 before A throws a total of six. The game stops as soon as either of the players wins. The probability of A winning the game is : (1) 5 (2) 31 31 61 (3) 5 (4) 30 6 61

Q71.The angle of elevation of the top of a hill from a point on the horizontal plane passing through the foot of the hill is found to be 45°. After walking a distance of 80 meters towards the top, up a slope inclined at angle of 30° to the horizontal plane the angle of elevation of the top of the hill becomes 75°. Then the height of the hill (in meters) is _____.