Practice Questions

Q60.If the system of linear equations 2x + 2ay + az = 0 2x + 3by + bz = 0 2x + 4cy + cz = 0, where a, b, c ∈R are non-zero and distinct; has a non-zero solution, then (1) a 1 , 1b , 1c are in A. P. (2) a, b, c are in G. P. (3) a + b + c = 0 (4) a, b, c are in A. P.

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

Q60.Let A be a 2 × 2 real matrix with entries from {0, 1} and |A| ≠0 . Consider the following two statements; (P) If A ≠l2 , then |A| = −1 (Q) If |A| = 1 , then tr(A) = 2 Where l2 denotes 2 × 2 identity matrix and tr(A) denotes the sum of the diagonal entries of A . Then (1) (P) is false and (Q) is true (2) Both (P) and (Q) are false (3) (P) is true and (Q) is false (4) Both (P) and (Q) are true

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.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

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}

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

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

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.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.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):

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.

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, ∞)

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.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

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 ______