Practice Questions

Q71.If x, y, z are in arithmetic progression with common difference d, x ≠3d, and the determinant of the matrix 3 4√2 x ⎡ ⎤ is zero, then the value of k2 is 4 5√2 y 5 k z ⎣ ⎦ (1) 72 (2) 12 (3) 36 (4) 6

Q71.Let [x] denote the greatest integer ≤x, where x ∈R. If the domain of the real valued function f(x) = is (−∞, a) ∪[b, c) ∪[4, ∞), a < b < c, then the value of a + b + c is: √|[x]|−2|[x]|−3 (1) 8 (2) 1 (3) −2 (4) −3

Q71.If sin−1a x = cos−1b x = tan−1c y ; 0 < x < 1, then the value of cos( a+bπc ) is: (1) 1−y2 (2) 1 −y2 1+y2 (3) 1−y2 (4) 1−y2 y√y 2y

Q71.Let Sk = ∑kr=1 tan−1( 22r+1+32r+16r ), then k→∞Sk (1) tan−1( 23 ) (2) π2 (3) cot−1( 23 ) (4) tan−1(3)

Q72.Let f be a real valued function, defined on R −{−1, 1} and given by f(x) = 3 loge x+1x−1 − x−12 . Then in which of the following intervals, function f(x) is increasing? (1) (−∞, −1) ∪([ 21 , ∞) −{1}) (2) (−∞, ∞) −{−1, 1} (3) (−1, 12 ] (4) (−∞, 21 ] −{−1} dx where [x] denotes the greatest integer less than or equal to x. Then the

Q72.The domain of the function, 𝑓𝑥= sin-13𝑥2 + cos-1 2 ( 𝑥- 1 𝑥+ 1 ) 1 1 (1) 0, (2) 0, 2 4 (3) 1 1 ∪0 (4) -2, 0 ∪1 1 4, 2 4, 2

Q72.The number of elements in the set {x ∈R : (|x| −3)|x + 4| = 6} is equal to (1) 3 (2) 2 (3) 4 (4) 1

Q72.If lim sin−1 x−tan−1 x is equal to L, then the value of (6L + 1) is x→0 3x3 (1) 1 (2) 1 6 2 (3) 6 (4) 2 JEE Main 2021 (18 Mar Shift 1) JEE Main Previous Year Paper Q73. 1 2 0 2 −1 5 Let A + 2B = ⎡ 6 −3 3⎤ and 2A −B = ⎡2 −1 6⎤ . If Tr(A) denotes the sum of all diagonal elements −5 3 1 0 1 2 ⎣ ⎦ ⎣ ⎦ of the matrix A, then Tr (A)−Tr (B) has value equal to (1) 1 (2) 2 (3) 0 (4) 3

Q72.If P = [ ], 2 1 (1) [125 10 ] (2) [10 501 ] (3) [10 251 ] (4) [150 10 ] is equal to:

Q72.The system of equations kx + y + z = 1, x + ky + z = k and x + y + zk = k2 has no solution if k is equal to: (1) 0 (2) 1 (3) −1 (4) −2

Q72.If (sin−1 x)2 −(cos−1 x)2 = a; 0 < x < 1, a ≠0, then the value of 2x2 −1 is (1) cos( 2aπ ) (2) sin( 2aπ ) (3) cos( 4aπ ) (4) sin( 4aπ )

Q72.A box open from top is made from a rectangular sheet of dimension a × b by cutting squares each of side x from each of the four corners and folding up the flaps. If the volume of the box is maximum, then x is equal to: (1) a+b+√a2+b2−ab (2) a+b−√a2+b2−ab 6 12 (3) a+b−√a2+b2−ab (4) a+b−√a2+b2+ab 6 6

Q72.A function f(x) is given by f(x) = 5x+55x , then the sum of the series f( 201 ) + f( 202 ) + f( 203 ) + … + f( 2039 ) is equal to: (1) 19 (2) 49 2 2 (3) 39 (4) 29 2 2

Q72.Let the system of linear equations 4x + λy + 2z = 0 2x −y + z = 0 μx + 2y + 3z = 0, λ, μ ∈R has a non-trivial solution. Then which of the following is true? JEE Main 2021 (18 Mar Shift 2) JEE Main Previous Year Paper (1) μ = 6, λ ∈R (2) λ = 2, μ ∈R (3) λ = 3, μ ∈R (4) μ = −6, λ ∈R

Q72.The triangle of maximum area that can be inscribed in a given circle of radius ' r' is : (1) An equilateral triangle having each of its side of (2) An isosceles triangle with base equal to 2r. length √3r. (3) An equilateral triangle of height 2r . (4) A right angle triangle having two of its sides of 3 length 2r and r. dt, then f(e) + f( 1e ) is equal to

Q72.The function 𝑓𝑥= 4𝑥3 - 3𝑥2 - 2sin𝑥+ 2𝑥- 1cos𝑥: 6 1 1 (1) increases in 2, ∞ (2) decreases in - ∞, 2 1 1 (3) decreases in 2, ∞ (4) increases in - ∞, 2

Q72. sin x −ex if x ≤0 ⎧ Let a function f : R →R be defined as, f(x) = a + [−x] if 0 < x < 1 ⎨ ⎩ 2x −b if x ≥1 JEE Main 2021 (20 Jul Shift 1) JEE Main Previous Year Paper Where [x] is the greatest integer less than or equal to x. If f is continuous on R, then (a + b) is equal to: (1) 4 (2) 3 (3) 2 (4) 5 Q73. ⎧ 1 , if i = j Let A = [aij] be a 3 × 3 matrix, where aij = −x , if |i −j| = 1 ⎨ ⎩2x + 1 , otherwise Let a function f : R →R be defined as f(x) =det (A). Then the sum of maximum and minimum values of f on R is equal to: (1) −2027 (2) 2788 (3) 27 20 (4) −8827

Q72.Let A = [−11 24 ]. If A−1 = αI + βA, α, β ∈R, I is a 2 × 2 identity matrix, then 4(α −β) is equal to : (1) 5 (2) 83 (3) 2 (4) 4 (1 + |sin x|) |sin x| , −π4 < x < 0Q73. ⎧ 3a b , x = 0 Let f : (−π4 , π4 ) →R be defined as, f(x) = ⎨ ⎩ ecot 4x/ cot 2x , 0 < x < π4 If f is continuous at x = 0 then the value of 6a + b2 is equal to: (1) 1 −e (2) e −1 (3) 1 + e (4) e

Q72.If the curve y = ax2 + bx + c, x ∈R, passes through the point (1, 2) and the tangent line to this curve at origin is y = x, then the possible values of a, b, c are: (1) a = −1, b = 1, c = 1 (2) a = 1, b = 0, c = 1 (3) a = 1, b = 1, c = 0 (4) a = 12 , b = 12 , c = 1

Q72.If the following system of linear equations 2x + y + z = 5 x −y + z = 3 x + y + a z = b has no solution, then : (1) a = −13 , b ≠73 (2) a ≠13 , b = 73 (3) a ≠−13 , b = 73 (4) a = 13 , b ≠73

Q72. x3 1+2xe−2x , x ≠0 (1−cos 2x)2 loge( (1−xe−x)2 ) Let f : R →R be defined as f(x) = { α , x = 0 If f is continuous at x = 0, then α is equal to: (1) 1 (2) 3 (3) 0 (4) 2

Q73.Let θ ∈(0, π2 ). If the system of linear equations (1 + cos2 θ)x + sin2 θy + 4 sin 3θz = 0 cos2 θx + (1 + sin2 θ)y + 4 sin 3θz = 0 cos2 θx + sin2 θy + (1 + 4 sin 3θ)z = 0 has a non-trivial solution, then the value of θ is: (1) 4π (2) 5π 9 18 (3) 7π (4) π 18 18 = tan−1 0 < x < 1. Then: x

Q73.Let x denote the total number of one-one functions from a set A with 3 elements to a set B with 5 elements and y denote the total number of one-one functions from the set A to the set A × B. Then : JEE Main 2021 (25 Feb Shift 2) JEE Main Previous Year Paper (1) y = 273x (2) 2y = 273x (3) 2y = 91x (4) y = 91x

Q73.If [x] denotes the greatest integer less than or equal to x, then the value of the integral ∫π/2−π/2[[x] −sin x]dx is equal to: (1) −π (2) π (3) 0 (4) 1

Q73.If Rolle's theorem holds for the function f(x) = x3 −ax2 + bx −4, x ∈[1, 2] with f ′( 43 ) = 0 , then ordered pair (a, b) is equal to : (1) (−5, −8) (2) (−5, 8) (3) (5, 8) (4) (5, −8) dθ is (where c is a constant of integration)