Practice Questions

Q72.The function 𝑓𝑥= 𝑥3 - 6𝑥2 + 𝑎𝑥+ 𝑏 is such that 𝑓2 = 𝑓4 = 0. Consider two statements: 𝑆1 there exists 𝑥1, 𝑥2 ∈2, 4, 𝑥1 < 𝑥2, such that 𝑓'𝑥1 = - 1 and 𝑓'𝑥2 = 0 . 𝑆2 there exists 𝑥3, 𝑥4 ∈2, 4, 𝑥3 < 𝑥4, such that 𝑓 is decreasing in 2, 𝑥4, increasing in 𝑥4, 4 and 2𝑓'𝑥3 = √3𝑓𝑥4 then (1) 𝑆1 is true and 𝑆2 is false (2) both 𝑆1 and 𝑆2 are false (3) both 𝑆1 and 𝑆2 are true (4) 𝑆1 is false and 𝑆2 is true JEE Main 2021 (01 Sep Shift 2) JEE Main Previous Year Paper Q73. 𝜋 sec2𝑥𝑓(𝑥)d𝑥 4 ∫2 Let f : R →R be a continuous function. Then lim 𝜋2 is equal to: 𝑥→𝜋/ 4 𝑥2 - 16 (1) 𝑓( 2 ) (2) 2𝑓( √2 ) (3) 2𝑓( 2 ) (4) 4𝑓( 2 )

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. 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

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.Let A and B be two 3 × 3 real matrices such that (A2 −B2) is invertible matrix. If A5 = B5 and A3 B2 = A2 B3, then the value of the determinant of the matrix A3 + B3 is equal to : (1) 2 (2) 4 (3) 1 (4) 0

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

Q72.The number of solutions of the equation sin−1[x2 + 13 ] + cos−1[x2 −23 ] = x2 for x ∈[−1, 1], and [x] denotes the greatest integer less than or equal to x, is : (1) 2 (2) 0 (3) 4 (4) Infinite . Then f is:

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.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.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.The sum of all the local minimum values of the twice differentiable function f : R →R defined by ′′(2) x + f ′′(1) is: f(x) = x3 −3x2 −3f 2 (1) −22 (2) 5 (3) −27 (4) 0

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 f, g : N →N such that f(n + 1) = f(n) + f(1) ∀ n ∈N and g be any arbitrary function. Which of the following statements is NOT true? (1) If f is onto, then f(n) = n∀n ∈N (2) If g is onto, then fog is one-one (3) f is one-one (4) If fog is one-one, then g is one-one

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.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 )

Q73.If cot−1(α) = cot−1 2 + cot−1 8 + cot−1 18 + cot−1 32 + … . upto 100 terms, then α is: JEE Main 2021 (17 Mar Shift 1) JEE Main Previous Year Paper (1) 1. 01 (2) 1. 00 (3) 1. 02 (4) 1. 03

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)

Q73.For x > 0 , if f(x) = ∫x1 (1+t)loge t (1) 0 (2) 21 (3) −1 (4) 1 x ∈R. Then f(x) equals :

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 the tangent to the curve 𝑦= 𝑥3 at the point 𝑃𝑡, 𝑡3 meets the curve again at 𝑄, then the ordinate of the point which divides 𝑃𝑄 internally in the ratio 1: 2 is: (1) 0 (2) -2𝑡3 (3) -𝑡3 (4) 2𝑡3

Q73.Consider the integral I = ∫100 [x]e[x]ex−1 value of I is equal to : (1) 9(e −1) (2) 45(e + 1) (3) 45(e −1) (4) 9(e + 1)

Q73.Let M and m respectively be the maximum and minimum values of the function f(x) = tan−1(sin x + cos x) in [0, π2 ]. Then the value of tan(M −m) is equal to: (1) 2 −√3 (2) 3 −2√2 (3) 3 + 2√2 (4) 2 + √3