Practice Questions

Q70.Let R1 and R2 be two relations defined on R by aR1b ⇔ab ≥0 and a R2b ⇔a ≥b, then JEE Main 2022 (27 Jul Shift 1) JEE Main Previous Year Paper (1) R1 is an equivalence relation but not R2 (2) R2 is an equivalence relation but not R1 (3) both R1 and R2 are equivalence relations (4) neither R1 nor R2 is an equivalence relation

Q71.The domain of the function 𝑓𝑥= sin-1 𝑥2 - 3𝑥+ 2 is 𝑥2 + 2𝑥+ 7 (1) [1, ∞) (2) ( - 1, 2] (3) [ - 1, ∞) (4) ( - ∞, 2]

Q71.The number of distinct real roots of x4 −4x + 1 = 0 is (1) 0 (2) 1 (3) 2 (4) 4

Q71. a −1 0 Let f(x) = ax a −1 , a ∈R. Then the sum of the squares of all the values of a for ax2 ax a 2f ′(10) −f ′(5) + 100 = 0 is (1) 117 (2) 106 (3) 125 (4) 136 is

Q71.If 0 < 𝑥< 1 and sin-1𝑥 = cos-1𝑥 , then a value of sin 2𝜋𝛼 is √2 𝛼 𝛽 𝛼+ 𝛽 (1) 4√1 - 𝑥2 1 - 2𝑥2 (2) 4𝑥√1 - 𝑥2 1 - 2𝑥2 (3) 2𝑥√1 - 𝑥2 1 - 4𝑥2 (4) 4√1 - 𝑥2 1 - 4𝑥2

Q71.The probability that a randomly chosen one-one function from the set {a, b, c, d} to the set {1, 2, 3, 4, 5} satisfied f(a) + 2 f(b) −f(c) = f(d) is (1) 1 (2) 1 24 40 (3) 1 (4) 1 30 20

Q71.If y = tan−1(sec x3 −tan x3), π2 < x3 < 3π2 , then (1) xy′′ + 2y′ = 0 (2) x2y′′ −6y + 3π2 = 0 (3) x2y′′ −6y + 3π = 0 (4) xy′′ −4y′ = 0

Q71.Let A = [aij] be a square matrix of order 3 such that aij = 2j−i , for all i, j = 1, 2, 3 . Then, the matrix A2 + A3 + … + A10 is equal to (1) ( 310−12 )A (2) ( 310+12 )A (3) ( 310+32 )A (4) ( 310−32 )A

Q71.Let A = (−21 −52 ). Let α, β ∈R be such that αA2 + βA = 2I . Then α + β is equal to (1) −10 (2) −6 (3) 6 (4) 10

Q71.The domain of the function f(x) = sin−1[2x2 −3] + log2(log (x2 −5x + 5)), where 2 integer function, is 2 , 5+√52 ) 2 , 5−√52 (1) (−√5 ) (2) ( 5−√5 (3) (1, 5−√52 ) (4) [1, 5+√52 )

Q71.If the function f(x) = loge(1−x+x2)+loge(1+x+x2) −π π sec x−cos x , x ∈( 2 , 2 ) −{0} is continuous at x = 0 , then k is equal { k , x = 0 to: (1) 1 (2) −1 (3) e (4) 0 are continuous on R, then and g(x) =

Q71.Let f(x) = x−1x+1 , x ∈R −{0, −1, 1) . If f n+1(x) = f(f n(x)) for all n ∈N , then f 6(6) + f 7(7) is equal to JEE Main 2022 (26 Jun Shift 1) JEE Main Previous Year Paper (1) 7 6 (2) −32 (3) 12 7 (4) −1112 Q72. + 3| , x < 0 f, g : R →R be two real valued function defined as f(x) = and {−|xex , x ≥0 + k1x , x < 0 g(x) = , where k1 and k2 are real constants. If gof is differentiable at x = 0, then {x24x + k2 , x ≥0 gof(−4)+gof(4) is equal to (1) 4(e4 + 1) (2) 2(2e4 + 1) (3) 4e4 (4) 2(2e4 −1)

Q71.The set of all values of k for which (tan−1 x)3 + (cot−1 x)3 = kπ3, x ∈R, is the interval (1) [ 321 , 87 ) (2) ( 241 , 1613 ) (3) [ 481 , 1613 ] (4) [ 321 , 89 ) x2−9 ) is

Q71.From the base of a pole of height 20 meter, the angle of elevation of the top of a tower is 60°. The pole subtends an angle 30° at the top of the tower. Then the height of the tower is: (1) 15√3 (2) 20√3 (3) 20 + 10√3 (4) 30 Q72. 2 −1 Let A = − . . . − 5C5(adj A)5 , then the sum of . If B = I −5C1(adj A) + 5C2(adj A)2 (0 2 ) all elements of the matrix B is: (1) −5 (2) −6 (3) −7 (4) −8

Q71.If the absolute maximum value of the function 𝑓𝑥= x2 - 2x + 7e4x3 - 12x2 - 180x + 31in the interval -3, 0 is 𝑓𝛼, then (1) 𝛼= 0 (2) 𝛼= - 3 (3) 𝛼∈-1, 0 (4) 𝛼∈-3, - 1

Q71.If the mean deviation about median for the number 3, 5, 7, 2k, 12, 16, 21, 24 arranged in the ascending order, is 6 then the median is (1) 11. 5 (2) 10. 5 (3) 12 (4) 11

Q71.Let 𝑓: 𝑁→𝑅 be a function such that 𝑓𝑥+ 𝑦= 2 𝑓𝑥 𝑓𝑦 for natural numbers 𝑥 and 𝑦. If 𝑓1 = 2, then the 10 512 value of 𝛼 for which ∑𝑘= 1 𝑓𝛼+ 𝑘= 3 220 - 1 holds, is (1) 3 (2) 4 (3) 5 (4) 6

Q71.The function f(x) = xex(1−x), x ∈R, is (1) increasing in (−12 , 1) (2) decreasing in ( 12 , 2) (3) increasing in (−1, −12 ) (4) decreasing in (−12 , 12 )

Q71.Considering the principal values of the inverse trigonometric functions, the sum of all the solutions of the equation cos-1𝑥- 2sin-1𝑥= cos-12𝑥 is equal to (1) 0 (2) 1 (3) 1 (4) -1 2 2

Q71.Let f : R →R be defined as f(x) = x −1 and g : R →{1, −1} →R be defined as g(x) = x2 . Then the x2−1 function fog is: (1) One-one but not onto (2) onto but not one-one (3) Both one-one and onto (4) Neither one-one nor onto

Q71.The system of equations -𝑘𝑥+ 3𝑦- 14𝑧= 25 -15𝑥+ 4𝑦- 𝑘𝑧= 3 -4𝑥+ 𝑦+ 3𝑧= 4 Question: is consistent for all 𝑘 in the set (1) 𝑅 (2) 𝑅- -11, 13 (3) 𝑅- -13 (4) 𝑅- -11, 11 - 1 4

Q71.The number of points, where the function f : R →R, f(x) = |x −1| cos|x −2| sin|x −1| + (x −3) x2 −5x + 4 , is NOT differentiable, is (1) 1 (2) 2 (3) 3 (4) 4

Q72.The number of real values of λ, such that the system of linear equations 2x −3y + 5z = 9 x + 3y −z = −18 3x −y + (λ2 −|λ|)z = 16 has no solutions, is (1) 0 (2) 1 (3) 2 (4) 4 JEE Main 2022 (25 Jul Shift 2) JEE Main Previous Year Paper

Q72.The value of cot(∑50n=1 tan−1( 1+n+n21 )) (1) 25 (2) 50 26 51 (3) 26 (4) 52 25 51 JEE Main 2022 (27 Jun Shift 2) JEE Main Previous Year Paper

Q72.Let f(x) = min{1, 1 + x sin x}, 0 ≤x ≤2π. If m is the number of points, where f is not differentiable and n is the number of points, where f is not continuous, then the ordered pair (m, n) is equal to (1) (2, 0) (2) (1, 0) (3) (1, 1) (4) (2, 1) JEE Main 2022 (26 Jun Shift 2) JEE Main Previous Year Paper