Practice Questions

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.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.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.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.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.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.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 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.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.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

Q72.The lengths of the sides of a triangle are 10 + x2 , 10 + x2 and 20 −2x2 . If for x = k, the area of the triangle is maximum, then 3k2 is equal to (1) 5 (2) 12 (3) 10 (4) 20 d3f dx = f(x)ex + C , where C is a constant, then at x = 1 is equal to Q73. ∫ (x2+1)ex dx3 (x+1)2 (1) 3 (2) 3 4 8 (3) −32 (4) 78 dx is equal to

Q72.If f(x) = {x|x+−4|,a, xx >≤00 { x(x+−4)21, + b, xx <≥00 (gof)(2) + (fog)(−2) is equal to: (1) −10 (2) 10 (3) 8 (4) −8 x > 1

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.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

Q72.The value of tan-1cos15𝜋 is equal to sin𝜋 4 𝜋 𝜋 (1) - (2) - 4 8 (3) -5𝜋 (4) -4𝜋 12 9

Q72. log𝑒1 + 5𝑥- log𝑒1 + 𝛼𝑥 if 𝑥≠0 Let the function 𝑓𝑥= 𝑥 be continuous at 𝑥= 0. Then 𝛼 is equal to 10 if 𝑥= 0 (1) 10 (2) -10 (3) 5 (4) -5

Q72.The number of real solutions of x7 + 5x3 + 3x + 1 = 0 is equal to _____. (1) 0 (2) 1 (3) 3 (4) 5

Q72.Let f, g : R →R be functions defined by , x < 0 f(x) = and {[x]|1 −x| , x ≥0 JEE Main 2022 (28 Jun Shift 2) JEE Main Previous Year Paper ex −x, x < 0 g(x) = { (x −1)2 −1, x ≥0 where [x] denote the greatest integer less than or equal to x. Then, the function fog is discontinuous at exactly (1) one point (2) two points (3) three points (4) four points

Q72.If for p ≠q ≠0 , then function f(x) = 7√p(729+x)−3 is continuous at x = 0 , then 3√729+qx−9 (1) 7pqf(0) −1 = 0 (2) 63qf(0) −p2 = 0 (3) 21qf(0) −p2 = 0 (4) 7pq f(0) −9 = 0

Q72.Let f(x) = 3(x2−2)3+4, x ∈R. Then which of the following statements are true? P : x = 0 is a point of local minima of f Q : x = √2 is a point of inflection of f R : f ′ is increasing for x > √2 (1) Only P and Q (2) Only P and R (3) Only Q and R (4) All P, Q and R π

Q72.The sum of the absolute maximum and absolute minimum values of the function f(x) = tan−1(sin x −cos x) in the interval [0, π] is (1) 0 (2) tan−1( √21 ) −π4 12 (3) cos−1( √31 ) −π4 (4) −π dt, n = 1, 2, 3, … . Then

Q72.If the system of linear equations 2x + y −z = 7 x −3y + 2z = 1 x + 4y + δz = k, where δ, k ∈R has infinitely many solutions, then δ + k is equal to (1) −3 (2) 3 (3) 6 (4) 9 1 ) 4x2−1