Practice Questions

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.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.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.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.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

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.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 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 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.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.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.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.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 )

Q72.Let f, g : N −{1} →N be functions defined by f(a) = α, where α is the maximum of the powers of those primes p such that pα divides a, and g(a) = a + 1, for all a ∈N −{1}. Then, the function f + g 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

Q72.Let 𝑓: 𝑅→𝑅 be defined as 𝑓𝑥= 𝑥3 + 𝑥- 5. If 𝑔𝑥 is a function such that 𝑓𝑔𝑥= 𝑥, ∀𝑥∈𝑅, then 𝑔'63 is equal to ______ (1) 49 (2) 1 49 43 3 (3) (4) 49 49

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