Practice Questions

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 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 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 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.Let 𝛼, 𝛽 and 𝛾 be three positive real numbers. Let 𝑓𝑥= 𝛼x5 + 𝛽x3 + 𝛾x, x ∈R and 𝑔: 𝑅→𝑅 be such that 𝑔𝑓𝑥= 𝑥 for all 𝑥∈𝑅. If 𝑎1, 𝑎2, 𝑎3, … , 𝑎𝑛 be in arithmetic progression with mean zero, then the value of 1 𝑛 𝑓𝑔 𝑛∑𝑖= 1 𝑓𝑎𝑖 is equal to (1) 0 (2) 3 (3) 9 (4) 27

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.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

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

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: 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.The value of d 𝑥 at 𝑥= 𝜋 is log𝑒2 dxlogcos𝑥cosec 4 (1) -2√2 (2) 2√2 (3) -4 (4) 4

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 distinct real roots of the equation x7 −7x −2 = 0 is (1) 5 (2) 7 (3) 1 (4) 3

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

Q73.Let f(x) = 2 + |x| −|x −1| + |x + 1|, x ∈R. Consider (S1) : f ′(−32 ) + f ′(−12 ) + f ′( 12 ) + f ′( 32 ) = 2 (S2) : ∫2−2 f(x)dx = 12 Then, (1) both (S1) and (S2) are correct (2) both (S1) and (S2) are wrong (3) only (S1) is correct (4) only (S2) is correct Q74. ∫20 ( 2x2 −3x + [x −12 ])dx, where [t] is the greatest integer function, is equal to (1) 7 (2) 19 6 12 (3) 31 (4) 3 12 2

Q73.Let f : R →R be a differentiable function such that f( π4 ) = √2, f( π2 ) = 0 and f ′( π2 ) = 1 and let π lim g(x) = ∫ x4 (f ′(t) sec t + tan t sec tf(t))dt for x ∈[ π4 , π2 ). Then π x→( 2 )−g(x) is equal to (1) 2 (2) 3 (3) 4 (4) −3

Q73.The domain of the function 2 sin−1( is π cos−1( ) , , ∞) ∞) (1) (−∞, −1√2 ] ∪[ √21 ∪{0} (2) (−∞, −1√2 ] ∪[ √21 ∪( 12 , ∞) ∪{0} (4) R −{−12 , 12 } (3) (−∞, −1√2 )

Q73.The number of bijective function f(1, 3, 5, 7, ⋯, 99) →(2, 4, 6, 8, ⋯, 100) if f(3) > f(5) > f(7) ⋯> f(99) is (1) 50C1 (2) 50C2 (3) 50! (4) 50C3 × 3! 2

Q73.For any real number 𝑥, let 𝑥 denote the largest integer less than or equal to 𝑥. Let 𝑓 be a real-valued function defined on the interval -10, 10 by 𝑥- 𝑥, if 𝑥 is odd 𝑓𝑥= 1 + 𝑥- 𝑥, if 𝑥 is even π2 10 Then, the value of 10 ∫-10 𝑓𝑥 cosπ𝑥𝑑𝑥 is (1) 4 (2) 2 (3) 1 (4) 0

Q73.Let λ* be the largest value of λ for which the function fλ(x) = 4λx3 −36λx2 + 36x + 48 is increasing for all x ∈R. Then fλ*(1) + fλ,*(−1) is equal to: (1) 36 (2) 48 (3) 64 (4) 72 π

Q73.For 𝐼𝑥= ∫sec2𝑥- 2022 if 𝐼𝜋 = 21011, then sin2022𝑥𝑑𝑥, 4 𝜋 𝜋 𝜋 𝜋 (1) 31010𝐼 - 𝐼 = 0 (2) 31010𝐼 - 𝐼 = 0 3 6 6 3 (3) 31011𝐼𝜋 - 𝐼𝜋 = 0 (4) 31011𝐼𝜋 - 𝐼𝜋 = 0 3 6 6 3 1

Q73.Let [t] denote the greatest integer less than or equal to t. Then, the value of the integral ∫10 [−8x2 + 6x −1]dx is equal to (1) −1 (2) −54 (3) √17−13 (4) √17−16 8 8

Q73.Considering only the principal values of the inverse trigonometric functions, the domain of the function 𝑥2 - 4𝑥+ 2 𝑓𝑥= cos-1 is 𝑥2 + 3 1 1 (1) - ∞, (2) - ∞ 4 4, (3) -1 ∞ (4) - ∞, 1 3, 3