Practice Questions

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 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.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.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.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

Q72.The value of d 𝑥 at 𝑥= 𝜋 is log𝑒2 dxlogcos𝑥cosec 4 (1) -2√2 (2) 2√2 (3) -4 (4) 4

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

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 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.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

Q73.If m and n respectively are the number of local maximum and local minimum points of the function dt, then the ordered pair (m, n) is equal to f(x) = ∫x20 t2−5t+42+et (1) (2, 3) (2) (3, 2) (3) (2, 2) (4) (3, 4) is equal to

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 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.The integral ∫ 0 2 3+2 sin1x+cos x dx is equal to: (1) tan−1(2) (2) tan−1(2) −π4 (3) 1 2 tan−1(2) −π8 (4) 21 α > 0, then f(e3) + f(e−3) is equal to

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.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.For the function f(x) = 4 loge(x −1) −2x2 + 4x + 5, x > 1 , which one of the following is NOT correct? JEE Main 2022 (24 Jun Shift 1) JEE Main Previous Year Paper (1) f(x) is increasing in (1, 2) and decreasing in (2) f(x) = −1 has exactly two solutions (2, ∞) (3) f ′(e) −f ′′(2) < 0 (4) f(x) = 0 has a root in the interval (e, e + 1)

Q73.The sum of the absolute minimum and the absolute maximum values of the function f(x) = 3x −x2 + 2 −x in the interval [−1, 2] is (1) √17+3 (2) √17+5 2 2 (3) 5 (4) 9−√17 2

Q73.Let f : R →R be a function defined by f(x) = (x −3)n1(x −5)n2, n1, n2 ∈N . The, which of the following is NOT true? (1) For n1 = 3, n2 = 4 , there exists α ∈(3, 5) (2) For n1 = 4, n2 = 3, there exists α ∈(3, 5) where f attains local maxima. where f attains local maxima. (3) For n1 = 3, n2 = 5 , there exists α ∈(3, 5) (4) For n1 = 4, n2 = 6, there exists α ∈(3, 5) where f attains local maxima. where f attains local maxima.

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.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

Q74.The area enclosed by the curves y = loge(x + e2), x = loge( 2y ) and (1) 2 + e −loge 2 (2) 1 + e −loge 2 (3) e −loge 2 (4) 1 + loge 2 dy +

Q74.If a = n→∞∑n (1) 2√2f( a2 ) = f ′( a2 ) (2) f( a2 )f ′( a2 ) = √2 (3) √2f( a2 ) = f ′( a2 ) (4) f( a2 ) = √2f ′( a2 )

Q74.If 𝑡 denotes the greatest integer ≤t, then the value of ∫0 2𝑥- 3𝑥2 - 5𝑥+ 2 + 1𝑑𝑥 is JEE Main 2022 (29 Jul Shift 2) JEE Main Previous Year Paper (1) √37 + √13 - 4 (2) √37 - √13 - 4 6 6 (3) -√37 - √13 + 4 (4) -√37 + √13 + 4 6 6