Practice Questions

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

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 domain of f(x) = cos−1(log(x2−3x+2)x2−5x+6 (1) x ∈[ −12 , 1) ∪(2, ∞) −{3} (2) x ∈[ −12 , 1] ∪(2, ∞) −{3} (3) x ∈( −12 , 1) ∪[2, ∞) −{3} (4) x ∈[ −12 , 1) ∪[2, ∞) −{3}

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

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

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.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

Q73.Let f(x) = { −2xx3 −x2+ log2(b2+ 10x −4),−7, x ≤1 Then the set of all values of b, for which f(x) has maximum value at x = 1 , is: (1) (−6, −2) (2) (2, 6) (3) [−6, −2) ∪(2, 6] (4) [−√6, −2) ∪(2, √6] , x ∈(0, 1), then: lim k=1 n2+k22n and f(x) = √1−cos1+cos xx

Q73.Let a function f : R →R be defined as: 0 (5 −|t −3|)dt, x > 4 f(x) = {∫xx2 + bx, x ≤4 where b ∈R. If f is continuous at x = 4, then which of the following statements is NOT true? (1) f is not differentiable at x = 4 (2) f ′(3) + f ′(5) = 354 (3) f is increasing in (−∞, 81 ) ∪(8, ∞) (4) f has a local minima at x = 81 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.Consider a cuboid of sides 2x, 4x and 5x and a closed hemisphere of radius r. If the sum of their surface areas is constant k, then the ratio x : r, for which the sum of their volumes is maximum, is (1) 2 : 5 (2) 19 : 45 (3) 3 : 8 (4) 19 : 15 dx = g(x) + c, g(1) = 0 , then g( 12 ) 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.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.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.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 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 In(x) = ∫x0 (t2+5)n1 (1) 50I6 −9I5 = xI 5′ (2) 50I6 −11I5 = xI 5′ (3) 50I6 −9I5 = I 5′ (4) 50I6 −11I5 = I 5′ x = loge 2 , above the line y = 1 is

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.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.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.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.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.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)