Practice Questions

Q73.Consider the function f : R →R defined by f(x) = { (2 −sin(0, x1 )) x , xx =≠00 (1) monotonic on (−∞, 0) ∪(0, ∞) (2) not monotonic on (−∞, 0) and (0, ∞) (3) monotonic on (0, ∞) only (4) monotonic on (−∞, 0) only

Q73.Let f : R →R be defined as f(x) = { −43 x3 +3xex2x2 + 3x,, xx >≤00 . Then f is increasing function in the interval (1) (−12 , 2) (2) (0, 2) (3) (−1, 23 ) (4) (−3, −1) , α ∈R where [x] is the greatest integer less than or equal to x, then the value of

Q73.Let f : R −{3} →R −{1} be defined by f(x) = x−3x−2 . Let g : R →R be given as g(x) = 2x −3 . Then, the sum of all the values of x for which f −1(x) + g−1(x) = 132 is equal to (1) 7 (2) 2 (3) 5 (4) 3

Q73.Let the functions f : R →R and g : R →R be defined as : + 2, x < 0 x < 1 f(x) = and g(x) = {xx2, x ≥0 {x3,3x −2, x ≥1 Then, the number of points in R where (fog)(x) is NOT differentiable is equal to : (1) 3 (2) 1 (3) 0 (4) 2

Q73.If [x] be the greatest integer less than or equal to x, then 100∑ [ (−1)nn2 ] n=8 (1) 0 (2) 4 (3) −2 (4) 2

Q73.Let f : R →R be defined as f(x + y) + f(x −y) = 2f(x)f(y), f( 21 ) = −1. Then the value of ∑20k=1 sin(k) sin(k+f(k))1 is equal to : (1) cosec2 (21) cos(20) cos(2) (2) sec2(1) sec(21) cos(20) (3) cosec2 (1) cosec (21) sin(20) (4) sec2(21) sin(20) sin(2) . Then which of

Q73.An angle of intersection of the curves, 𝑥2 + 𝑦2 = 1 and 𝑥2 + 𝑦2 = 𝑎𝑏, 𝑎> 𝑏, is : 𝑎2 𝑏2 (1) tan-12√𝑎𝑏 (2) tan-1𝑎+ 𝑏 √𝑎𝑏 (3) tan-1𝑎- 𝑏 (4) tan-1 𝑎- 𝑏 √𝑎𝑏 2√𝑎𝑏

Q73.A wire of length 20 m is to be cut into two pieces. One of the pieces is to be made into a square and the other into a regular hexagon. Then the length of the side (in meters) of the hexagon, so that the combined area of the square and the hexagon is minimum, is (1) 10 (2) 5 2+3√3 3+√3 (3) 10 (4) 5 3+2√3 2+√3 + … + n2

Q73.Let 𝑓𝑥= 3sin4𝑥+ 10sin3𝑥+ 6sin2𝑥- 3, 𝑥∈- 6, 2. Then, 𝑓 is : (1) increasing in -𝜋 𝜋 (2) decreasing in 0, 𝜋 6, 2 2 𝜋 𝜋 (3) increasing in - 6, 0 (4) decreasing in - 6, 0

Q73.Let x denote the total number of one-one functions from a set A with 3 elements to a set B with 5 elements and y denote the total number of one-one functions from the set A to the set A × B. Then : JEE Main 2021 (25 Feb Shift 2) JEE Main Previous Year Paper (1) y = 273x (2) 2y = 273x (3) 2y = 91x (4) y = 91x

Q73.Let θ ∈(0, π2 ). If the system of linear equations (1 + cos2 θ)x + sin2 θy + 4 sin 3θz = 0 cos2 θx + (1 + sin2 θ)y + 4 sin 3θz = 0 cos2 θx + sin2 θy + (1 + 4 sin 3θ)z = 0 has a non-trivial solution, then the value of θ is: (1) 4π (2) 5π 9 18 (3) 7π (4) π 18 18 = tan−1 0 < x < 1. Then: x

Q73.Let [t] denote the greatest integer less than or equal to t. Let f(x) = x −[x], g(x) = 1 −x + [x], and h(x) = min{f(x), g(x)}, x ∈[−2, 2]. Then h is : (1) continuous in [−2, 2] but not differentiable at (2) Continous in [−2, 2] but not differentiable at more than four points in (−2, 2) exactly three poionts in (−2, 2) (3) not continuous at exactly four points in [−2, 2] (4) not continuous at exactly three points in [−2, 2] is

Q73.The value of the integral, ∫31 [x2 −2x −2]dx, where [x] denotes the greatest integer less than or equal to x, is (1) −4 (2) −5 (3) −√2 −√3 + 1 (4) −√2 −√3 −1

Q73.The maximum slope of the curve y = 21 x4 −5x3 + 18x2 −19x occurs at the point (1) (3, 212 ) (2) (2, 2) (3) (2, 9) (4) (0, 0)

Q73.The function f(x) = x2 −2x −3 ⋅e9x2−12x+4 is not differentiable at exactly : (1) Four points (2) Two points (3) three points (4) one point 1 1+ xaQ74. , x < 0 ⎧ x loge( 1−xb ) If the function f(x) = k , x = 0 ⎨ cos2 x−sin2 x−1 , x > 0 ⎩ √x2+1−1 is continuous at x = 0, then a1 + 1b + k4 is equal to : (1) 4 (2) 5 (3) −4 (4) −5

Q74.The value of the integral ∫10 (1+x)(1+3x)(3+x)√xdx is: (1) π 4 (1 −√32 ) (2) π8 (1 −√36 ) (3) π 8 (1 −√32 ) (4) π4 (1 −√36 )

Q74.Let f(x) = ∫x0 etf(t)dt + ex be a differentiable function for all (1) e(ex−1) (2) eex −1 (3) 2eex −1 (4) 2e(ex−1) −1

Q74.Let α, β, γ be the real roots of the equation, x3 + ax2 + bx + c = 0, ( a, b, c ∈R and a, b ≠0). If the system of equations (in, u, v, w) given by αu + βv + γw = 0, βu + γv + αw = 0, γu + αv + βw = 0 has non-trivial solution, then the value of a2 is b (1) 5 (2) 3 (3) 1 (4) 0

Q74.Consider function f : A →B and g : B →C(A, B, C ⊆R) such that (gof)−1 exists, then: (1) f and g both are one-one (2) f and g both are onto (3) f is one-one and g is onto (4) f is onto and g is one-one JEE Main 2021 (25 Jul Shift 2) JEE Main Previous Year Paper ∫x0 (5 + |1 −t|)dt, , then

Q74.Let f : [0, ∞) →[0, 3] be a function defined by f(x) = {max{sin2 + cos x,t :x0>≤tπ ≤π}, x ∈[0, π] the following is true ? (1) f is continuous everywhere but not differentiable (2) f is differentiable everywhere in (0, ∞) exactly at one point in (0, ∞) (3) f is not continuous exactly at two points in (4) f is continuous everywhere but not differentiable (0, ∞) exactly at two points in (0, ∞)

Q74.Let 1 / 2 𝑥𝑛 ∀𝑛> 𝑚 and 𝑛, 𝑚∈𝑁. Consider a matrix 𝐴= where 𝐽𝑛, 𝑚= ∫0 𝑥𝑚- 1𝑑𝑥, 𝑎𝑖𝑗3 × 3 J6 + 𝑖, 3 - J𝑖+ 3, 3 , 𝑖≤𝑗 a𝑖𝑗= Then adj A-1 is : 0 , 𝑖> 𝑗. (1) (15 ) 2 × 234 (2) (15 ) 2 × 242 (3) (105 ) 2 × 236 (4) (105 ) 2 × 238

Q74.Let a be a real number such that the function f(x) = ax2 + 6x −15, x ∈R is increasing in (−∞, 43 ) and decreasing in ( 34 , ∞) . Then the function g(x) = ax2 −6x + 15, x ∈R has a (1) local maximum at x = −34 (2) local minimum at x = −34 (3) local maximum at x = 34 (4) local minimum at x = 34

Q74.If f : R →R is given by f(x) = x + 1, then the value of lim n1 [f(0) + f( n5 ) + f( 10n ) + … . . +f( 5(n−1)n )] is: n→∞ (1) 3 (2) 5 2 2 (3) 1 (4) 7 2 2 + √x2 + x ∈R. Then which one of the

Q74.If Un = (1 + n2 2 n −4 n2 1 )(1 22 ) (1 n2 ) , then n→∞(Un)lim n2 is equal to (1) 16e2 (2) 4e (3) e24 (4) 16e2 dx is equal to Q75. ∫166 loge x2+loge(x2−44x+484)loge x2 (1) 5 (2) 10 (3) 8 (4) 6

Q74.If ∫cos𝑥- sin𝑥 𝑑𝑥= 𝑎sin-1sin𝑥+ cos𝑥 + 𝑐, where 𝑐 is a constant of integration, then the ordered pair 𝑎, 𝑏 is √8 - sin2𝑥 𝑏 equal to: (1) 1, - 3 (2) 3, 1 (3) -1, 3 (4) 1, 3 Q75. ∫0𝑥2 sin√𝑡𝑑𝑡 lim is equal to: 𝑥→0 𝑥3 JEE Main 2021 (24 Feb Shift 1) JEE Main Previous Year Paper 2 (1) 0 (2) 3 (3) 3 (4) 1 2 15