Practice Questions

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.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 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.For x > 0 , if f(x) = ∫x1 (1+t)loge t (1) 0 (2) 21 (3) −1 (4) 1 x ∈R. Then f(x) equals :

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.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.Consider the integral I = ∫100 [x]e[x]ex−1 value of I is equal to : (1) 9(e −1) (2) 45(e + 1) (3) 45(e −1) (4) 9(e + 1)

Q73.Let M and m respectively be the maximum and minimum values of the function f(x) = tan−1(sin x + cos x) in [0, π2 ]. Then the value of tan(M −m) is equal to: (1) 2 −√3 (2) 3 −2√2 (3) 3 + 2√2 (4) 2 + √3

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.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

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.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

Q74.The value of lim n1 ∑nj=1 (2j−1)+4n(2j−1)+8n is equal to: n→∞ (1) 5 + loge( 32 ) (2) 2 −loge( 23 ) (3) 3 + 2 loge( 23 ) (4) 1 + 2 loge( 32 ) π dx is equal to : cos x)(sin4 x+cos4 x)

Q74.The sum of possible values of x for tan−1(x + 1) + cot−1( x−11 ) = tan−1( 318 ) is: (1) −324 (2) −314 (3) −304 (4) −334

Q74.The local maximum value of the function, f(x) = ( x2 )x2 , x > 0, e (1) 1 (2) ( √e4 ) 4 e (3) (e) 2e (4) (2√e) 1 π x x )dx is :

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.The area of the region: R = {(x, y) : 5x2 ≤y ≤2x2 + 9} is (1) 9√3 square units (2) 12√3 square units (3) 11√3 square units (4) 6√3 square units

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.The number of real roots of the equation 𝑒6𝑥- 𝑒4𝑥- 2𝑒3𝑥- 12𝑒2𝑥+ 𝑒𝑥+ 1 = 0 is: (1) 2 (2) 4 (3) 6 (4) 1 𝑑𝑥 is

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.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 f : R →R be a function defined as , if x < 0 ⎧ sin(a+1)x+sin2x 2x f(x) = ⎨ b , if x = 0 √x+bx3−√x , if x > 0 ⎩ bx5/2 If f is continuous at x = 0 , then the value of a + b is equal to : (1) −52 (2) −2 (3) −3 (4) −32

Q74.Let P(x) = x2 + bx + c be a quadratic polynomial with real coefficients such that ∫10 P(x)dx = 1 and P(x) leaves remainder 5 when it is divided by (x −2) Then the value of 9(b + c) is equal to: (1) 9 (2) 15 (3) 7 (4) 11

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

Q74.The range of a ∈R for which the function f(x) = (4a −3)(x + loge 5) + 2(a −7) cot( x2 ) sin2( x2 ), x ≠2nπ, n ∈N , has critical points, is : (1) (−3, 1) (2) [−43 , 2] (3) [1, ∞) (4) (−∞, −1] JEE Main 2021 (16 Mar Shift 1) JEE Main Previous Year Paper