Practice Questions

Q72.The sum of all the local minimum values of the twice differentiable function f : R →R defined by ′′(2) x + f ′′(1) is: f(x) = x3 −3x2 −3f 2 (1) −22 (2) 5 (3) −27 (4) 0

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.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.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 →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.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.If the tangent to the curve 𝑦= 𝑥3 at the point 𝑃𝑡, 𝑡3 meets the curve again at 𝑄, then the ordinate of the point which divides 𝑃𝑄 internally in the ratio 1: 2 is: (1) 0 (2) -2𝑡3 (3) -𝑡3 (4) 2𝑡3

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.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 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.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 [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.If [x] denotes the greatest integer less than or equal to x, then the value of the integral ∫π/2−π/2[[x] −sin x]dx is equal to: (1) −π (2) π (3) 0 (4) 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.If cot−1(α) = cot−1 2 + cot−1 8 + cot−1 18 + cot−1 32 + … . upto 100 terms, then α is: JEE Main 2021 (17 Mar Shift 1) JEE Main Previous Year Paper (1) 1. 01 (2) 1. 00 (3) 1. 02 (4) 1. 03

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.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 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.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.If Rolle's theorem holds for the function f(x) = x3 −ax2 + bx −4, x ∈[1, 2] with f ′( 43 ) = 0 , then ordered pair (a, b) is equal to : (1) (−5, −8) (2) (−5, 8) (3) (5, 8) (4) (5, −8) dθ is (where c is a constant of integration)

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)

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.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.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