Practice Questions

Q71.Let 𝑓: 𝑁→𝑁 be a function such that 𝑓𝑚+ 𝑛= 𝑓𝑚+ 𝑓𝑛 for every 𝑚, 𝑛∈𝑁. If 𝑓6 = 18 then 𝑓2 · 𝑓3 is equal to : (1) 54 (2) 6 (3) 36 (4) 18 JEE Main 2021 (31 Aug Shift 2) JEE Main Previous Year Paper + 𝑥- 1 𝑥- 1 is:

Q71.Let 𝑓: 𝑅→𝑅 be defined as 𝜆𝑥2 - 5𝑥+ 6 𝑥< 2 𝜇5𝑥- 𝑥2 - 6 𝑓𝑥= tan ( 𝑥- 2 ) 𝑒 𝑥- [𝑥] 𝑥> 2 𝜇 𝑥= 2 where 𝑥 is the greatest integer less than or equal to 𝑥. If 𝑓 is continuous at 𝑥= 2, then 𝜆+ 𝜇 is equal to : (1) 𝑒( - 𝑒+ 1 ) (2) 𝑒( 𝑒- 2 ) (3) 1 (4) 2𝑒- 1

Q71.Let N be the set of natural numbers and a relation R on N be defined by R = {(x, y) ∈N × N : x3 −3x2y −xy2 + 3y3 = 0}. Then the relation R is (1) symmetric but neither reflexive nor transitive (2) reflexive but neither symmetric nor transitive (3) reflexive and symmetric, but not transitive (4) an equivalence relation

Q71.If the matrix A = [K0 −12 ] (1) 21 (2) 1 (3) −1 (4) −12

Q71.Out of all the patients in a hospital 89% are found to be suffering from heart ailment and 98% are suffering from lungs infection. If K% of them are suffering from both ailments, then K can not belong to the set: (1) {79, 81, 83, 85} (2) {84, 87, 90, 93} (3) {80, 83, 86, 89} (4) {84, 86, 88, 90} Q72. 1 2 √5 √5 1 0 If A = ⎡ ⎤ , B = i = √−1, and Q = ATBA, then the inverse of the matrix AQ2021AT is −2 1 [ i 1 ], √5 √5 ⎣ ⎦ equal to: (1) [ 10 −20211 ] (2) [ −2021i1 10 ] (3) 1 −2021 (4) 1 0 √5 ⎡ ⎤ [ 2021 i 1 ] 2021 1 √5 ⎣ ⎦

Q71.If A = 0 sin α and det(A2 −12 I) = 0, [sin α 0 ] (1) π (2) π 2 3 (3) π (4) π 4 6

Q71. cosec [2 cot−1(5) + cos−1( 54 )] is equal to: (1) 65 (2) 75 56 56 (3) 65 (4) 56 33 33

Q71.Let [x] denote the greatest integer ≤x, where x ∈R. If the domain of the real valued function f(x) = is (−∞, a) ∪[b, c) ∪[4, ∞), a < b < c, then the value of a + b + c is: √|[x]|−2|[x]|−3 (1) 8 (2) 1 (3) −2 (4) −3

Q71.If the domain of the function f(x) = cos−1 √x2−x+1 is the interval (α, β], then α + β is equal to: √sin−1( 2x−12 ) (1) 3 (2) 2 2 (3) 1 (4) 1 2

Q71.For the four circles M, N, O and P, following four equations are given: Circle M : x2 + y2 = 1 Circle N : x2 + y2 −2x = 0 Circle O : x2 + y2 −2x −2y + 1 = 0 Circle P : x2 + y2 −2y = 0 If the centre of circle M is joined with centre of the circle N, further centre of circle N is joined with centre of the circle O, centre of circle O is joined with the centre of circle P and lastly, centre of circle P is joined with centre of circle M, then these lines form the sides of a (1) Rhombus (2) Square (3) Rectangle (4) Parallelogram

Q71.If sin−1a x = cos−1b x = tan−1c y ; 0 < x < 1, then the value of cos( a+bπc ) is: (1) 1−y2 (2) 1 −y2 1+y2 (3) 1−y2 (4) 1−y2 y√y 2y

Q71.The range of the function 𝑓(𝑥) = + cos 3𝜋 + 𝑥+ cos 𝜋 + 𝑥+ cos 𝜋 - 𝑥- cos 3𝜋 - 𝑥 is : log√53 4 4 4 4 1 (1) √5, √5 (2) [0, 2] (3) (0, √5 ) (4) [ - 2, 2]

Q71.If ∑50r=1 tan−1 2r21 = p, then the value of tan p is : (1) 100 (2) 5051 (3) 50 (4) 101 51 102 JEE Main 2021 (26 Aug Shift 2) JEE Main Previous Year Paper

Q71.Let Sk = ∑kr=1 tan−1( 22r+1+32r+16r ), then k→∞Sk (1) tan−1( 23 ) (2) π2 (3) cot−1( 23 ) (4) tan−1(3)

Q71.Let f : R −{ α6 } →R be defined by f(x) = ( 6x−α5x+3 ). Then the value of α for which (fof)(x) = x, for all x ∈R −{ α6 }, is (1) No such α exists (2) 5 (3) 8 (4) 6

Q71.If the mean and variance of the following data: 6, 10, 7, 13, a, 12, b, 12 are 9 and 374 respectively, then (a −b)2 is equal to: (1) 24 (2) 12 (3) 32 (4) 16

Q71.If y(x) cot−1( √1+sin√1+sin x+√1−sinx−√1−sin xx ), (1) 0 (2) −1 (3) −1 (4) 1 2 2

Q71.Let f : S →S where S = (0, ∞) be a twice differentiable function such that f(x + 1) = xf(x). If g : S →R be defined as g(x) = loge f(x), then the value of |g′′(5) −g′′(1)| is equal to : (1) 205 (2) 197 144 144 (3) 187 (4) 1 144 JEE Main 2021 (16 Mar Shift 2) JEE Main Previous Year Paper

Q72.The number of elements in the set {x ∈R : (|x| −3)|x + 4| = 6} is equal to (1) 3 (2) 2 (3) 4 (4) 1

Q72.Let 𝑓: [0, ∞) →[0, ∞) be defined as 𝑓𝑥= 𝑥𝑦𝑑𝑦 where [𝑥] is the greatest integer less than or equal to 𝑥. ∫0 Which of the following is true? (1) 𝑓 is continuous at every point in [0, ∞) and (2) 𝑓 is both continuous and differentiable except at differentiable except at the integer points. the integer points in [0, ∞) . (3) 𝑓 is continuous everywhere except at the integer (4) 𝑓 is differentiable at every point in [0, ∞) . points in [0, ∞) . 𝜋 𝜋

Q72.If the following system of linear equations 2x + y + z = 5 x −y + z = 3 x + y + a z = b has no solution, then : (1) a = −13 , b ≠73 (2) a ≠13 , b = 73 (3) a ≠−13 , b = 73 (4) a = 13 , b ≠73

Q72.A function f(x) is given by f(x) = 5x+55x , then the sum of the series f( 201 ) + f( 202 ) + f( 203 ) + … + f( 2039 ) is equal to: (1) 19 (2) 49 2 2 (3) 39 (4) 29 2 2

Q72.If the curve y = ax2 + bx + c, x ∈R, passes through the point (1, 2) and the tangent line to this curve at origin is y = x, then the possible values of a, b, c are: (1) a = −1, b = 1, c = 1 (2) a = 1, b = 0, c = 1 (3) a = 1, b = 1, c = 0 (4) a = 12 , b = 12 , c = 1

Q72.Let f be any function defined on R and let it satisfy the condition: |f(x) −f(y)| ≤(x −y)2 , ∀(x, y) ∈R. If f(0) = 1, then : (1) f(x) = 0, ∀x ∈R (2) f(x) can take any value in R (3) f(x) < 0, ∀x ∈R (4) f(x) > 0, ∀x ∈R

Q72.The domain of the function cosec−1 ( 1+xx ) is : (1) [−12 , ∞) −{0} (2) (−1, −12 ] ∪(0, ∞) (3) [−12 , 0) ∪[1, ∞) (4) (−12 , ∞) −{0}