Practice Questions

Q70.Let f(x) = sin−1 x and g(x) = x2−x−2 . If g(2) = lim g(x), then the domain of the function fog is 2x2−x−6 x→2 (1) (−∞, −1] ∪[2, ∞) (2) (−∞, −2] ∪[−32 , ∞) (3) (−∞, −2] ∪[−43 , ∞) (4) (−∞, −2] ∪[−1, ∞) Q71. 2 sin(−πx2 ), if x < −1 ⎧ Let f : R→R be defined as f(x) = ax2 + x + b , if −1 ≤x ≤1 ⎨ ⎩sin(πx), if x > 1 If f(x) is continuous on R, then a + b equals : (1) 1 (2) 3 (3) −3 (4) −1

Q70.The following system of linear equations 2x + 3y + 2z = 9 3x + 2y + 2z = 9 x −y + 4z = 8 (1) has infinitely many solutions (2) has a unique solution (3) has a solution (α, β, γ) satisfying (4) does not have any solution α + β2 + γ 3 = 12

Q70.Let sinsin BA = sin(C−B)sin(A−C) , where A, B, C are angles of a triangle ABC. If the lengths of the sides opposite these angles are a, b, c respectively, then (1) b2, c2, a2 are in A.P. (2) c2, a2, b2 are in A.P. (3) b2 −a2 = a2 + c2 (4) a2, b2, c2 are in A.P. satisfies A(A3 + 3I) = 2I, then the value of K is

Q70.If 𝛼+ 𝛽+ 𝛾= 2𝜋, then the system of equations 𝑥+ cos𝛾𝑦+ cos𝛽𝑧= 0 cos𝛾𝑥+ 𝑦+ cos𝛼𝑧= 0 cos𝛽𝑥+ cos𝛼𝑦+ 𝑧= 0 has : (1) infinitely many solutions (2) a unique solution (3) no solution (4) exactly two solutions

Q70.The number of real roots of the equation tan−1 √x(x + 1) + sin−1 √x2 + x + 1 = π4 is: (1) 1 (2) 2 (3) 4 (4) 0

Q70. (a + 1)(a + 2) a + 2 1 The value of (a + 2)(a + 3) a + 3 1 is (a + 3)(a + 4) a + 4 1 (1) 0 (2) (a + 2)(a + 3)(a + 4) (3) −2 (4) (a + 1)(a + 2)(a + 3)

Q70.Two fair dice are thrown. The numbers on them are taken as λ and μ, and a system of linear equations x + y + z = 5 x + 2y + 3z = μ x + 3y + λz = 1 is constructed. If p is the probability that the system has a unique solution and q is the probability that the system has no solution, then: (1) p = 16 and q = 365 (2) p = 65 and q = 361 (3) p = 16 and q = 361 (4) p = 65 and q = 365

Q70.Let 𝑓: 𝑅→𝑅 be defined as 𝑓𝑥= 2 𝑥- 1 and 𝑔: 𝑅- 1 →𝑅. be defined as 𝑔𝑥= 𝑥- 𝑥- 1. function 𝑓𝑔𝑥 is: (1) neither one-one nor onto (2) one-one but not onto (3) onto but not one-one (4) both one-one and onto

Q70.For which of the following curves, the line x + √3y = 2√3 is the tangent at the point ( 3√32 , 12 )? (1) 2x2 −18y2 = 9 (2) y2 = 1 x 6√3 (3) x2 + 9y2 = 9 (4) x2 + y2 = 7

Q71.Let f : R →R be defined as ⎧−55x, if x < −5 f(x) = −120x, if −5 ≤x ≤4 ⎨2x3 −3x2 ⎩2x3 −3x2 −36x −336, if x > 4 Let A = {x ∈R : f is increasing}. Then A is equal to: (1) (−5, ∞) (2) (−5, −4) ∪(4, ∞) (3) (−∞, −5) ∪(−4, ∞) (4) (−∞, −5) ∪(4, ∞)

Q71.Define a relation R over a class of n × n real matrices A and B as " ARB iff there exists a non-singular matrix P such that PAP −1 = B". Then which of the following is true ? (1) R is symmetric, transitive but not reflexive (2) R is reflexive, symmetric but not transitive (3) R is an equivalence relation (4) R is reflexive, transitive but not symmetric

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

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 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.A man is observing, from the top of a tower, a boat speeding towards the tower from a certain point A , with uniform speed. At that point, angle of depression of the boat with the man's eye is 30° (Ignore man's height). After sailing for 20 seconds, towards the base of the tower (which is at the level of water), the boat has reached a point B, where the angle of depression is 45°. Then the time taken (in seconds) by the boat from B to reach the base of the tower is : JEE Main 2021 (25 Feb Shift 1) JEE Main Previous Year Paper (1) 10 (2) 10(√3 −1) + (3) 10√3 (4) 10(√3 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 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.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.The first of the two samples in a group has 100 items with mean 15 and standard deviation 3. If the whole group has 250 items with mean 15. 6 and standard deviation √13. 44, then the standard deviation of the second sample is: (1) 8 (2) 6 (3) 4 (4) 5 1 0 50 then P is: 1

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.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.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 𝑓: 𝑅→𝑅 is a function defined by 𝑓𝑥= 𝑥- 1cos2𝑥- 1 𝜋, where · denotes the greatest integer function, then 𝑓 2 is: (1) discontinuous only at 𝑥= 1 (2) discontinuous at all integral values of 𝑥 except at 𝑥= 1 (3) continuous only at 𝑥= 1 (4) continuous for every real 𝑥