Practice Questions

Q70.Choose the correct statement about two circles whose equations are given below: x2 + y2 −10x −10y + 41 = 0 x2 + y2 −22x −10y + 137 = 0 (1) circles have same centre (2) circles have no meeting point (3) circles have only one meeting point (4) circles have two meeting points

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

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

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 x, y, z are in arithmetic progression with common difference d, x ≠3d, and the determinant of the matrix 3 4√2 x ⎡ ⎤ is zero, then the value of k2 is 4 5√2 y 5 k z ⎣ ⎦ (1) 72 (2) 12 (3) 36 (4) 6

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 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.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.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 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.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.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.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.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. 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. a1 a2 a3 If ar = cos 2rπ9 + i sin 2rπ9 , r = 1, 2, 3, … , i = √−1, then the determinant a4 a5 a6 is equal to : a7 a8 a9 (1) a9 (2) a1a9 −a3a7 (3) a5 (4) a2a6 −a4a8

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.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.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

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.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 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.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