Practice Questions

Q69.Let R1 = {(a, b) ∈N × N : |a −b| ≤13} and R2 = {(a, b) ∈N × N : |a −b| ≠13} Then on N : (1) Both R1 and R2 are equivalence relations (2) Neither R1 nor R2 is an equivalence relation (3) R1 is an equivalence relation but R2 is not (4) R2 is an equivalence relation but R1 is not

Q69.For 𝛼∈𝑁, consider a relation 𝑅 on 𝑁 given by 𝑅= {𝑥, 𝑦: 3𝑥+ 𝛼𝑦 is a multiple of 7}. The relation 𝑅 is an equivalence relation if and only if (1) 𝛼= 14 (2) 𝛼 is a multiple of 4 (3) 4is the remainder when 𝛼 is divided by 10 (4) 4 is the remainder when 𝛼 is divided by 7 Q70. 0 1 0 Let the matrix 𝐴= 1 0 0 and the matrix 𝐵0 = 𝐴49 + 2𝐴98. If 𝐵𝑛= Adj𝐵𝑛- 1 for all 𝑛≥1, then det 𝐵4 is 0 0 1 equal to (1) 328 (2) 330 (3) 332 (4) 336

Q70.The negation of the Boolean expression ~𝑞∧𝑝⇒~𝑝∨𝑞 is logically equivalent to (1) 𝑝⇒𝑞 (2) 𝑞⇒𝑝 (3) ~𝑝⇒𝑞 (4) ~𝑞⇒𝑝

Q70.If the system of equations 𝑥+ 𝑦+ 𝑧= 6 2𝑥+ 5𝑦+ 𝛼𝑧= 𝛽 𝑥+ 2𝑦+ 3𝑧= 14 has infinitely many solutions, then 𝛼+ 𝛽 is equal to (1) 8 (2) 36 (3) 44 (4) 48

Q70.Consider the following statements: P : Ramu is intelligent. Q : Ramu is rich. R : Ramu is not honest. The negation of the statement "Ramu is intelligent and honest if and only if Ramu is not rich" can be expressed as: (1) ((P ∧(~R)) ∧Q) ∧((~Q) ∧((~P) ∨R)) (2) ((P ∧R) ∧Q) ∨((~Q) ∧((~P) ∨(~R))) (3) ((P ∧R) ∧Q) ∧((~Q) ∧((~P) ∨(~R))) (4) ((P ∧(~R)) ∧Q) ∨((~Q) ∧((~P) ∧R))

Q70.Let R1 and R2 be two relations defined on R by aR1b ⇔ab ≥0 and a R2b ⇔a ≥b, then JEE Main 2022 (27 Jul Shift 1) JEE Main Previous Year Paper (1) R1 is an equivalence relation but not R2 (2) R2 is an equivalence relation but not R1 (3) both R1 and R2 are equivalence relations (4) neither R1 nor R2 is an equivalence relation

Q70.Let A and B be two 3 × 3 matrices such that AB = I and |A| = 18 then |adj(Badj(2A))| is equal to (1) 128 (2) 32 (3) 64 (4) 102

Q70.The number of values of α for which the system of equations x + y + z = α αx + 2αy + 3z = −1 x + 3αy + 5z = 4 is inconsistent, is (1) 0 (2) 1 (3) 2 (4) 3

Q70.The ordered pair (a, b), for which the system of linear equations 3x −2y + z = b 5x −8y + 9z = 3 2x + y + az = −1 has no solution, is (1) (3, 13 ) (2) (−3, 31 ) (3) (−3, −13 ) (4) (3, −13 )

Q70.Let f : R →R be a continuous function such that f(3x) −f(x) = x. If f(8) = 7 , then f(14) is equal to: (1) 4 (2) 10 (3) 11 (4) 16

Q70.Let A and B be two 3 × 3 non-zero real matrices such that AB is a zero matrix. Then (1) The system of linear equations AX = 0 has a (2) The system of linear equations AX = 0 has unique solution infinitely many solutions (3) B is an invertible matrix (4) adj(A) is an invertible matrix

Q70.The number of values of a ∈N such that the variance of 3, 7, 12, a, 43 −a is a natural number is: (1) 0 (2) 2 (3) 5 (4) infinite

Q70.The value of n→∞6lim tan{∑nr=1 tan−1( r2+3r+31 )} is equal to (1) 1 (2) 2 (3) 3 (4) 6

Q70.If the inverse trigonometric functions take principal values, then cos−1( 103 cos(tan−1( 43 )) + 25 sin(tan−1( 43 ))) is equal to (1) 0 (2) π4 (3) π (4) π 3 6

Q70.The probability that a randomly chosen 2 × 2 matrix with all the entries from the set of first 10 primes, is singular, is equal to (1) 133 (2) 19 104 103 (3) 18 (4) 271 103 104

Q70.The total number of functions, 𝑓: 1, 2, 3, 4 →1, 2, 3, 4, 5, 6 such that 𝑓1 + 𝑓2 = 𝑓3, is equal to (1) 60 (2) 90 (3) 108 (4) 126

Q70.Let A = (α4 −2β ) (1) −18 (2) 18 (3) −50 (4) 50 1 [t] is the greatest

Q70.If cos−1( 2y ) = loge ( x5 ) 5, |y| < 2, then (1) x2y′′ + xy′ −25y = 0 (2) x2y′′ −xy′ −25y = 0 (3) x2y′′ −xy′ + 25y = 0 (4) x2y′′ + xy′ + 25y = 0

Q71.The set of all values of k for which (tan−1 x)3 + (cot−1 x)3 = kπ3, x ∈R, is the interval (1) [ 321 , 87 ) (2) ( 241 , 1613 ) (3) [ 481 , 1613 ] (4) [ 321 , 89 ) x2−9 ) is

Q71.If the absolute maximum value of the function 𝑓𝑥= x2 - 2x + 7e4x3 - 12x2 - 180x + 31in the interval -3, 0 is 𝑓𝛼, then (1) 𝛼= 0 (2) 𝛼= - 3 (3) 𝛼∈-1, 0 (4) 𝛼∈-3, - 1

Q71.If y = tan−1(sec x3 −tan x3), π2 < x3 < 3π2 , then (1) xy′′ + 2y′ = 0 (2) x2y′′ −6y + 3π2 = 0 (3) x2y′′ −6y + 3π = 0 (4) xy′′ −4y′ = 0

Q71.The system of equations -𝑘𝑥+ 3𝑦- 14𝑧= 25 -15𝑥+ 4𝑦- 𝑘𝑧= 3 -4𝑥+ 𝑦+ 3𝑧= 4 Question: is consistent for all 𝑘 in the set (1) 𝑅 (2) 𝑅- -11, 13 (3) 𝑅- -13 (4) 𝑅- -11, 11 - 1 4

Q71.The domain of the function 𝑓𝑥= sin-1 𝑥2 - 3𝑥+ 2 is 𝑥2 + 2𝑥+ 7 (1) [1, ∞) (2) ( - 1, 2] (3) [ - 1, ∞) (4) ( - ∞, 2]

Q71.If the mean deviation about median for the number 3, 5, 7, 2k, 12, 16, 21, 24 arranged in the ascending order, is 6 then the median is (1) 11. 5 (2) 10. 5 (3) 12 (4) 11

Q71.Let A = (−21 −52 ). Let α, β ∈R be such that αA2 + βA = 2I . Then α + β is equal to (1) −10 (2) −6 (3) 6 (4) 10