Practice Questions

Q69.Which of the following matrices can NOT be obtained from the matrix -1 2 by a single elementary row 1 -1 operation? (1) 0 1 (2) 1 -1 1 -1 -1 2 (3) -1 2 (4) -1 2 -2 7 -1 3

Q69.Let a set A = A1 ∪A2 ∪… ∪Ak , where Ai ∩Aj = ϕ for i ≠j; 1 ≤i, j ≤k. Define the relation R from A to A by R ={ (x, y) : y ∈Ai if and only if x ∈Ai, 1 ≤i ≤k}. Then, R is: (1) reflexive, symmetric but not transitive (2) reflexive, transitive but not symmetric (3) reflexive but not symmetric and transitive (4) an equivalence relation JEE Main 2022 (29 Jun Shift 1) JEE Main Previous Year Paper

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.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.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.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.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

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

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.From the base of a pole of height 20 meter, the angle of elevation of the top of a tower is 60°. The pole subtends an angle 30° at the top of the tower. Then the height of the tower is: (1) 15√3 (2) 20√3 (3) 20 + 10√3 (4) 30 Q72. 2 −1 Let A = − . . . − 5C5(adj A)5 , then the sum of . If B = I −5C1(adj A) + 5C2(adj A)2 (0 2 ) all elements of the matrix B is: (1) −5 (2) −6 (3) −7 (4) −8

Q71.The probability that a randomly chosen one-one function from the set {a, b, c, d} to the set {1, 2, 3, 4, 5} satisfied f(a) + 2 f(b) −f(c) = f(d) is (1) 1 (2) 1 24 40 (3) 1 (4) 1 30 20

Q71.The number of points, where the function f : R →R, f(x) = |x −1| cos|x −2| sin|x −1| + (x −3) x2 −5x + 4 , is NOT differentiable, is (1) 1 (2) 2 (3) 3 (4) 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.Let A = [aij] be a square matrix of order 3 such that aij = 2j−i , for all i, j = 1, 2, 3 . Then, the matrix A2 + A3 + … + A10 is equal to (1) ( 310−12 )A (2) ( 310+12 )A (3) ( 310+32 )A (4) ( 310−32 )A

Q71.Let f : R →R be defined as f(x) = x −1 and g : R →{1, −1} →R be defined as g(x) = x2 . Then the x2−1 function fog is: (1) One-one but not onto (2) onto but not one-one (3) Both one-one and onto (4) Neither one-one nor onto