Practice Questions

Q69.Let x × y = x2 + y3 and (x × 1) × 1 = x × (1 × 1). Then a value of 2 sin−1( x4+x2−2x4+x2+2 ) is (1) π (2) π 4 3 (3) π (4) π 6 JEE Main 2022 (24 Jun Shift 2) JEE Main Previous Year Paper Q70. , x ∈(−2, −1) ⎧ sin(x−[x])x−[x] Let f(x) = max(2x, 3[|x|]), |x| < 1 ⎨ ⎩1, otherwise where [t] denotes greatest integer ≤t. If m is the number of points where f is not continuous and n is the number of points where f is not differentiable, the ordered pair (m, n) is: (1) (3, 3) (2) (2, 4) (3) (2, 3) (4) (3, 4)

Q69.The function f : R →R defined by f(x) = lim cos(2πx)−x2n sin(x−1) is continuous for all x in n→∞ 1+x2n+1−x2n (1) R −{−1} (2) R −{−1, 1} (3) R −{1} (4) R −{0} Q70. π 1+( dxdy ) 2 π Let x(t) = 2√2 cos t√sin 2t and y(t) = 2√2 sin t√sin 2t, t ∈(0, 2 ). Then d2y at t = 4 is equal to dx2 (1) −2√2 (2) 2 3 3 (3) 1 (4) −2 3 3

Q69.Let 𝐴= 0 -2 . If 𝑀 and 𝑁 are two matrices given by 𝑀= ∑𝑘=10 1 𝐴2𝑘 and 𝑁= ∑𝑘=10 1 𝐴2𝑘- 1 then 𝑀𝑁2 2 0 is (1) a non-identity symmetric matrix (2) a skew-symmetric matrix (3) neither symmetric nor skew-symmetric matrix (4) an identity matrix JEE Main 2022 (25 Jun Shift 1) JEE Main Previous Year Paper Q70. 1 1 1 -1 0 1 Let 𝐴 be a 3 × 3 real matrix such that 𝐴 1 = 1 ; 𝐴 0 = 0 and 𝐴 0 = 1 . If 𝑋= 𝑥1 𝑥2 𝑥3𝑇 0 0 1 1 1 2 4 and 𝐼 is an identity matrix of order 3, then the system 𝐴- 2𝐼𝑋= 1 has 1 (1) no solution (2) infinitely many solutions (3) unique solution (4) exactly two solutions

Q69.Let a vertical tower AB of height 2h stands on a horizontal ground. Let from a point P on the ground a man can see upto height h of the tower with an angle of elevation 2α. When from P , he moves a distance d in the −→ direction of AP , he can see the top B of the tower with an angle of elevation α. If d = √7h , then tan α is equal to (1) √5 −2 (2) √3 −1 (3) √7 −2 (4) √7 −√3

Q69. sin−1(sin 2π3 ) + cos−1(cos 7π6 ) + tan−1(tan 3π4 ) is equal to JEE Main 2022 (27 Jun Shift 1) JEE Main Previous Year Paper (1) 11π (2) 17π 12 12 (3) 31π 12 (4) −3π4

Q69.Let A be a 3 × 3 invertible matrix. If |adj(24A)| =adj (3 adj (2A))|, then |A|2 is equal to (1) 26 (2) 212 (3) 512 (4) 66

Q69.Negation of the Boolean expression 𝑝↔𝑞→𝑝 is (1) ~𝑝∧𝑞 (2) 𝑝∧~𝑞 (3) ~𝑝∨~q (4) ~𝑝∧~𝑞 Q70. 1 92 -102 112 Let 𝐴= 1 and 𝐵= 122 132 -142 , then the value of 𝐴'𝐵𝐴 is; 1 -152 162 172 (1) 1224 (2) 1042 (3) 540 (4) 539

Q69.If the system of linear equations 2x + 3y −z = −2 x + y + z = 4 x −y + |λ|z = 4λ −4 where λ ∈R, has no solution, then (1) λ = 7 (2) λ = −7 (3) λ = 8 (4) λ2 = 1 Q70. ⎡ 2n, n = 2, 4, 6, 8, … . . Let a function f : N →N be defined by f(n) = n −1, n = 3, 7, 11, 15, … . . n+1 ⎣ 2 , n = 1, 5, 9, 13, … . . then, f is (1) One-one and onto (2) One-one but not onto (3) Onto but not one-one (4) Neither one-one nor onto JEE Main 2022 (28 Jun Shift 1) JEE Main Previous Year Paper Q71. ⎡[ex], x < 0 aex + [x −1], 0 ≤x < 1 Let f : R →R be defined as f(x) = b + [sin(πx)], 1 ≤x < 2 ⎣[e−x] −c, x ≥2 where a, b, c ∈R and [t] denotes greatest integer less than or equal to t. Then, which of the following statements is true? (1) There exists a, b, c ∈R such that f is continuous (2) If f is discontinuous at exactly one point, then of R. a + b + c = 1 (3) If f is discontinuous at exactly one point, then (4) f is discontinuous at atleast two points, for any a + b + c ≠1 . values of a, b and c.

Q69. tan(2 tan−1 51 + sec−1 √52 + 2 tan−1 18 ) is equal to: (1) 1 (2) 2 (3) 1 (4) 5 4 4

Q69.The angle of elevation of the top P of a vertical tower PQ of height 10 from a point A on the horizontal ground is 45° . Let R be a point on AQ and from a point B, vertically above R, the angle of elevation of P is 60° . If ∠BAQ = 30°, AB = d and the area of the trapezium PQRB is α, then the ordered pair (d, α) is (1) (10(√3 −1), 25) (2) (10(√3 −1), 252 ) + + (3) (10(√3 1), 25) (4) (10(√3 1), 252 ) . If A2 + γA + 18I = O, then det (A) is equal to _______.

Q69.Let R be a relation from the set {1, 2, 3 … … … , 60} to itself such that R ={ (a, b) : b = pq , where p, q ≥3 are prime numbers}. Then, the number of elements in R is (1) 600 (2) 660 (3) 540 (4) 720

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

Q69.If the system of equations αx + y + z = 5, x + 2y + 3z = 4, x + 3y + 5z = β. Has infinitely many solutions, then the ordered pair (α, β) is equal to (1) (1, −3) (2) (−1, 3) (3) (1, 3) (4) (−1, −3)

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

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

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 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.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 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