Practice Questions

Q68.Let 𝛽= lim 𝛼𝑥- 𝑒3𝑥- 1 for some 𝛼∈ℝ. Then the value of 𝛼+ 𝛽 is: 𝑥→0 𝛼𝑥𝑒3𝑥- 1 14 3 (1) (2) 5 2 (3) 5 (4) 7 2 2

Q68.Let 𝑎, 𝑏 and 𝑐 be the length of sides of a triangle 𝐴𝐵𝐶 such that 𝑎+ 𝑏 = 𝑏+ 𝑐 = 𝑐+ 𝑎 . If 𝑟 and 𝑅 are the radius of 7 8 9 𝑅 incircle and radius of circumcircle of the triangle 𝐴𝐵𝐶, respectively, then the value of is equal to 𝑟 (1) 2 (2) 3 5 (3) 5 (4) 1 2

Q68.The mean of the numbers a, b, 8, 5, 10 is 6 and their variance is 6. 8. If M is the mean deviation of the numbers about the mean, then 25M is equal to (1) 60 (2) 55 (3) 50 (4) 75

Q68.Let the system of linear equations x + 2y + z = 2, αx + 3y −z = α, −αx + y + 2z = −α be inconsistent. Then α is equal to (1) 2 5 (2) −52 (3) 2 7 (4) −72

Q68.Which of the following statement is a tautology? (1) ((~q) ∧p) ∧q (2) ((~q) ∧p) ∧(p ∧(~p)) (3) ((~q) ∧p) ∨(p ∨(~p)) (4) (p ∧q) ∧(~(p ∧q))

Q68.If the system of linear equations. JEE Main 2022 (26 Jul Shift 1) JEE Main Previous Year Paper 8x + y + 4z = −2 x + y + z = 0 λx −3y = μ has infinitely many solutions, then the distance of the point (λ, μ, −12 ) from the plane 8x + y + 4z + 2 = 0 is: (1) 3√5 (2) 4 (3) 26 (4) 10 9 3

Q68.The number of choices for Δ ∈{∧, ∨, ⇒, ⇔} , such that (pΔq) ⇒((pΔ~q) ∨((~p)Δq)) is a tautology, is (1) 1 (2) 2 (3) 3 (4) 4 Q69. ⎡ 1 0 a ⎤ Let S ={ √n : 1 ⩽n ⩽50 and n is odd}. Let a ∈S and A = −1 1 0 . If Σ det (adj A) = 100λ, then λ ⎣−a 0 1 ⎦ a∈S is equal to (1) 218 (2) 221 (3) 663 (4) 1717

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 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)

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 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.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.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.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. 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.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.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.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. 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 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.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

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.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 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.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