Practice Questions

Q68.The line 𝑦= 𝑥+ 1 meets the ellipse 𝑥2 + 𝑦2 = 1 at two points 𝑃 and 𝑄. If 𝑟 is the radius of the circle with 𝑃𝑄 4 2 as diameter then 3𝑟2 is equal to (1) 20 (2) 12 (3) 11 (4) 8 Q69. 12 12 lim tan2𝑥2sin2𝑥+ 3sin𝑥+ 4 - sin2𝑥+ 6sin𝑥+ 2 is equal to 𝑥→𝜋 2 1 1 (1) (2) - 12 18 (3) - 1 (4) 1 12 6

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.Let A be a matrix of order 3 × 3 and det(A) = 2 . Then det(det (A) adj (5 adj (A3)) is equal to _____. (1) 256 × 106 (2) 1024 × 106 (3) 512 × 106 (4) 256 × 1011

Q68.Let the foci of the ellipse x2 coincide. Then the length of the 16 + 7 = 1 and the hyperbola 144x2 −y2α = 251 latus rectum of the hyperbola is: (1) 32 (2) 18 9 5 (3) 27 (4) 27 4 10 8√2−(cos x+sin x)7

Q68.Let the operations * , ⊙∈∧, ∨. If 𝑝* 𝑞⊙𝑝⊙~𝑞 is a tautology, then the ordered pair * , ⊙ is (1) ∨, ∧ (2) ∨, ∨ (3) ∧, ∧ (4) ∧, ∨ JEE Main 2022 (28 Jul Shift 1) JEE Main Previous Year Paper

Q68.Let the system of linear equations x + y + az = 2 3x + y + z = 4 x + 2z = 1 have a unique solution ( x∗, y∗, z∗). If ( (a, x∗), (y∗, α) and ( x∗, −y∗) are collinear points, then the sum of absolute values of all possible values of α is: (1) 4 (2) 3 (3) 2 (4) 1

Q68.Let the mean of 50 observations is 15 and the standard deviation is 2 . However, one observation was wrongly recorded. The sum of the correct and incorrect observations is 70 . If the mean of the correct set of observations is 16 , then the variance of the correct set is equal to (1) 10 (2) 36 (3) 43 (4) 60

Q69.The mean and variance of the data 4, 5, 6, 6, 7, 8, x, y where x < y are 6 and 49 respectively. Then x4 + y2 is equal to (1) 320 (2) 420 (3) 162 (4) 674

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 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. 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.Negation of the Boolean statement (p ∨q) ⇒((~r) ∨p) is equivalent to: (1) p ∧(~q) ∧r (2) (~p) ∧(~q) ∧r (3) (~p) ∧q ∧r (4) p ∧q ∧(~r)

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. is equal to lim x→π4 √2−√2 sin 2x (1) 14 (2) 7 (3) 14√2 (4) 7√2

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 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 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.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

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.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. 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.The number of 𝜃∈0, 4𝜋 for which the system of linear equations 3sin3𝜃𝑥- 𝑦+ 𝑧= 2 3cos2𝜃𝑥+ 4𝑦+ 3𝑧= 3 6𝑥+ 7𝑦+ 7𝑧= 9 has no solution is (1) 6 (2) 7 (3) 8 (4) 9

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.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