Practice Questions

Q70.The minimum number of elements that must be added to the relation 𝑅= ( 𝑎, 𝑏) , ( 𝑏, c ) on the set {a, b, c} so that it becomes symmetric and transitive is: (1) 4 (2) 7 (3) 5 (4) 3 𝑚 𝑛

Q70.Let 𝐴= 𝑎𝑖𝑗2 × 2, where 𝑎𝑖𝑗≠0 for all 𝑖, 𝑗 and 𝐴2 = 𝐼, Let a be the sum of all diagonal elements of 𝐴 and 𝑏= 𝐴 Then 3𝑎2 + 4𝑏2 is equal to (1) 4 (2) 14 (3) 7 (4) 3

Q70.Let 𝛼 be a root of the equation 𝑎- 𝑐𝑥2 + 𝑏- 𝑎𝑥+ 𝑐- 𝑏= 0 where 𝑎, 𝑏, 𝑐 are distinct real numbers such that 𝛼2 𝛼1 𝑎- 𝑐2 𝑏- 𝑎2 𝑐- 𝑏2 the matrix 1 1 1 is singular. Then the value of is 𝑏- 𝑎𝑐- 𝑏+ 𝑎- 𝑐𝑐- 𝑏+ 𝑎- 𝑐𝑏- 𝑎 𝑎 𝑏 𝑐 (1) 6 (2) 3 (3) 9 (4) 12

Q70.Let 𝐴 be a 2 × 2 matrix with real entries such that 𝐴' = 𝛼𝐴+ 1, where 𝛼∈ℝ- -1, 1., If det 𝐴2 - 𝐴= 4, the sum of all possible values of 𝛼 is equal to 3 (1) 0 (2) 2 (3) 2 (4) 5 2

Q70.Let 𝜇 be the mean and 𝜎 be the standard deviation of the distribution 𝑋𝑖 0 1 2 3 4 5 𝑓𝑖 𝑘+ 2 2𝑘 𝑘2 - 1 𝑘2 - 1 𝑘2 + 1 𝑘- 3 where 𝛴𝑓𝑖= 62. If 𝑥 denotes the greatest integer ≤𝑥, then𝜇2 + 𝜎2 is equal to (1) 9 (2) 8 (3) 7 (4) 6

Q70.If the tangents at the points P and Q on the circle x2 + y2 −2x + y = 5 meet at the point R( 94 , 2), then the area of the triangle PQR is (1) 5 (2) 13 4 8 (3) 5 (4) 13 8 4

Q70.The equations of sides AB and AC of a triangle ABC are (λ + 1)x + λy = 4 and λx + (1 −λ)y + λ = 0 respectively. Its vertex A is on the y−axis and its orthocentre is (1, 2). The length of the tangent from the point C to the part of the parabola y2 = 6x in the first quadrant is (1) √6 (2) 2√2 (3) 2 (4) 4 JEE Main 2023 (24 Jan Shift 2) JEE Main Previous Year Paper

Q70.Let O be the origin and OP and OQ be the tangents to the circle x2 + y2 −6x + 4y + 8 = 0 at the points P and Q on it. If the circumcircle of the triangle OPQ passes through the point (α, 12 ), then a value of α is (1) 3 2 (2) −12 (3) 5 (4) 1 2

Q70.Let 𝑅 be a relation on ℝ, given by 𝑅= {𝑎, 𝑏: 3𝑎- 3𝑏+ √7 is an irrational number }. Then 𝑅 is (1) Reflexive but neither symmetric nor transitive (2) Reflexive and transitive but not symmetric (3) Reflexive and symmetric but not transitive (4) An equivalence relation

Q70.Let the determinant of a square matrix A of order m be m −n , where m and n satisfy 4m + n = 22 and 17m + 4n = 93 . If det(n adj(adj(mA))) = 3a5b6c , then a + b + c is equal to (1) 84 (2) 96 (3) 101 (4) 109

Q70.If sin-1 𝛼 + cos-14 - tan-177 = 0, 0 < 𝛼< 13, then sin-1sin𝛼+ cos-1cos𝛼 is equal to 17 5 36 (1) 𝜋 (2) 16 (3) 0 (4) 16 - 5𝜋 1 1

Q70.Let H be the hyperbola, whose foci are (1 ± √2, 0) and eccentricity is √2 . Then the length of its latus rectum is: (1) 3 (2) 52 (3) 2 (4) 32

Q70.Let B and C be the two points on the line y + x =0 such that B and C are symmetric with respect to the origin. Suppose A is a point on y −2x = 2 such that ΔABC is an equilateral triangle. Then, the area of the ΔABC is (1) 3√3 (2) 2√3 (3) 8 (4) 10 √3 √3

Q70.The negation of the statement ((A ∧(B ∨C)) ⇒(A ∨B)) ⇒A is (1) equivalent to ~C (2) equivalent to B ∨~C (3) a fallacy (4) equivalent to ~A

Q70.Let A be a point on the x-axis. Common tangents are drawn from A to the curves x2 + y2 = 8 and y2 = 16x . If one of these tangents touches the two curves at Q and R, then (QR)2 is equal to (1) 64 (2) 76 (3) 81 (4) 72

Q71.Let P(x0, y0) be the point on the hyperbola 3x2 −4y2 = 36 , which is nearest to the line 3x + 2y = 1 . Then √2(y0 −x0) is equal to : (1) −3 (2) 9 (3) −9 (4) 3

Q71.Let 𝐴= 𝑑= 𝐴≠0 and 𝐴- d Adj 𝐴= 0. Then 𝑝 𝑞, (1) 1 + 𝑑2 = 𝑚+ 𝑞2 (2) 1 + 𝑑2 = 𝑚+ 𝑞2 (3) 1 + 𝑑2 = 𝑚2 + 𝑞2 (4) 1 + 𝑑2 = 𝑚2 + 𝑞2

Q71.If the system of equations 𝑥+ 𝑦+ 𝑎𝑧= 𝑏 2𝑥+ 5𝑦+ 2𝑧= 6 𝑥+ 2𝑦+ 3𝑧= 3 has infinitely many solutions, then 2𝑎+ 3𝑏 is equal to (1) 25 (2) 20 (3) 23 (4) 28 1 1 2

Q71.A square piece of tin of side 30 cm is to be made into a box without top by cutting a square from each corner and folding up the flaps to form a box. If the volume of the box is maximum, then its surface area (in cm2) is equal to (1) 800 (2) 675 (3) 1025 (4) 900

Q71.Let P( 2√3√7 √7 perpendicular and pass through the origin. If 1 + 1 = pq , where p and q are coprime, then p + q is (PQ)2 (RS)2 equal to (1) 147 (2) 143 (3) 137 (4) 157

Q71.Let R be the focus of the parabola y2 = 20x and the line y = mx + c intersect the parabola at two points P and Q. Let the points G(10, 10) be the centroid of the triangle PQR . If c −m = 6 , then PQ2 is (1) 296 (2) 325 (3) 317 (4) 346

Q71.Let 𝑓𝑥= 𝑥2 - 𝑥+ -𝑥+ 𝑥, where 𝑥∈ℝ and 𝑡 denotes the greatest integer less than or equal to 𝑡. Then, 𝑓 is (1) continuous at 𝑥= 0, but not continuous at 𝑥= 1 (2) continuous at 𝑥= 1, but not continuous at 𝑥= 0 (3) continuous at 𝑥= 0 and 𝑥= 1 (4) not continuous at 𝑥= 0 and 𝑥= 1 1

Q71.Let 𝑆 denote the set of all real values of 𝜆 such that the system of equations 𝜆𝑥+ 𝑦+ 𝑧= 1 𝑥+ 𝜆𝑦+ 𝑧= 1 𝑥+ 𝑦+ 𝜆𝑧= 1 is inconsistent, then ∑𝜆∈𝑆𝜆2 + 𝜆 is equal to (1) 2 (2) 12 (3) 4 (4) 6 - 1

Q71.Let the system of linear equations –x + 2y −9z = 7 −x + 3y + 7z = 9 −2x + y + 5z = 8 −3x + y + 13z = λ has a unique solution x = α, y = β, z = γ . Then the distance of the point (α, β, γ) from the plane 2x −2y + z = λ is (1) 11 (2) 7 (3) 9 (4) 13

Q71.The set of values of a for which x→a([xlim −5] −[2x + 2]) = 0 , where, [ζ] denotes the greatest integer less than or equal to ζ is equal to (1) (−7. 5, −6. 5) (2) (−7. 5, −6. 5] (3) [−7. 5, −6. 5] (4) [−7. 5, −6. 5)