Practice Questions

Q69.For a triangle 𝐴𝐵𝐶, the value of cos2𝐴+ cos2𝐵+ cos2𝐶 is least. If its inradius is 3 and incentre is 𝑀, then which of the following is NOT correct? (1) Perimeter of ∆𝐴𝐵𝐶 is 18√3 (2) sin2𝐴+ sin2𝐵+ sin2𝐶= sin𝐴+ sin𝐵+ sin𝐶 (3) →MA · →MB = - 18 (4) area of ∆𝐴𝐵𝐶 is 27√3 2

Q69.The value of tan 9 o −tan 27 o −tan 63 o + tan 81 o is _____.

Q69.Among the statements: 𝑆1: 𝑝∨𝑞⇒𝑟⇔𝑝⇒𝑟 𝑆2: 𝑝∨𝑞⇒𝑟⇔𝑝⇒𝑟∨𝑞⇒𝑟 (1) Only ( 𝑆1 ) is a tautology (2) Neither ( 𝑆1 ) nor ( 𝑆2 ) is a tautology (3) Only ( 𝑆2 ) is a tautology (4) Both ( 𝑆1 ) and ( 𝑆2 ) are tautologies

Q69.Let S be the set of all a ∈N such that the area of the triangle formed by the tangent at the point P(b, c), b, c ∈N , on the parabola y2 = 2ax and the lines x = b, y = 0 is 16 unit2 , then ∑a∈S a is equal to _____ .

Q69.If m and n respectively are the numbers of positive and negative value of θ in the interval [−π, π] that satisfy the equation cos 2θ cos 2θ = cos 3θ cos 9θ2 , then mn is equal to _____ .

Q69.If the x-intercept of a focal chord of the parabola y2 = 8x + 4y + 4 is 3 , then the length of this chord is equal to _____ .

Q69.The mean and standard deviation of 10 observations are 20 and 8 respectively. Later on, it was observed that one observation was recorded as 50 instead of 40. Then the correct variance is (1) 11 (2) 13 (3) 12 (4) 14

Q69.Let C(α, β) be the circumcentre of the triangle formed by the lines 4x + 3y = 69 , 4y −3x = 17 , and x + 7y = 61 . Then (α −β)2 + α + β is equal to (1) 18 (2) 17 (3) 15 (4) 16

Q69.For the system of linear equations 𝑥+ 𝑦+ 𝑧= 6 𝛼𝑥+ 𝛽𝑦+ 7𝑧= 3 𝑥+ 2𝑦+ 3𝑧= 14 which of the following is NOT true ? (1) If 𝛼= 𝛽= 7, then the system has no solution (2) If 𝛼= 𝛽 and 𝛼≠7 then the system has a unique solution. (3) There is a unique point ( 𝛼, 𝛽) on the line (4) For every point ( 𝛼, 𝛽) ≠( 7, 7 ) on the line 𝑥+ 2𝑦+ 18 = 0 for which the system has x - 2y + 7 = 0, the system has infinitely many infinitely many solutions solutions.

Q69.Let 𝑁 denote the number that turns up when a fair die is rolled. If the probability that the system of equations 𝑥+ 𝑦+ 𝑧= 12𝑥+ 𝑁𝑦+ 2𝑧= 23𝑥+ 3𝑦+ 𝑁𝑧= 3 has unique solution is 𝑘 then the sum of value of 𝑘 and all possible values of 𝑁 is 6, JEE Main 2023 (24 Jan Shift 1) JEE Main Previous Year Paper (1) 18 (2) 19 (3) 20 (4) 21

Q69.For the system of linear equations 2𝑥- 𝑦+ 3𝑧= 5 3𝑥+ 2𝑦- 𝑧= 7 4𝑥+ 5𝑦+ 𝛼𝑧= 𝛽, which of the following is NOT correct? (1) The system has infinitely many solutions for (2) The system has infinitely many solutions for 𝛼= – 5 and 𝛽= 9 𝛼= - 6 and 𝛽= 9 (3) The system in inconsistent for 𝛼= – 5 and (4) The system has a unique solution for 𝛼≠– 5 𝛽= 8 and 𝛽= 8

Q69.An organization awarded 48 medals in event '𝐴', 25 in event '𝐵' and 18 in event '𝐶'. If these medals went to total 60 men and only five men got medals in all the three events, then, how many received medals in exactly two of three events? (1) 15 (2) 21 (3) 10 (4) 9

Q69.The set of all values of λ for which the equation cos2 2x −2 sin4 x −2 cos2 x = λ (1) [−2, −1] (2) [−2, −32 ] (3) [−1, −12 ] (4) [−32 , −1]

Q69.The distance of the point (6, −2√2) from the common tangent and x = 1 + y2 is (1) 1 (2) 5 3 (3) 14 (4) 5√3 3

Q70.If the radius of the largest circle with centre (2, 0) inscribed in the ellipse x2 + 4y2 = 36 is r, then 12 r2 is equal to (1) 115 (2) 92 (3) 69 (4) 72

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.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 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

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.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 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.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.The vertices of a hyperbola H are (±6, 0) and its eccentricity is √52 . Let N be the normal to H at a point in the first quadrant and parallel to the line √2x + y = 2√2 . If d is the length of the line segment of N between H and the y -axis then d2 is equal to _____ .

Q70.A triangle is formed by X -axis, Y -axis and the line 3x + 4y = 60 . Then the number of points P(a, b) which lie strictly inside the triangle, where a is an integer and b is a multiple of a, is _____ .

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