Practice Questions

Q69.The parabolas : ax2 + 2bx + cy = 0 and d2 + 2ex + fy = 0 intersect on the line y = 1. If a, b, c, d, e, f are positive real numbers and a, b, c are in G. P., then (1) d, e, f are in A.P. (2) ad , eb , fc are in G.P. (3) a d , eb , fc are in A.P. (4) d, e, f are in G.P.

Q69.The locus of the middle points of the chords of the circle C1 : (x −4)2 + (y −5)2 = 4 which subtend an angle θi at the centre of the circle Ci , is a circle of radius ri . If θ1 = π3 , θ3 = 2π3 and r12 = r22 + r32 , then θ2 is equal to π 3π (1) (2) 4 4 (3) π (4) π 6 2

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

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.In a triangle ABC , if cos A + 2 cos B + cos C = 2 and the lengths of the sides opposite to the angles A and C are 3 and 7 respectively, then cos A −cos C is equal to (1) 9 (2) 10 7 7 (3) 5 (4) 3 7 7

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.Let m1 and m2 be the slopes of the tangents drawn from the point P(4, 1) to the hyperbola H : 25y2 −x216 = 1 If Q is the point from which the tangents drawn to H have slopes |m1| and |m2| and they make positive (PQ)2 intercepts α and β on the x− axis, then αβ is equal to _______.

Q69.From the top 𝐴 of a vertical wall 𝐴𝐵 of height 30 m, the angles of depression of the top 𝑃 and bottom 𝑄 of a vertical tower 𝑃𝑄 are 15∘ and 60∘ respectively, 𝐵 and 𝑄 are on the same horizontal level. If 𝐶 is a point on 𝐴𝐵 such that 𝐶𝐵= 𝑃𝑄, then the area (in m2) of the quadrilateral 𝐵𝐶𝑃𝑄 is equal to JEE Main 2023 (06 Apr Shift 1) JEE Main Previous Year Paper (1) 300 ( √3 - 1 ) (2) 300 ( √3 + 1 ) (3) 600 ( √3 - 1 ) (4) 200 ( √3 - 1 )

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.A light ray emits from the origin making an angle 30° with the positive x -axis. After getting reflected by the line x + y = 1 , if this ray intersects x-axis at Q, then the abscissa of Q is (1) 2 (2) 2 (√3−1) 3+√3 (3) 2 (4) √3 3−√3 2(√3+1) JEE Main 2023 (29 Jan Shift 1) JEE Main Previous Year Paper

Q69.The statement ( 𝑝∧( ~𝑞) ) ∨( ( ~𝑝) ∧ 𝑞) ∨( ( ~𝑝) ∧ ( ~𝑞) ) is equivalent to _____ (1) ~𝑝∨𝑞 (2) ~𝑝∨~𝑞 (3) 𝑝∨~𝑞 (4) 𝑝∨𝑞 Q70. 1 2 3 Let for 𝐴= 𝛼3 1 , 𝐴= 2. If |2 adj ( 2 adj ( 2𝐴) ) | = 32𝑛, then 3𝑛+ 𝛼 is equal to 1 1 2 (1) 9 (2) 11 (3) 12 (4) 10

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.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.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.

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 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.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 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.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.A circle with centre (2, 3) and radius 4 intersects the line x + y = 3 at the points P and Q. If the tangents at P and Q intersect at the point S(α, β), then 4α −7β is equal to

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.Consider a circle C1 : x 2 + y2 – 4x – 2y = α – 5. Let its mirror image in the line y = 2x + 1 be another circle C2 : 5x2 + 5y2 –10fx – 10gy + 36 = 0. Let r be the radius of C2 . Then α + r is equal to ________

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 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.If 𝑓𝑥= tan1°𝑥+ log𝑒123 𝑥> 0, then the least value of 𝑓𝑓𝑥+ 𝑓𝑓4 is 𝑥 𝑥 log𝑒1234 - tan1°, (1) 0 (2) 8 (3) 2 (4) 4