Practice Questions

Q61.Let 𝑆 be the set of positive integral values of 𝑎 for which 𝑎𝑥2 + 2𝑎+ 1𝑥+ 9𝑎+ 4 < 0, ∀𝑥∈ℝ. Then, the number 𝑥2 - 8𝑥+ 32 of elements in 𝑆 is: (1) 1 (2) 0 (3) ∞ (4) 3

Q61.If 2 and 6 are the roots of the equation ax2 + bx + 1 = 0, then the quadratic equation, whose roots are 2a+b1 and 1 , is : 6a+b (1) 2x2 + 11x + 12 = 0 (2) x2 + 8x + 12 = 0 (3) 4x2 + 14x + 12 = 0 (4) x2 + 10x + 16 = 0

Q61.If z1, z2 are two distinct complex number such that z1−2z21 = 2, then 2 −z1¯z2 (1) z1 lies on a circle of radius 21 and z2 lies on a (2) both z1 and z2 lie on the same circle. both z1 and circle of radius 1 . z2 lie on the same circle. (3) either z1 lies on a circle of radius 21 or z2 lies on (4) either z1 lies on a circle of radius 1 or z2 lies on a a circle of radius 1 . circle of radius 1 . 2

Q61.Let 𝛼 and 𝛽 be the roots of the equation 𝑝𝑥2 + 𝑞𝑥−𝑟= 0, where 𝑝≠0. If 𝑝, 𝑞 and 𝑟 be the consecutive terms of a non-constant G.P and 1 1 3 then the value of 𝛼−𝛽2 is: 𝛼+ 𝛽= 4, (1) 80 (2) 9 9 20 (3) (4) 8 3

Q61.Let S1 = {z ∈C : |z| ≤5}, S2 = {z ( z+1−√3i1−√3i ) ≥0} area of the region S1 ∩S2 ∩S3 is : (1) 125π (2) 125π 12 4 (3) 125π (4) 125π 24 6 Q62.60 words can be made using all the letters of the word BHBJO, with or without meaning. If these words are written as in a dictionary, then the 50th word is : (1) JBBOH (2) OBBJH (3) OBBHJ (4) HBBJO

Q61.Consider the following two statements : Statement I : For any two non-zero complex numbers z1, z2 , (|z1| + |z2|) z1 + z2 ≤2 (|z1| + |z2|), and |z1| |z2| Statement II : If x, y, z are three distinct complex numbers and a, b, c are three positive real numbers such that |y−z| a = |z−x|b = |x−y|c , then y−za2 + z−xb2 + x−yc2 = 1. Between the above two statements, (1) Statement I is correct but Statement II is (2) both Statement I and Statement II are correct. incorrect. (3) both Statement I and Statement II are incorrect. (4) Statement I is incorrect but Statement II is correct.

Q61.If 𝛼, 𝛽 are the roots of the equation, x2 - x - 1 = 0 and Sn = 2023𝛼n + 2024𝛽n, then (1) 2 S12 = S11 + S10 (2) S12 = S11 + S10 (3) 2 S11 = S12 + S10 (4) S11 = S10 + S12 4! ! 5! ( ) (

Q61.If S = z ∈C : |z −i| = |z + i| = |z −1|, then, n(S) is: (1) 1 (2) 0 (3) 3 (4) 2

Q61.Let α, β be the distinct roots of the equation x2 −(t2 −5t + 6)x + 1 = 0, t ∈R and an = αn + βn . Then the minimum value of a2023+a2025 is a2024 (1) −1/4 (2) −1/4 (3) −1/2 (4) 1/4

Q62.Let α and β be the sum and the product of all the non-zero solutions of the equation (¯z)2 + |z| = 0, z ∈ C. Then 4 (α2 + β2) is equal to : (1) 6 (2) 8 (3) 2 (4) 4

Q62.The number of common terms in the progressions 4, 9, 14, 19, … …, up to 25th term and 3, 6, 9, 12,.... up to 37th term is : (1) 9 (2) 5 (3) 7 (4) 8 n

Q62.Let z be a complex number such that the real part of z−2i is zero. Then, the maximum value of |z −(6 + 8i)| z+2i is equal to (1) 12 (2) 10 (3) 8 (4) ∞

Q62.Let Sa denote the sum of first n terms an arithmetic progression. If S20 = 790 and S10 = 145, then S15−S5 is : JEE Main 2024 (30 Jan Shift 1) JEE Main Previous Year Paper (1) 395 (2) 390 (3) 405 (4) 410

Q62.In an A.P., the sixth term a6 = 2. If the a1a4a5 is the greatest, then the common difference of the A.P., is equal to (1) 3 (2) 8 2 5 (3) 2 (4) 5 3 8

Q62.The number of triangles whose vertices are at the vertices of a regular octagon but none of whose sides is a side of the octagon is (1) 48 (2) 56 (3) 24 (4) 16

Q62.Let 𝑎 and 𝑏 be two distinct positive real numbers. Let 11th term of a GP, whose first term is 𝑎 and third term is 𝑏, is equal to 𝑝th term of another GP, whose first term is 𝑎 and fifth term is 𝑏. Then 𝑝 is equal to (1) 20 (2) 25 (3) 21 (4) 24

Q62.Let 𝑧1 and 𝑧2 be two complex number such that 𝑧1 + 𝑧2 = 5 and 𝑧13 + 𝑧23 = 20 + 15𝑖. Then 𝑧14 + 𝑧24 equals- (1) 30√3 (2) 75 (3) 15√15 (4) 25√3

Q62.Let 𝛼= and 𝛽= 3! )4!.! Then : 4! 5! ( ( ) ) (1) 𝛼∈N and 𝛽∉N (2) 𝛼∉N and 𝛽∈N (3) 𝛼∈N and 𝛽∈N (4) 𝛼∉N and 𝛽∉N

Q62.If the sum of the series 1 + 1 + … + 1 is equal to 5 , then 50 d is equal to : 1⋅(1+d) (1+d)(1+2 d) (1+9 d)(1+10 d) (1) 10 (2) 5 (3) 15 (4) 20

Q62.If 𝑧 is a complex number such that 𝑧≤1, then the minimum value of 𝑧+ 1 + 4𝑖 is: 23 5 (1) 2 (2) 2 3 (3) (4) 3 2

Q62.The number of ways five alphabets can be chosen from the alphabets of the word MATHEMATICS, where the chosen alphabets are not necessarily distinct, is equal to : (1) 179 (2) 177 (3) 181 (4) 175

Q62.The value of 1×22+2×32+…+100×(101)2 is 12×2+22×3+….+1002×101 (1) 32 (2) 31 31 30 (3) 306 (4) 305 305 301 JEE Main 2024 (04 Apr Shift 2) JEE Main Previous Year Paper

Q62.Let z be a complex number such that |z + 2| = 1 and Im ( z+2 ) = 5 . Then the value of |Re(z + 2)| is (1) 2√6 (2) 24 5 5 (3) 1+√6 (4) √6 5 5

Q62.Number of ways of arranging 8 identical books into 4 identical shelves where any number of shelves may remain empty is equal to (1) 18 (2) 16 (3) 12 (4) 15

Q62.For 0 < 𝑐< 𝑏< 𝑎, let ( 𝑎+ 𝑏– 2𝑐) 𝑥2 + ( 𝑏+ 𝑐– 2𝑎) 𝑥+ ( 𝑐+ 𝑎– 2𝑏) = 0 and 𝛼≠1 be one of its root. Then, among the two statements (I) If 𝛼∈-1, 0, then 𝑏 cannot be the geometric mean of 𝑎 and 𝑐. (II) If 𝛼∈0, 1, then 𝑏 may be the geometric mean of 𝑎 and 𝑐. (1) Both (I) and (II) are true (2) Neither (I) nor (II) is true (3) Only (II) is true (4) Only (I) is true 1 2 3