Practice Questions

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 α, β be the roots of the equation x2 + 2√2x −1 = 0. The quadratic equation, whose roots are α4 + β4 and 1 (α6 + β6), is : 10 (1) x2 −190x + 9466 = 0 (2) x2 −180x + 9506 = 0 (3) x2 −195x + 9506 = 0 (4) x2 −195x + 9466 = 0

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.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.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.If 1 + 1 + … + 1 = m and 1⋅21 + 2⋅31 + … + 99⋅1001 = n , then the point (m, n) lies on the √1+√2 √2+√3 √99+√100 line (1) 11(x −1) −100(y −2) = 0 (2) 11x −100y = 0 (3) 11(x −2) −100(y −1) = 0 (4) 11(x −1) −100y = 0

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.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.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.Let 𝑆= 𝑧∈𝐶: 𝑧−1 = 1 and √2 −1𝑧+ ¯𝑧- 𝑖𝑧- ¯𝑧= 2√2. Let 𝑧1, 𝑧2 ∈𝑆 be such that 𝑧1 = max𝑧∈𝑠𝑧 and 2 𝑧2 = min𝑧∈𝑠𝑧. Then √2𝑧1 −𝑧2 equals: (1) 1 (2) 4 (3) 3 (4) 2

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.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.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.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

Q62.Let 0 ≤r ≤n. If n+1Cr+1 : nCr : n−1Cr−1 = 55 : 35 : 21, then 2n + 5r is equal to: JEE Main 2024 (06 Apr Shift 2) JEE Main Previous Year Paper (1) 50 (2) 62 (3) 55 (4) 60

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.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 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

Q63.Let A = {n ∈[100, 700] ∩N : n is neither a multiple of 3 nor a multiple of 4 }. Then the number of elements in A is (1) 290 (2) 280 (3) 300 (4) 310

Q63.In an increasing geometric progression of positive terms, the sum of the second and sixth terms is 70 and the 3 product of the third and fifth terms is 49 . Then the sum of the 4th , 6th and 8th terms is equal to : (1) 96 (2) 91 (3) 84 (4) 78

Q63.If in a G.P. of 64 terms, the sum of all the terms is 7 times the sum of the odd terms of the G.P, then the common ratio of the G.P. is equal to (1) 7 (2) 4 (3) 5 (4) 6

Q63.There are 5 points P1, P2, P3, P4, P5 on the side AB, excluding A and B, of a triangle ABC . Similarly there are 6 points P6, P7, … , P11 on the side BC and 7 points P12, P13, … , P18 on the side CA of the triangle. The number of triangles, that can be formed using the points P1, P2, … , P18 as vertices, is : (1) 776 (2) 796 (3) 751 (4) 771

Q63.If A denotes the sum of all the coefficients in the expansion of (1 −3x + 10x2) and B denotes the sum of all the coefficients in the expansion of (1 + x2)n , then : (1) A = B3 (2) 3 A = B (3) B = A3 (4) A = 3 B

Q63.Let three real numbers a, b, c be in arithmetic progression and a + 1, b, c + 3 be in geometric progression. If a > 10 and the arithmetic mean of a, b and c is 8, then the cube of the geometric mean of a, b and c is (1) 128 (2) 316 (3) 120 (4) 312