Practice Questions

Q90.If the probability that the random variable X takes values x is given by P(X = x) = k(x + 1)3−x , x = 0, 1, 2, 3, … … , where k is a constant, then P(X ≥2) is equal to (1) 7 (2) 7 27 18 (3) 11 (4) 20 18 27 JEE Main 2023 (08 Apr Shift 2) JEE Main Previous Year Paper

Q90.Three dice are rolled. If the probability of getting different numbers on the three dice is p q , where p and q are co-prime, then q −p is equal to (1) 2 (2) 1 (3) 3 (4) 4 JEE Main 2023 (06 Apr Shift 2) JEE Main Previous Year Paper

Q61.If the sum of the squares of the reciprocals of the roots α and β of the equation 3x2 + λx −1 = 0 is 15 , then 6(α3 + β3)2 is equal to (1) 46 (2) 36 (3) 24 (4) 18

Q61.The sum of all real roots of equation (e2x −4)(6e2x −5ex + 1) = 0 is (1) ln 4 (2) −ln 3 (3) ln 3 (4) ln 5

Q61.Let a circle 𝐶 in complex plane pass through the points 𝑧1 = 3 + 4𝑖, 𝑧2 = 4 + 3𝑖 and 𝑧3 = 5𝑖. If 𝑧≠𝑧1 is a point on 𝐶 such that the line through 𝑧 and 𝑧1 is perpendicular to the line through 𝑧2 and 𝑧3, then arg𝑧 is equal to 2 (1) tan-124 - 𝜋 (2) tan-1 - 𝜋 7 √5 (3) tan-13 - 𝜋 (4) tan-13 - 𝜋 4 JEE Main 2022 (25 Jun Shift 1) JEE Main Previous Year Paper 1 1 1 𝐾

Q61.Let α be a root of the equation 1 + x2 + x4 = 0. Then the value of α1011 + α2022 −α3033 is equal to: (1) 1 (2) α (3) 1 + α (4) 1 + 2α

Q61.Let A = {z ∈C : z−1z+1 < 1} and B = {z ∈C : arg( z+1z−1 ) = 2π3 }. Then A ∩B is (1) a portion of a circle centred at (0, −1√3 ) that (2) a portion of a circle centred at (0, −1√3 ) that lies in the second and third quadrants only lies in the second quadrant only (3) an empty set (4) a portion of a circle of radius 2 that lies in the √3 third quadrant only

Q61.Let f(x) be a quadratic polynomial such that f(−2) +f(3) = 0. If one of the roots of f(x) = 0 is −1, then the sum of the roots of f(x) = 0 is equal to (1) 11 (2) 7 3 3 (3) 12 (4) 14 3 3

Q61.The number of points of intersection |z −(4 + 3i)| = 2| and |z| + |z −4| = 6, z ∈C is (1) 1 (2) 2 (3) 3 (4) 4

Q61.If 𝛼, 𝛽, 𝛾, 𝛿 are the roots of the equation 𝑥4 + 𝑥3 + 𝑥2 + 𝑥+ 1 = 0, then 𝛼2021 + 𝛽2021 + 𝛾2021 + 𝛿2021 is equal to (1) 4 (2) 1 (3) -4 (4) -1

Q61.Let 𝐴= 𝑥∈𝑅: 𝑥+ 1 < 2 and 𝐵= 𝑥∈𝑅: 𝑥- 1 ≥2. Then which one the following statements is NOT true? (1) 𝐴- 𝐵= -1, 1 (2) 𝐵- 𝐴= 𝑅- -3, 1 (3) 𝐴∩𝐵= ( - 3, - 1] (4) 𝐴∪𝐵= 𝑅- [1, 3 )

Q61.If 𝑧≠0 be a complex number such that 𝑧- 𝑧= 2, then the maximum value of 𝑧 is (1) √2 (2) 1 (3) √2 - 1 (4) √2 + 1

Q61.If z = 2 + 3i, then z5 + (z)5 is equal to: (1) 244 (2) 224 (3) 245 (4) 265

Q61.If A = ∑∞n=1 (3+(−1)n)n and B = ∑∞n=1 (3+(−1)n)n , then B is equal to (1) 11 (2) 1 9 (3) −119 (4) −113 Q62. 16 sin(20°) sin(40°) sin(80°) is equal to (1) √3 (2) 2√3 (3) 3 (4) 4√3 y2

Q62.Let A = {z ∈C : 1 ⩽|z −(1 + i)| ⩽2} and B = {z ∈A : |z −(1 −i)| = 1} . Then, B (1) is an empty set (2) contains exactly two elements (3) contains exactly three elements (4) is an infinite set

Q62.If (20−a)(40−a) 1 + (40−a)(60−a)1 + … … + (180−a)(200−a)1 = 2561 , then the maximum value of a is (1) 198 (2) 202 (3) 212 (4) 218

Q62.The remainder when (2021)2023 is divided by 7 is JEE Main 2022 (26 Jun Shift 1) JEE Main Previous Year Paper (1) 2 (2) 3 (3) 4 (4) 5

Q62.Suppose a1, a2, … , an, … be an arithmetic progression of natural numbers. If the ratio of the sum of the first five terms to the sum of first nine terms of the progression is 5 : 17 and 110 < a15 < 120 , then the sum of the JEE Main 2022 (27 Jul Shift 1) JEE Main Previous Year Paper first ten terms of the progression is equal to (1) 290 (2) 380 (3) 460 (4) 510

Q62.Let x, y > 0 . If x3y2 = 215 , then the least value of 3x + 2y is JEE Main 2022 (24 Jun Shift 2) JEE Main Previous Year Paper (1) 30 (2) 32 (3) 36 (4) 40

Q62.If the minimum value of 𝑓𝑥= 5𝑥2 + 𝛼 𝑥> 0, is 14, then the value of 𝛼 is equal to 2 𝑥5, (1) 32 (2) 64 (3) 128 (4) 256 2

Q62.Let A1, A2, A3, … … be an increasing geometric progression of positive real numbers. If A1 A3 A5 A7 = 12961 and A2 + A4 = 367 , then, the value of A6 + A8 + A10 is equal to (1) 43 (2) 33 (3) 37 (4) 48 JEE Main 2022 (28 Jun Shift 1) JEE Main Previous Year Paper α ∈R, then the value of 16α is equal to

Q62.Let α, β be the roots of the equation x2 −√2x + √6 = 0 and 1 + 1, 1 + 1 be the roots of the equation α2 β2 x2 + ax + b = 0 . Then the roots of the equation x2 −(a + b −2)x + (a + b + 2) = 0 are : (1) non-real complex numbers (2) real and both negative (3) real and both positive (4) real and exactly one of them is positive

Q62.Let for some real numbers α and β, a = α −iβ . If the system of equations 4ix + (1 + i)y = 0 and ¯8(cos 2π3 + i sin 2π3 )x + ay = 0 has more than one solution then αβ is equal to (1) 2 −√3 (2) 2 + √3 (3) −2 + √3 (4) −2 −√3

Q62.Let 𝑆= 𝑧= 𝑥+ 𝑖𝑦: 𝑧- 1 + 𝑖≥𝑧, 𝑧< 2, 𝑧+ 𝑖= 𝑧- 1. Then the set of all values of 𝑥, for which 𝑤= 2𝑥+ 𝑖𝑦∈𝑆 for some 𝑦∈ℝ, is 1 1 1 (2) - (1) -√2, 4 2√2 √2, (3) -√2, 1 (4) - 1 1 2 √2, 2√2

Q62.Let {an}∞n=0 be a sequence such that a0 = a1 = 0 and an+2 = 2an+1 −an + 1 for all n ≥0 . Then, ∑∞n=2 an7n is equal to (1) 6 (2) 7 343 216 (3) 8 (4) 49 343 216 5 10