Practice Questions

Q90.Let y = y(x) be the solution of the differential equation x+2 ) + (y + = (x + 2)dy, y(1) = 1. If the domain of y = y(x) is an open interval (α, β), + 2)e( 1))dx ((x y+1 then |α + β| is equal to ___________. JEE Main 2021 (22 Jul Shift 1) JEE Main Previous Year Paper

Q90.A line l passing through origin is perpendicular to the lines l1 :→r= (3 + t)ˆi + (−1 + 2t)ˆj + (4 + 2t)ˆk l2 :→r= (3 + 2s)ˆi + (3 + 2s)ˆj + (2 + s)ˆk If the co-ordinates of the point in the first octant on l2 at a distance of √17 from the point of intersection of l and l1 are (a, b, c), then 18(a + b + c) is equal to ___ . JEE Main 2021 (25 Feb Shift 2) JEE Main Previous Year Paper

Q90.Let 𝐵𝑖𝑖= 1, 2, 3 be three independent events in a sample space. The probability that only 𝐵1 occur is 𝛼, only 𝐵2 occurs is 𝛽 and only 𝐵3 occurs is 𝛾. Let 𝑝 be the probability that none of the events 𝐵𝑖 occurs and these 4 probabilities satisfy the equations 𝛼- 2𝛽𝑝= 𝛼𝛽 and 𝛽- 3𝛾𝑝= 2𝛽𝛾 (All the probabilities are assumed to lie in 𝑃𝐵1 the interval 0, 1 Then is equal to______. 𝑃𝐵3 JEE Main 2021 (24 Feb Shift 1) JEE Main Previous Year Paper

Q90.If the shortest distance between the lines r1 = αˆi + 2ˆj + 2ˆk + λ(ˆi −2ˆj 2ˆk), → μ ∈R is 9, then α is equal to_____. r2 = −4ˆi −ˆk + μ(3ˆi −2ˆj −2ˆk), JEE Main 2021 (20 Jul Shift 1) JEE Main Previous Year Paper

Q51.If the equation x2 + bx + 45 = 0, b ∈R has conjugate complex roots and they satisfy |z + 1| = 2√10, then (1) b2 −b = 30 (2) b2 + b = 72 (3) b2 −b = 42 (4) b2 + b = 12

Q51.Let S , be the set of all real roots of the equation, 3x(3x −1) + 2 = |3x −1| + |3x −2|, then (1) contains exactly two elements. (2) is a singleton. (3) is an empty set. (4) contains at least four elements.

Q51.Let [t] denote the greatest integer ≤t. Then the equation in x, [x]2 + 2[x + 2] −7 = 0 has : (1) exactly two solutions (2) exactly four integral solutions (3) no integral solution (4) infinitely many solutions

Q51.If α and β are the roots of the equation 2x(2x + 1) = 1, then β is equal to : (1) 2α(α + 1) (2) −2α(α + 1) (3) 2α(α −1) (4) 2α2

Q51.Let λ ≠0 be in R. If α and β are the roots of the equation, x2 −x + 2λ = 0 and α and γ are the roots of the equation, 3x2 −10x + 27λ = 0, then βγλ is equal to: (1) 27 (2) 18 (3) 9 (4) 36 a + b is equal to:

Q51.If A = {x ∈R : |x| < 2} and B = {x ∈R : |x −2| ≥3}; then (1) A ∩B = (−2, −1) (2) B −A = R −(−2, 5) (3) A ∪B = R −(2, 5) (4) A −B = [−1, 2)

Q51.The number of real roots of the equation, e4x + e3x −4e2x + ex + 1 = 0 is: (1) 1 (2) 3 (3) 2 (4) 4

Q51.Consider the two sets: A = {m ∈R : both the roots of x2 −(m + 1)x + m + 4 = 0 are real } and B = [−3, 5) Which of the following is not true? (1) A −B = (−∞, −3) ∪(5, ∞) (2) A ∩B = {−3} (3) B −A = (−3, 5) (4) A ∪B = R

Q51.If α and β are the roots of the equation, 7x2 −3x −2 = 0, then the value of α + β is equal to: 1−α2 1−β2 (1) 27 (2) 1 32 24 (3) 3 (4) 27 8 16

Q51.If α and β be two roots of the equation x2 −64x + 256 = 0. Then the value of 1 1 + ( β5 ) ( α5 ) JEE Main 2020 (06 Sep Shift 1) JEE Main Previous Year Paper (1) 2 (2) 3 (3) 1 (4) 4

Q51.Let α and β be the roots of the equation, 5x2 + 6x −2 = 0. If Sn = αn + βn, n = 1, 2, 3, . . . . , then (1) 6S6 + 5S5 = 2S4 (2) 5S6 + 6S5 + 2S4 = 0 (3) 5S6 + 6S5 = 2S4 (4) 6S6 + 5S5 + 2S4 = 0 1+sin 9 +i cos

Q51.The product of the roots of the equation 9x2 −18 x + 5 = 0 is : (1) 59 (2) 2581 (3) 275 (4) 259 ¯¯

Q51.The set of all real values of λ for which the quadratic equation (λ2 + 1)x2 −4λx + 2 = 0 always have exactly one root in the interval (0, 1) is : (1) (−3, −1) (2) (0, 2) (3) (1, 3] (4) (2, 4]

Q51.Let f(x) be a quadratic polynomial such that f(–1) + f(2) = 0. If one of the roots of f(x) = 0 is 3 , then its other root lies in (1) (−1, 0) (2) (1, 3) (3) (–3, –1) (4) (0, 1) 1 1 2 2 +

Q51.Let α and β be two real roots of the equation (k + 1)tan2x −√2 ⋅λ tan x = (1 −k), where k(≠−1) and λ are real numbers. If tan2(α + β) = 50, then a value of λ is (1) 10√2 (2) 10 (3) 5 (4) 5√2

Q52.Let α = −1+i√32 . If a = (1 + α) ∑100k=0 α2k and b = ∑100k=0 α3k , then a and b, are the roots of the quadratic equation. (1) x2 + 101x + 100 = 0 (2) x2 −102x + 101 = 0 (3) x2 −101x + 100 = 0 (4) x2 + 102x + 101 = 0

Q52.The value of 2π 2π 3 2π 2π ( 1+sin 9 −i cos 99 ) is (1) 1 (2) 1 2 (1 −i√3) 2 (√3 −i) (3) −12 (√3 −i) (4) −12 (1 −i√3)

Q52.If Re( 2z+iz−1 ) = 1, where z = x + iy, then the point (x, y) lies on a (1) circle whose centre is at (−12 , −32 ) (2) straight line whose slope is −23 (3) straight line whose slope is 3 2 (4) circle whose diameter is √52

Q52.If z1, z2 are complex numbers such that Re (z1) = |z1 −1| and Re (z2) = |z2 −1| and arg(z1 −z2) = π6 , then Im(z1 + z2) is equal to : (1) 2√3 (2) √3 2 (3) 1 (4) 2 √3 √3

Q52.Let z be a complex number such that z+2i z−i = 1 and |z| = 52 . Then, the value of |z + 3i| is (1) √10 (2) 72 (3) 15 (4) 2√3 4

Q52.If 3+isinθ , θ ∈[0 ,2 π], is a real number, then an argument of sinθ + icosθ is 4−icosθ (1) π −tan−1( 34 ) (2) π −tan−1( 43 ) (3) −tan−1( 43 ) (4) tan−1( 43 )