Practice Questions

Q89.Let an = ∫n−1(1 + x2 + x23 + … + xn−1n )dx for every {n ∈N : an ∈(2, 30)} is _________. , y(1) = 1. If for some

Q89.Let S = {1, 2, 3, 4} . Then the number of elements in the set { f : S × S →S : f is onto and f(a, b) = f(b, a) ≥a∀(a, b) ∈S × S } is

Q89.Let y = y(x) be the solution curve of the differential equation = 0, 0 < x < √π2 sin(2x2) loge(tan x2)dy + (4xy −4√2x sin(x2 −π4 ))dx , which passes through the point (√π6 , 1). Then y(√π3 ) is equal to _______. y−2

Q89.Let p and p + 2 be prime numbers and let p! (p + 1)! (p + 2)! Δ = (p + 1)! (p + 2)! (p + 3)! (p + 2)! (p + 3)! (p + 4)! Then the sum of the maximum values of α and β , such that pα and (p + 2)β divide Δ , is _______.

Q89.Let a line having direction ratios 1, −4, 2 intersect the lines x−73 = y−1−1 = z+21 and x2 = y−73 = 1z at the points A and B. Then (AB)2 is equal to + + = + + +

Q89.Let the solution curve y = y(x) of the differential equation (4 + x2)dy −2x(x2 + 3y + 4)dx = 0 pass through the origin. Then y(2) is equal to _____.

Q89.Let a curve y = y(x) pass through the point (3, 3) and the area of the region under this curve, above the x-axis y 3 and between the abscissae 3 and x(> 3) be ( x ) . If this curve also passes through the point (α, 6√10) in the first quadrant, then α is equal to _______. y+2

Q89.Let d be the distance between the foot of perpendiculars of the points P(1, 2 −1) and Q(2, −1, 3) on the plane −x + y + z = 1 . Then d2 is equal to ______. JEE Main 2022 (29 Jun Shift 1) JEE Main Previous Year Paper = 4 be a plane. Let P2 be another plane which passes through the points

Q89.Let f be a differentiable function satisfying f(x) = 2 ∫√30 f( λ2x3 )dλ, √3 passes through the point (α, 6), then α is equal to _______. → → → →

Q89.The largest value of 𝑎, for which the perpendicular distance of the plane containing the lines →𝑟= ^𝑖+ ^𝑗+ 𝜆 ^𝑖+ 𝑎 ^𝑗- ^𝑘and →𝑟= ^𝑖+ ^𝑗+ 𝜇- ^𝑖+ ^𝑗- 𝑎𝑘 from the point 2, 1, 4 is √3, is ______.

Q89.If 𝛼√2 + 𝛽√3, where 𝛼, 𝛽 are integers, then 𝛼+ 𝛽 is equal to ∫0 √1 + 𝑥2 + √1 + 𝑥23𝑑𝑥= 56 43 111

Q89.If 𝑙𝑖𝑚 lim ( 𝑛𝑘+ 1 ) + ( 𝑛𝑘+ 2 ) + … + 𝑛𝑘+ 𝑛= 33 . 1 1 · 1𝑘+ 2𝑘+ 3𝑘+ … + 𝑛𝑘, then the 𝑛𝑘+ 𝑛𝑘+ 𝑛→∞ 𝑛→∞ integral value of 𝑘 is equal to _____ . 𝑥- 2 𝑦- 1 𝑧 𝑥- 3 𝑦- 5 𝑧- 1

Q89.The area (in sq. units) of the region enclosed between the parabola y2 = 2x and the line x + y = 4 is ______.

Q89.Let →𝑏= ^𝑖+ ^𝑗+ 𝜆 ^𝑘, 𝜆∈ℝ. If →𝑎 is a vector such that →𝑎× →𝑏= 13 ^𝑖- ^𝑗- 4 ^𝑘 and →𝑎· →𝑏+ 21 = 0, then →𝑏- →𝑎· ^𝑘- ^𝑗+ →𝑏+ →𝑎· ^𝑖- ^𝑘 is equal to 1 1

Q89.Let →𝑎 and →𝑏 be two vectors such that →𝑎+ →𝑏 = →𝑎 + 2 →𝑏 , →𝑎· →𝑏= 3 and →𝑎× →𝑏 = 75. Then →𝑎 is equal to ______.

Q89.Let A1 = {(x, y) : |x| ≤y2, |x| + 2y ≤8} and A2 = {(x, y) : |x| + |y| ≤k}. If 27 (Area A1 ) = 5 (Area A2 ), then k is equal to

Q90.If →a = 2ˆi + ˆj + 3ˆk, b = 3ˆi + 3ˆj + ˆk and →c= c1ˆi + c2ˆj + c3ˆk are coplanar vectors and →a⋅→c= 5, b ⊥→c, then 122(c1 + c2 + c3) is equal to ______. JEE Main 2022 (28 Jun Shift 1) JEE Main Previous Year Paper

Q90.Let P1 :→r⋅(2ˆi + ˆj −3ˆk) (2, − 3, 2)(2, −2, −3) and (1, −4, 2). If the direction ratios of the line of intersection of P1 and P2 be 16 , α, β, then the value of α + β is equal to ______. JEE Main 2022 (29 Jun Shift 1) JEE Main Previous Year Paper

Q90.If the shortest distance between the lines→r= (−ˆi 3ˆk) λ(ˆi −aˆj) and →r (−ˆj 2ˆk) μ(ˆi −ˆj ˆk) is , then the integral value of a is equal to _____ √23 JEE Main 2022 (24 Jun Shift 1) JEE Main Previous Year Paper

Q90.Let →a, b,→cbe three non-coplanar vectors such that →a× b = 4→c, b ×→c= 9→a and →c×→a = αb, α > 0 → If →a + b + →c = 36, then α is equal to _______. JEE Main 2022 (27 Jul Shift 2) JEE Main Previous Year Paper

Q90.If the probability that a randomly chosen 6 -digit number formed by using digits 1 and 8 only is a multiple of 21 is p, then 96p is equal to _____. JEE Main 2022 (26 Jun Shift 2) JEE Main Previous Year Paper

Q90.Let a line with direction ratios a, −4 a, −7 be perpendicular to the lines with direction ratios 3, −1, 2b and b, a, −2. If the point of intersection of the line x+1 = y−2 = 1z and the plane x −y + z = 0 is (α, β, γ), a2+b2 a2−b2 then α + β + γ is equal to ________. JEE Main 2022 (29 Jul Shift 1) JEE Main Previous Year Paper

Q90.Let the image of the point P(1, 2, 3) in the line L : x−63 = y−12 = z−23 be Q. let R(α, β, γ) be a point that divides internally the line segment PQ in the ratio 1 : 3 . Then the value of 22(α + β + γ) is equal to JEE Main 2022 (28 Jun Shift 2) JEE Main Previous Year Paper

Q90.Let 𝑙1 be the line in 𝑥𝑦-plane with 𝑥 and 𝑦 intercepts 8 and 4√2 respectively, and 𝑙2 be the line in 𝑧𝑥-plane with 𝑥 and 𝑧 intercepts -1 and - 1 respectively. If 𝑑 is the shortest distance between the line 𝑙1 and 𝑙2, then 𝑑-2 8 6√3 is equal to _____. JEE Main 2022 (25 Jun Shift 2) JEE Main Previous Year Paper

Q90.Let S = (0, 2π) −{ π2 , 3π4 , 3π2 , 7π4 }. Let y = y(x), x ∈S , be the solution curve of the differential equation dy dx = 1+sin1 2x , y( π4 ) = 21 . If the sum of abscissas of all the points of intersection of the curve y = y(x) with the curve y = √2 sin x is kπ12 , then k is equal to _____. JEE Main 2022 (26 Jun Shift 1) JEE Main Previous Year Paper