Practice Questions

Q89.If the solution curve, of the differential equation 𝑑𝑦 𝑥+ 𝑦- 2 𝑑𝑥= 𝑥- 𝑦 passing through the point ( 2, 1 ) is tan-1𝑦- 1 - 1 𝑦- 1 2 = 1, then 5𝛽+ 𝛼 is equal to 𝑥- 1 𝛽log𝑒𝛼+ 𝑥- 1 log𝑒𝑥- x - 2 y z - 7 x + 3 y + 2 z + 2

Q89.The least positive integral value of α, for which the angle between the vectors αˆi −2ˆj + 2ˆk and αˆi + 2αˆj −2ˆk is acute, is _____.

Q89.Let →𝑎= 3 ^𝑖+ 2 ^𝑗+ ^𝑘, →𝑏= 2 ^𝑖− ^𝑗+ 3 ^𝑘 and →𝑐 be a vector such that →𝑎+ →𝑏× →𝑐= 2→𝑎× →𝑏+ 24 ^𝑗−6 ^𝑘 and → 2 →𝑎− 𝑏+ ^𝑖. →𝑐= −3. Then →𝑐 is equal to _______.

Q89.Let →𝑎 and 𝑏 be two vectors such that →𝑎= 1, 𝑏= 4 and →𝑎⋅ 𝑏= 2. If →𝑐= 2 →𝑎× 𝑏−3 𝑏 and the angle between →𝑏 and→𝑐 is 𝛼, then 192sin2𝛼 is equal to _________

Q89.Let y = y(x) be the solution of the differential equation (1 −x2)dy = [xy + (x3 + 2)√3(1 −x2)]dx −1 < x < 1, y(0) = 0. If y( 21 ) = mn , m and n are coprime numbers, then m + n is equal to __________.

Q89.The area of the region enclosed by the parabolas y = x2 −5x and y = 7x −x2 is → →

Q89.Let the set of all values of p, for which f(x) = (p2 −6p + 8) (sin2 2x −cos2 2x) + 2(2 −p)x + 7 does not have any critical point, be the interval (a, b). Then 16ab is equal to _______

Q89.Let →a = 9^i −13^j + 25^k,→b = 3^i + 7^j −13^k and →c = 17^i −2^j + ^k be three given vectors. If →r is a vector such |593→r+67→a|2 is equal to___________ that →r × →a = (→b + →c) × →a and →r ⋅(→b −→c) = 0 , then (593)2

Q89.Let the area of the region {(x, y) : 0 ≤x ≤3, 0 ≤y ≤min{x2 + 2, 2x + 2}} be A . Then 12A is equal to ______.

Q89.Let the set of all positive values of λ , for which the point of local minimum of the function (1 + x (λ2 −x2)) satisfies x2+x+2 < 0, be (α, β). Then α2 + β2 is equal to _________ x2+5x+6

Q89.If the shortest distance between the lines x−λ 3 = y−2−1 = z−11 and x+2−3 = y+52 = z−44 is √3044 , then the largest possible value of |λ| is equal to _________

Q89.Let y = y(x) be the solution of the differential equation dx + (1+x2)2 − 1 Then the area enclosed by the curve f(x) = y(x)e (1+x2) and the line y −x = 4 is__________

Q89.Let α|x| = |y|exy−β, α, β ∈N be the solution of the differential equation x dy −y dx + xy(x dy + y dx) = 0, y(1) = 2. Then α + β is equal to ________ β + γ is equal

Q89.Let ABC be a triangle of area 15√2 and the vectors AB→ = ^i + 2^j −7^k, BC→ = a^i + b^j + ck and −→ AC = 6^i + d^j −2^k, d > 0. Then the square of the length of the largest side of the triangle ABC is _______

Q89.If 𝑑𝑥 1 + 𝑥−𝑦2 , 𝑥1 = 1, then 5𝑥2 is equal to: 𝑑𝑦= 𝑦

Q90.A line passes through 𝐴4, −6, −2 and 𝐵16, −2, 4. The point 𝑃𝑎, 𝑏, 𝑐 where 𝑎, 𝑏, 𝑐 are non-negative integers, on the line 𝐴𝐵 lies at a distance of 21 units, from the point 𝐴. The distance between the points 𝑃𝑎, 𝑏, 𝑐 and 𝑄4, −12, 3 is equal to ______. JEE Main 2024 (31 Jan Shift 2) JEE Main Previous Year Paper

Q90.A line with direction ratio 2, 1, 2 meets the lines x = y + 2 = z and x + 2 = 2y = 2z respectively at the point P and Q. if the length of the perpendicular from the point (1, 2, 12) to the line PQ is l, then l2 is JEE Main 2024 (29 Jan Shift 1) JEE Main Previous Year Paper

Q90.Let a, b and c denote the outcome of three independent rolls of a fair tetrahedral die, whose four faces are marked 1, 2, 3, 4. If the probability that ax2 + bx + c = 0 has all real roots is mn , gcd(m, n) = 1, then m + n is equal to ________ JEE Main 2024 (09 Apr Shift 1) JEE Main Previous Year Paper

Q90.The lines = = and = = intersect at the point P. If the distance of P from the line 2 -2 16 4 3 1 x + 1 y - 1 = = z - 1 is 𝑙, then 14𝑙2 is equal to _____. 2 3 1 JEE Main 2024 (27 Jan Shift 2) JEE Main Previous Year Paper

Q90.From a lot of 12 items containing 3 defectives, a sample of 5 items is drawn at random. Let the random variable X denote the number of defective items in the sample. Let items in the sample be drawn one by one without replacement. If variance of X is mn , where gcd(m, n) = 1, then n −m is equal to _________ JEE Main 2024 (06 Apr Shift 2) JEE Main Previous Year Paper

Q90.Let →𝑎= ^𝑖+ ^𝑗+ ^𝑘, →𝑏= − ^𝑖−8 ^𝑗+ 2 ^𝑘 and →𝑐= 4 ^𝑖+ 𝑐2 ^𝑗+ 𝑐3 ^𝑘 be three vectors such that →𝑏× →𝑎= →𝑐× →𝑎. If the angle between the vector →𝑐 and the vector 3 ^𝑖+ 4 ^𝑗+ ^𝑘 is 𝜃, then the greatest integer less than or equal to tan2𝜃 is: JEE Main 2024 (01 Feb Shift 2) JEE Main Previous Year Paper

Q90.In a tournament, a team plays 10 matches with probabilities of winning and losing each match as 1 and 2 3 3 respectively. Let x be the number of matches that the team wins, and y be the number of matches that team loses. If the probability P(|x −y| ≤ 2) is p , then 39p equals ______ JEE Main 2024 (04 Apr Shift 2) JEE Main Previous Year Paper

Q90.Let P(α, β, γ) be the image of the point Q(1, 6, 4) in the line x1 = y−12 = z−23 . Then 2α + to_______ JEE Main 2024 (08 Apr Shift 2) JEE Main Previous Year Paper

Q90.A fair die is tossed repeatedly until a six is obtained. Let X denote the number of tosses required and let a = P(X = 3), b = P(X ≥3) and c = P(X ≥6 ∣X > 3). Then b+ca is equal to JEE Main 2024 (27 Jan Shift 1) JEE Main Previous Year Paper

Q90.Let the line of the shortest distance between the lines 𝐿1: →𝑟= ^𝑖+ 2 ^𝑗+ 3 ^𝑘+ 𝜆 ^𝑖− ^𝑗+ ^𝑘 and 𝐿2: →𝑟= 4 ^𝑖+ 5 ^𝑗+ 6 ^𝑘+ 𝜇 ^𝑖+ ^𝑗− ^𝑘 intersect 𝐿1 and 𝐿2 at 𝑃 and 𝑄 respectively. If 𝛼, 𝛽, 𝛾 is the midpoint of the line segment 𝑃𝑄, then 2𝛼+ 𝛽+ 𝛾 is equal to___________ JEE Main 2024 (01 Feb Shift 1) JEE Main Previous Year Paper