Practice Questions

Q88.If ∫ π3 √1 −sin 2xdx = α + β√2 + γ√3, where α, β and γ are rational numbers, then 3α + 4β −γ is equal 6 to _____.

Q88.Let the solution y = y(x) of the differential equation dydx −y = 1 + 4 sin x satisfy y(π) = 1. Then y ( π2 ) + 10 is equal to ______ −−

Q88.If the area of the region ( x, y ) : 0 ≤y ≤min2x, 6x - x2 is A, then 12 A is equal to _______.

Q88.The value 9 ∫90 [√10x ] ,

Q88.Let 𝑦= 𝑦𝑥 be the solution of the differential equation sec2𝑥𝑑𝑥+ 𝑒2𝑦tan2𝑥+ tan𝑥𝑑𝑦= 0, 0 < 𝑥< 𝜋 𝑦𝜋 = 0. 2, 4 𝜋 If 𝑦 = 𝛼, then 𝑒8𝛼 is equal to ______. 6

Q88.If the solution of the differential equation (2x + 3y −2)dx + (4x + 6y −7)dy = 0, y(0) = 3, is αx + βy + 3 loge |2x + 3y −γ| = 6, then α + 2β + 3γ is equal to ______.

Q88.Let the area of the region enclosed by the curve y = min{sin x, cos x} and the x axis between x = −π to x = π be A . Then A2 is equal to ___________

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

Q89.Consider a line L passing through the points P(1, 2, 1) and Q(2, 1, −1). If the mirror image of the point A(2, 2, 2) in the line L is (α, β, γ), then α + β + 6γ is equal to _______

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

Q89.If the solution curve y = y(x) of the differential equation (1 + y2)(1 + loge x)dx + xdy = 0, x > 0 passes α−tan( 23 ) through the point (1, 1) and y(e) = 3 , then α + 2β is β+tan( 2 )

Q89.If 𝑥= 𝑥𝑡 is the solution of the differential equation 𝑡+ 1𝑑𝑥= 2𝑥+ 𝑡+ 14𝑑𝑡, 𝑥0 = 2, then 𝑥1 equals ________

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 →a = 2^i −3^j + 4^k,→b = 3^i + 4^j −5^k and a vector →c be such that →a × (→b + →c) + →b × →c = ^i + 8^j + 13^k . If →a ⋅→c = 13 , then (24 −→b ⋅→c) is equal to_______

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 _______

Q90.Let 𝑄 and 𝑅 be the feet of perpendiculars from the point 𝑃𝑎, 𝑎, 𝑎 on the lines 𝑥= 𝑦, 𝑧= 1 and 𝑥= −𝑦, 𝑧= −1 respectively. If ∠𝑄𝑃𝑅 is a right angle, then 12𝑎2 is equal to ________ JEE Main 2024 (31 Jan Shift 1) JEE Main Previous Year Paper

Q90.Let a line passing through the point ( - 1, 2, 3 ) intersect the lines 𝐿1: 𝑥- 1 = 𝑦- 2 = 𝑧+ 1 at 𝑀( 𝛼, 𝛽, 𝛾) and 3 2 -2 𝑥+ 2 𝑦- 2 𝑧- 1 ( 𝛼+ 𝛽+ 𝛾) 2 equals ________________. = = at 𝑁( 𝑎, 𝑏, 𝑐) . Then the value of 𝐿2: -3 -2 4 ( 𝑎+ 𝑏+ 𝑐) 2 JEE Main 2024 (30 Jan Shift 2) JEE Main Previous Year Paper

Q90.Let P be the point (10, −2, −1) and Q be the foot of the perpendicular drawn from the point R(1, 7, 6) on the line passing through the points (2, −5, 11) and (−6, 7, −5). Then the length of the line segment PQ is equal to ________ JEE Main 2024 (06 Apr Shift 1) 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.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.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