Practice Questions

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

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

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

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

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.The square of the distance of the image of the point (6, 1, 5) in the line x−13 = 2y = z−24 , from the origin is _________ JEE Main 2024 (09 Apr Shift 2) JEE Main Previous Year Paper

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.If d1 is the shortest distance between the lines x + 1 = 2 y = −12 z, x = y + 2 = 6 z −6 and d2 is the shortest distance between the lines x−1 2 = y+8−7 = z−45 , x−12 = y−21 = z−6−3 , then the value of 32√3d2 d1 is : JEE Main 2024 (30 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.Three balls are drawn at random from a bag containing 5 blue and 4 yellow balls. Let the random variables X and Y respectively denote the number of blue and yellow balls. If ¯X and ¯Y are the means of X and Y respectively, then 7¯X + 4¯Y is equal to________ JEE Main 2024 (08 Apr Shift 1) JEE Main Previous Year Paper

Q90.Let O be the origin, and M and N be the points on the lines x−5 4 = y−41 = z−53 and x+812 = y+25 = z+119 −−→ → respectively such that MN is the shortest distance between the given lines. Then OM ⋅ON is equal to _________. JEE Main 2024 (29 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.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 →a = ^i −3^j + 7^k, b = 2^i −^j + ^k and→cbe a vector such that (→a+ 2b) ×→c= 3(→c×→a) . If →a ⋅→c = 130 , then →b ⋅→c is equal to _______ JEE Main 2024 (05 Apr Shift 1) JEE Main Previous Year Paper