Practice Questions

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.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 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.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.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

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.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

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.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.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.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 the point (−1, α, β) lie on the line of the shortest distance between the lines x+2−3 = y−24 = z−52 and y+6 x+2 −1 = 2 = z−10 . Then (α −β)2 is equal to___________ JEE Main 2024 (05 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 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 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.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 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

Q1. In an experiment of measuring the refractive index of a glass slab using travelling microscope in physics lab, a student measures real thickness of the glass slab as 5. 25 mm and apparent thickness of the glass slab at 5. 00 mm . Travelling microscope has 20 divisions in one cm on main scale and 50 divisions on Vernier scale is equal to 49 divisions on main scale. The estimated uncertainty in the measurement of refractive index of the slab is x × 10−3 , where x is ______ 10

Q1. If P = 3ˆi + √3ˆj + 2ˆk and Q = 4ˆi + √3ˆj + 2. 5ˆk then, the unit vector in the direction of P × Q is x is x 1 (√3ˆi +ˆj −2√3ˆk). The value of

Q2. For a train engine moving with speed of 20 ms–1 , the driver must apply brakes at a distance of 500 m before the station for the train to come to rest at the station. If the brakes were applied at half of this distance, the train engine would cross the station with speed √x ms−1 . The value of x is ______. (Assuming same retardation is produced by brakes)

Q2. A tennis ball is dropped on to the floor from a height of 9. 8 m. It rebounds to a height 5. 0 m. Ball comes in contact with the floor for 0. 2 s . The average acceleration during contact is ______ m s−2 . [Given g = 10 m s−2 ]

Q3. As per given figure, a weightless pulley 𝑃 is attached on a double inclined frictionless surface. The tension in the string (massless) will be (if 𝑔= 10 m s-2) (1) 4√3 + 1 N (2) 4√3 + 1 N (3) 4√3 - 1 N (4) 4√3 - 1 N

Q4. A car is moving on a circular path of radius 600 m such that the magnitudes of the tangential acceleration and centripetal acceleration are equal. The time taken by the car to complete first quarter of revolution, if it is 2 ) s . The value of t is ______. moving with an initial speed of 54 km h−1 is t(1–e– π

Q4. Two bodies are projected from ground with same speeds 40 m s−1 at two different angles with respect to horizontal. The bodies were found to have same range. If one of the body was projected at an angle of 60° , with horizontal then sum of the maximum heights, attained by the two projectiles, is _____ m. (Given g = 10 m s−2 )

Q4. A stone tied to 180 cm long string at its end is making 28 revolutions in horizontal circle in every minute. The magnitude of acceleration of stone is 1936 x m s−2 . The value of x ______. [Take π = 227 ]