Practice Questions

Q80.If the shortest distance between the lines x−41 = y+12 = −3z and x−λ2 = y+14 = z−2−5 is √56 , then the sum of all possible values of λ is : (1) 5 (2) 8 (3) 7 (4) 10

Q80.A coin is biased so that a head is twice as likely to occur as a tail. If the coin is tossed 3 times, then the probability of getting two tails and one head is- 2 1 (1) (2) 9 9 (3) 2 (4) 1 27 27

Q80.Let Ajay will not appear in JEE exam with probability 𝑝= 2 while both Ajay and Vijay will appear in the 7, exam with probability 𝑞= 15. Then the probability, that Ajay will appear in the exam and Vijay will not appear is: 9 18 (1) (2) 35 35 (3) 24 (4) 3 35 35

Q80.A bag contains 8 balls, whose colours are either white or black. 4 balls are drawn at random without replacement and it was found that 2 balls are white and other 2 balls are black. The probability that the bag contains equal number of white and black balls is: (1) 2 (2) 2 5 7 1 1 (3) (4) 7 5

Q80.A company has two plants A and B to manufacture motorcycles. 60% motorcycles are manufactured at plant A and the remaining are manufactured at plant B.80% of the motorcycles manufactured at plant A are rated of the standard quality, while 90% of the motorcycles manufactured at plant B are rated of the standard quality. A motorcycle picked up randomly from the total production is found to be of the standard quality. If p is the probability that it was manufactured at plant B, then 126p is (1) 54 (2) 66 (3) 64 (4) 56

Q80.Bag 𝐴 contains 3 white, 7 red balls and bag 𝐵 contains 3 white, 2 red balls. One bag is selected at random and a ball is drawn from it. The probability of drawing the ball from the bag A, if the ball drawn in white, is : 1 1 (1) (2) 4 9 (3) 1 (4) 3 3 10

Q80.There are three bags X, Y and Z . Bag X contains 5 one-rupee coins and 4 five-rupee coins; Bag Y contains 4 one-rupee coins and 5 five-rupee coins and Bag Z contains 3 one-rupee coins and 6 five-rupee coins. A bag is selected at random and a coin drawn from it at random is found to be a one-rupee coin. Then the probability, that it came from bag Y, is : (1) 1 (2) 1 4 2 (3) 5 (4) 1 12 3

Q80.Let the sum of two positive integers be 24 . If the probability, that their product is not less than 3 times their 4 greatest possible product, is m , where gcd(m, n) = 1, then n −m equals n (1) 10 (2) 9 (3) 11 (4) 8

Q80.Let P be the point of intersection of the lines x−21 = y−45 = z−21 and x−32 = y−23 = z−32 . Then, the shortest distance of P from the line 4x = 2y = z is (1) 5√14 (2) 3√14 7 7 (3) √14 (4) 6√14 7 7

Q80.The coefficients a, b, c in the quadratic equation ax2 + bx + c = 0 are from the set {1, 2, 3, 4, 5, 6}. If the probability of this equation having one real root bigger than the other is p, then 216 p equals : (1) 57 (2) 76 (3) 38 (4) 19

Q80.Three rotten apples are accidently mixed with fifteen good apples. Assuming the random variable 𝑥 to be the number of rotten apples in a draw of two apples, the variance of 𝑥 is 37 57 (1) (2) 153 153 47 40 (3) (4) 153 153

Q80.An integer is chosen at random from the integers 1 , 2, 3, . . . . . , 50. The probability that the chosen integer is a multiple of atleast one of 4, 6 and 7 is (1) 8 (2) 21 25 50 (3) 9 (4) 14 50 25 is equal to _______. +

Q80.Two integers x and y are chosen with replacement from the set {0, 1, 2, 3, … . . , 10}. Then the probability that |x −y| > 5 is : (1) 30 (2) 62 121 121 (3) 60 (4) 31 121 121

Q81.The number of integers, between 100 and 1000 having the sum of their digits equals to 14 , is _________

Q81.The number of real solutions of the equation x|x + 5| + 2|x + 7| −2 = 0 is_________

Q81.The sum of the square of the modulus of the elements in the set {z = a + ib : a, b ∈Z, z ∈C, |z −1| ≤1, |z −5| ≤|z −5i|} is ________

Q81.Let α, β be the roots of the equation x2 −x + 2 = 0 with Im (α) >Im (β). Then α6 + α4 + β4 −5α2 is equal to

Q81.The number of 3-digit numbers, formed using the digits 2, 3, 4, 5 and 7 , when the repetition of digits is not allowed, and which are not divisible by 3 , is equal to__________ JEE Main 2024 (08 Apr Shift 1) JEE Main Previous Year Paper

Q81.Let 𝑃= 𝑧∈ℂ: 𝑧+ 2 −3𝑖≤1 and 𝑄= 𝑧∈ℂ: 𝑧1 + 𝑖+ ¯𝑧1 −𝑖≤−8. Let in 𝑃∩𝑄, 𝑧−3 + 2𝑖 be maximum and minimum at 𝑧1 and 𝑧2 respectively. If 𝑧12 + 2𝑧2 = 𝛼+ 𝛽√2, where 𝛼, 𝛽 are integers, then 𝛼+ 𝛽 equals __________

Q82.All the letters of the word GTWENTY are written in all possible ways with or without meaning and these words are written as in a dictionary. The serial number of the word GTWENTY IS 11C2 11C9

Q61.The number of real roots of the equation √𝑥2 - 4𝑥+ 3 + √𝑥2 - 9 = √4𝑥2 - 14𝑥+ 6, is: (1) 0 (2) 1 (3) 3 (4) 2

Q61.The number of integral solution 𝑥 of 7 ≥0 is log𝑥+ 2𝑥- 3 2 (1) 7 (2) 8 (3) 6 (4) 5

Q61.The equation e4x + 8e3x + 13e2x −8ex + 1 = 0, x ∈R has : (1) four solutions two of which are negative (2) two solutions and both are negative (3) no solution (4) two solutions and only one of them is negative

Q61.Let λ ≠0 be a real number. Let α, β be the roots of the equation 14x2 −31x + 3λ = 0 and α, γ be the roots of the equation 35x2 −53x + 4λ = 0. Then 3αβ and 4αγ are the roots of the equation : (1) 7x2 + 245x −250 = 0 (2) 7x2 −245x + 250 = 0 (3) 49x2 −245x + 250 = 0 (4) 49x2 + 245x + 250 = 0

Q61.Let α, β be the roots of the quadratic equation x2 + √6x + 3 = 0. Then α15+β15+α10+β10α23+β23+α14+β14 (1) 81 (2) 9 (3) 72 (4) 729