Practice Questions

Q80.Let A be a set of all 4 -digit natural numbers whose exactly one digit is 7. Then the probability that a randomly chosen element of A leaves remainder 2 when divided by 5 is: (1) 1 (2) 122 5 297 (3) 97 (4) 2 297 9

Q80.Four dice are thrown simultaneously and the numbers shown on these dice are recorded in 2 × 2 matrices. The probability that such formed matrices have all different entries and are non-singular, is: (1) 45 (2) 23 162 81 (3) 22 (4) 43 81 162

Q80.Words with or without meaning are to be formed using all the letters of the word EXAMINATION. The probability that the letter M appears at the fourth position in any such word is: JEE Main 2021 (20 Jul Shift 1) JEE Main Previous Year Paper (1) 1 (2) 1 66 11 (3) 1 (4) 2 9 11

Q81.Let z1 and z2 be two complex numbers such that arg(z1 −z2) = π4 and z1, z2 satisfy the equation |z −3| =Re (z). Then the imaginary part z1 + z2 is equal to

Q81.Let z1, z2 be the roots of the equation z2 + az + 12 = 0 and z1, z2 form an equilateral triangle with origin. Then, the value of |a| is

Q81.The total number of numbers, lying between 100 and 1000 that can be formed with the digits 1, 2, 3, 4, 5, if the repetition of digits is not allowed and numbers are divisible by either 3 or 5, is

Q81.If the least and the largest real values of 𝛼, for which the equation 𝑧+ 𝛼𝑧- 1 + 2𝑖= 0 𝑧∈𝐶 and 𝑖= √-1 has a solution, are 𝑝 and 𝑞 respectively; then 4𝑝2 + 𝑞2 is equal to_______.

Q81.Let z and w be two complex numbers such that w = zz −2z + 2, z−3iz+i = 1 and Re (w) has minimum value. Then, the minimum value of n ∈N for which wn is real, is equal to _______.

Q82.Let {an}∞n=1 be a sequence such that a1 = 1, a2 = 1 and an+2 = 2an+1 + an for all n ≥1. Then the value of 47 ∑∞n=1( 23nan ) is equal to ________.

Q82.Consider an arithmetic series and a geometric series having four initial terms from the set {11, 8, 21, 16, 26, 32, 4}. If the last terms of these series are the maximum possible four digit numbers, then the number of common terms in these two series is equal to _______.

Q82.There are 5 students in class 10, 6 students in class 11 and 8 students in class 12 . If the number of ways, in which 10 students can be selected from them so as to include at least 2 students from each class and at most 5 students from the total 11 students of class 10 and 11 is 100𝑘, then 𝑘 is equal to + … . upto ∞ 2 + 2 6 10 log0 . 25 3 + 3 33

Q82.Let S = {1, 2, 3, 4, 5, 6, 9} . Then the number of elements in the set T = {A ⊆S : A ≠ϕ and the sum of all the elements of A is not a multiple of 3} is Q83. 3 × 722 + 2 × 1022 −44 when divided by 18 leaves the remainder

Q82.All the arrangements, with or without meaning, of the word FARMER are written excluding any word that has two 𝑅 appearing together. The arrangements are listed serially in the alphabetic order as in the English dictionary. Then the serial number of the word FARMER in this list is _____ .

Q82.The sum of all the elements in the set {n ∈{1, 2, … … , 100} ∣ H.C.F. of n and 2040 is 1} is equal to __________.

Q82.A number is called a palindrome if it reads the same backward as well as forward. For example 285582 is a six digit palindrome. The number of six digit palindromes, which are divisible by 55, is ________.

Q82.Let Sn(x) = loga1/2 x + loga1/3 x + loga1/6 x + loga1/11 x + loga1/18 x + loga1/27 x + … up to n-terms, where a > 1 . If S24(x) = 1093 and S12(2x) = 265, then value of a is equal to _____ . k )

Q82.Let z be those complex numbers which satisfy |z + 5| ≤4 and z(1 + i) + z(1 −i) ⩾−10, i = √−1. If the maximum value of |z + 1|2 is α + β√2 , then the value of (α + β) is

Q83.Let n ∈N and [x] denote the greatest integer less than or equal to x. If the sum of (n + 1) terms of nC0, 3 ⋅nC1, 5 ⋅nC2, 7 ⋅nC3, … is equal to 2100 ⋅101, then 2[ n−12 ] is equal to n is equal to :

Q83.The total number of 4 -digit numbers whose greatest common divisor with 18 is 3 is _____.

Q83.The sum of all 3 -digit numbers less than or equal to 500, that are formed without using the digit 1 and they all are multiple of 11, is ______.

Q83.Let n be a non-negative integer. Then the number of divisors of the form 4n + 1 of the number (10)10 ⋅(11)11 ⋅(13)13 is equal to _____.

Q83.Let 𝐴= {𝑛∈𝑁: 𝑛 is a 3 - digit number } 𝐵= 9𝑘+ 2: 𝑘∈𝑁 and 𝐶= 9𝑘+ 𝑙: 𝑘∈𝑁 for some 𝑙0 < 𝑙< 9. If the sum of all the elements of the set 𝐴∩𝐵∪𝐶 is 274 × 400, then 𝑙 is equal to Q84. 3 -1 -2 Let 𝑃= 2 0 𝛼 , where 𝛼∈𝑅. Suppose 𝑄= 𝑞𝑖𝑗 is a matrix satisfying 𝑃𝑄= 𝑘𝐼3 for 3 -5 0 𝑘 𝑘2 some non-zero 𝑘∈𝑅. If 𝑞23 = - 8 and 𝑄= 2 , then 𝛼2 + 𝑘2 is equal to_________.

Q83.Let tan α, tan β and tan γ; α, β, γ ≠(2n−1)π2 , OC, respectively, where O is origin. If circumcentre of ΔABC coincides with origin and its orthocentre lies 2 on y-axis, then the value of ( coscos3α+cosα⋅cos3β+cosβ⋅cos γ 3γ ) is equal to :

Q83.The locus of the point of intersection of the lines (√3)kx + ky −4√3 = 0 and √3x −y −4(√3)k conic, whose eccentricity is a −b −tan( 2θ )

Q83.A line is a common tangent to the circle (x −3)2 + y2 = 9 and the parabola y2 = 4x. If the two points of contact (a, b) and (c, d) are distinct and lie in the first quadrant, then 2(a + c) is equal to ___ . JEE Main 2021 (25 Feb Shift 2) JEE Main Previous Year Paper