Practice Questions

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.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 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 _______.

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

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.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 {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 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

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.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 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

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.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

Q83.The students S1, S2, … , S10 are to be divided into 3 groups A, B and C such that each group has at least one student and the group C has at most 3 students. Then the total number of possibilities of forming such groups is __________.

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

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.If A = [20 −13 ], JEE Main 2021 (17 Mar Shift 1) JEE Main Previous Year Paper

Q83.For k ∈N, let α(α+1)(α+2)…….(α+20) 2 1 = ∑20K=0 α+kAk , where α > 0. Then the value of 100( A14+A15A13 ) is equal to ____________.

Q83.Let ABCD be a square of side of unit length. Let a circle C1 centered at A with unit radius is drawn. Another circle C2 which touches C1 and the lines AD and AB are tangent to it, is also drawn. Let a tangent line from the point C to the circle C2 meet the side AB at E . If the length of EB is α + √3β, where α, β are integers, then α + β is equal to ________. JEE Main 2021 (16 Mar Shift 1) JEE Main Previous Year Paper aex−b cos x+ce−x

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 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 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.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_________.