Practice Questions

Q81.If ∫0.15−0.15 100x2 −1

Q81.A person forgets his 4-digit ATM pin code. But he remembers that in the code all the digits are different, the greatest digit is 7 and the sum of the first two digits is equal to the sum of the last two digits. Then the maximum number of trials necessary to obtain the correct code is________.

Q81.If 𝑎 and 𝑏 are the roots of the equation 𝑥2 - 7𝑥- 1 = 0, then the value of 𝑎21 + 𝑏21 𝑎19 + 𝑏19

Q81.Let 𝜆∈ℝ and let the equation 𝐸 be |𝑥| 2 - 2 | 𝑥| + | 𝜆- 3 | = 0. Then the largest element in the set 𝑆= {𝑥+ 𝜆: 𝑥 is an integer solution of 𝐸} is ______

Q81.The value of the integral ∫21 ( t4+1t6+1 )dt is : (1) tan−1 12 + 31 tan−1 8 −π3 (2) tan−1 2 −13 tan−1 8 + π3 (3) tan−1 2 + 13 tan−1 8 −π3 (4) tan−1 21 −13 tan−1 8 + π3 dx is equal to

Q81.Let 𝑎, 𝑏, 𝑐 be the three distinct positive real numbers such that 2𝑎log𝑒𝑎= 𝑏𝑐log𝑒𝑏 and 𝑏log𝑒2 = 𝑎log𝑒𝑐 Then 6𝑎+ 5𝑏𝑐 is equal to ______.

Q81.The number of ways of giving 20 distinct oranges to 3 children such that each child gets at least one orange is _____ 1 15

Q81.If ∫3 m n2 1 |loge x|dx = n loge( e ), where 3 _____ .

Q81.The number of words, with or without meaning, that can be formed using all the letters of the word ASSASSINATION so that the vowels occur together, is _____ .

Q82.Number of 4-digit numbers (the repetition of digits is allowed) which are made using the digits 1, 2, 3 and 5 , and are divisible by 15 , is equal to Q83. ∞ 𝑛3 ( (2𝑛)!) + (2𝑛- 1) (𝑛!) 𝑏 ∞ 1 ∑𝑛= 0 ( 𝑛! ) ( ( 2𝑛) ! ) = 𝑎𝑒+ 𝑒+ 𝑐 where 𝑎, 𝑏, 𝑐 ∈ℤ and 𝑒= ∑𝑛= 0 𝑛! Then 𝑎2 - 𝑏+ 𝑐 is equal to _______ JEE Main 2023 (30 Jan Shift 1) JEE Main Previous Year Paper

Q82.Let 𝑎1, 𝑎2, … … , 𝑎𝑛 be in A.P. If 𝑎5 = 2𝑎7 and 𝑎11 = 18, then 12 + + + + … . . + √𝑎10 √𝑎11 √𝑎11 √𝑎12 √𝑎17 √𝑎18 is equal to _____ .

Q82.A boy needs to select five courses from 12 available courses, out of which 5 courses are language courses. If he can choose at most two language courses, then the number of ways he can choose five courses is

Q82.The number of permutations, of the digits 1, 2, 3, … , 7 without repetition, which neither contain the string 153 nor the string 2467, is _______ .

Q82.Let for x ∈R, S0(x) = x, Sk(x) = Ckx + k ∫x0 Sk−1(t)dt where k = 1, 2, 3, … Then S2(3) + 6C3 is equal to _______. C0 = 1, Ck = 1 −∫10 Sk−1(x)dx,

Q82.Let 𝑎1 = 8, 𝑎2, 𝑎3, … . 𝑎𝑛 be an A.P. If the sum of its first four terms is 50 and the sum of its last four terms is 170 , then the product of its middle two terms is _____ .

Q82.Suppose 𝑎1, 𝑎2, 2, 𝑎3, 𝑎4 be in an arithmetico-geometric progression. If the common ratio of the corresponding geometric progression is 2 and the sum of all 5 terms of the arithmetico-geometric progression is 49 , then 𝑎4 is 2 equal to ______________

Q82.If ∫π0 5cos x(1+cos x cos 3x+cos21+5cos xx+cos3 x cos 3x)dx = JEE Main 2023 (01 Feb Shift 2) JEE Main Previous Year Paper

Q82.Let [t] denote the greatest integer ≤t. Then π 5π 6

Q82.The coefficient of 𝑥18 in the expansion of 𝑥4 - is ____________ 𝑥3

Q82.Let [t] denote the greatest integer function. If α + β√2 + γ√3 + δ√5, then α + β + γ + δ is equal to ∫2.40 [x2]dx =

Q82.Let f be a differentiable function defined on [0, π2 ] 2 e ∀x f(x) + ∫x0 f(t)√1 −(loge(f(t)))2dt = ∈[0, π2 ], then {6 loge(f( π6 ))} is equal to

Q82.If the sum of the series + − 1 + 1 − + − 1 + 1 − 1 + . . . . . is αβ 22⋅3 2⋅32 33 23⋅3 22⋅32 2⋅33 34 ( 21 −13 ) + ( 221 − 2⋅31 + 321 ) ( 231 1 ) ( 241 1 )+. , where α and β are co-prime, then α + 3β is equal to ________.

Q82.In an examination, 5 students have been allotted their seats as per their roll numbers. The number of ways, in which none of the students sits on the allotted seat, is

Q83.Let 𝑆= 109 + 108 + 107 + … . + 2 + 1 Then the value of 16𝑆- ( 25 -54 is equal to 5 52 5107 5108. ) 1 1 680 4 is equal to

Q83.Let the equations of two adjacent sides of a parallelogram 𝐴𝐵𝐶𝐷 be 2𝑥- 3𝑦= - 23 and 5𝑥+ 4𝑦= 23. If the equation of its one diagonal 𝐴𝐶 is 3𝑥+ 7𝑦= 23 and the distance of 𝐴 from the other diagonal is 𝑑, then 50𝑑2 is equal to ______________