Practice Questions

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

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

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

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.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

Q83.Let the area of the region {(x, y) : |2x −1| ≤y ≤x2 −x , 0 ≤x ≤1} be A . Then (6A + 11)2 is equal to _____ .

Q83.Let 𝑓𝑥= ∑𝑘=10 1 𝑘· 𝑥𝑘, 𝑥∈ℝ, if 2𝑓2 + 𝑓'2 = 1192𝑛+ 1 then 𝑛 is equal to ______.

Q83.Number of 4-digit numbers that are less than or equal to 2800 and either divisible by 3 or by 11 , is equal to _____ . JEE Main 2023 (31 Jan Shift 1) JEE Main Previous Year Paper 2 30

Q83.If the area of the region bounded by the curves y2 −2y = −x and x + y = 0 is A , then 8A =

Q83.The number of 3-digit numbers, that are divisible by either 2 or 3 but not divisible by 7 is _____ .

Q83.If A is the area in the first quadrant enclosed by the curve C : 2x2 −y + 1 = 0 , the tangent to C at the point (1, 3) and the line x + y = 1 , then the value of 60A is................ (x5+1)2 y = , x > 0 . If y(1) = 2, then x7

Q83.The number of 9 digit numbers, that can be formed using all the digits of the number 123412341 so that the even digits occupy only even places, is ______ 1

Q83.If the area enclosed by the parabolas P1 : 2y = 5x2 and P2 : x2 −y + 6 = 0 is equal to the area enclosed by P1 and y = αx, α > 0, then α3 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 ______________

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

Q84.The number of integral terms in the expansion of 3 2 + 5

Q84.Let the solution curve x = x(y), 0 < y < π2 , of the differential equation (loge(cos y))2 cos y dx −(1 + 3x loge(cos y)) sin y dy = 0 satisfy x( π3 ) = 2 loge1 2 . If x( π6 ) = loge m−loge1 n , where m and n are coprime, then mn is equal to −−−

Q84.If the solution curve of the differential equation (y −2 loge x)dx + (x loge x2)dy = 0, x > 1 passes through the points (e, 34 ) and (e4, α) , then α is equal to _______