Practice Questions

Q81.Among (S1) : lim 1 + 4 + 6 + … + = 1 n→∞ n2 (2 2n) JEE Main 2023 (13 Apr Shift 1) JEE Main Previous Year Paper (S2) : lim 1 (115 + 215 + 315 + … + n15) = 161 n16 n→∞ (1) Both (S1) and (S2) are true (2) Only (S1) is true (3) Both (S1) and (S2) are false (4) Only (S2) is true

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 α > 0 . If ∫α0 √x+α−√xx (1) 2 (2) 2√2 (3) 4 (4) √2 = sin t ∫xπ x > 0 then ϕ′( 4 ) is equal to √x

Q81.Let 𝑧= 1 + 𝑖 and 𝑧1 = 1 · Then 𝜋 arg𝑧1 is equal to ¯𝑧(1 - 𝑧) + 𝑧

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 _____ .

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 value of the integral ∫ −π4 2−cos 2x (1) π2 (2) π2 6 12√3 (3) π2 (4) π2 3√3 6√3 kπ , then k is equal to _____ . 16

Q81.Let [x] denote the greatest integer ≤x. Consider the function f(x) = max{x2, 1 + [x]}. Then the value of the integral ∫20 f(x)dx is : (1) 5+4√2 (2) 8+4√2 3 3 (3) 1+5√2 (4) 4+5√2 3 3 and y) ∈R2 : y ≥0, 2x ≤y ≤√4 −(x −1)2}

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 ______

Q82.The value of the integral ∫21/2 tan−1x x (1) π loge 2 (2) 21 loge 2 (3) π 4 loge 2 (4) π2 loge 2

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.The area of the region enclosed by the curve y = x3 and its tangent at the point (–1, –1) is (1) 19 (2) 23 4 4 (3) 31 (4) 27 4 4

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.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.Let 𝛼 denote the greatest integer ≤𝛼. Then √1 + √2 + √3 + . . . . . . . . . . . . . + √120 is equal to

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.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.Let q be the maximum integral value of p in [0, 10] for which the roots of the equation x2 −px + 45 p = 0 are rational. Then the area of the region {(x, y) : 0 ≤y ≤(x −q)2, 0 ≤x ≤q} is (1) 243 (2) 25 (3) 125 (4) 164 3

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 = 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.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 A = {(x, . Then the ratio of the area of A to the area of B −(x B = y) ∈R × R : 0 ≤y −1)2}} {(x, ≤min{2x, √4 is (1) π−1 (2) π π+1 π−1 (3) π (4) π+1 π+1 π−1 −21 sin−1 2 ) is

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

Q82.Let T and C respectively, be the transverse and conjugate axes of the hyperbola 16x2 −y2 + 64x + 4y + 44 = 0 . Then the area of the region above the parabola x2 = y + 4 , below the transverse axis T and on the right of the conjugate axis C is: (1) 4√6 + 443 (2) 4√6 + 283 (3) 4√6 −443 (4) 4√6 −283