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 f(x) be a function satisfying f(x) + f(π −x) = π2, ∀x ∈R. Then ∫π0 f(x) sin (1) π2 (2) 2π2 4 (3) π2 (4) π2 2

Q81.Let 5 digit numbers be constructed using the digits 0, 2, 3, 4, 7, 9 with repetition allowed, and are arranged in ascending order with serial numbers. Then the serial number of the number 42923 is _____ . 1 1 1

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

Q81.If ∫0.15−0.15 100x2 −1

Q81. lim n3 {4 + (2 + n1 )2 + (2 + n2 )2 + … + (3 −1n )2} is equal to n→∞ (1) 12 (2) 193 (3) 0 (4) 19 JEE Main 2023 (30 Jan Shift 2) JEE Main Previous Year Paper

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.The sum of all the four-digit numbers that can be formed using all the digits 2, 1, 2, 3 is equal to ____.

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

Q81.Let f(x) = ∫ (x2+1)(x2+3)2x dx. If f(3) = 21 (loge 5 −loge 6), then f(4) is equal to (1) 1 2 (loge 17 −logc 19) (2) loge 17 −loge 18 (3) 1 2 (logc 19 −logc 17) (4) logc 19 −logc 20

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.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 I(x) = ∫ x+1 dx, x > 0. If lim = 0 then I(1) is equal to x(1+xex)2 x→∞I(x) (1) e+1 e+2 −loge(e + 1) (2) e+1e+2 + loge(e + 1) (3) e+2 e+1 −loge(e + 1) (4) e+2e+1 + loge(e + 1) 6 (8[cosec x] −5[cot x])dx is equal to _______ 2 ∫ π

Q82.Let 𝛼 denote the greatest integer ≤𝛼. Then √1 + √2 + √3 + . . . . . . . . . . . . . + √120 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.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.If f : R →R be a continuous function satisfying ∫ 0π π 2 f(sin 2x) sin x dx + α ∫ 04 f(cos 2x) cos x dx = 0 , then the value of α is JEE Main 2023 (11 Apr Shift 2) JEE Main Previous Year Paper (1) √2 (2) −√3 (3) √3 (4) −√2

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.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.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.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