Practice Questions

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.The integral ∫(( x2 ) x + ( x2 ) x) log2 C (1) ( x2 ) x + ( x2 ) x + C (2) ( x2 ) x −( x2 ) x + C (3) ( x2 ) x log2( x2 ) + C (4) ( x2 ) x log2( x2 ) +

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.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.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.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 ______.

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

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.The minimum value of the function f(x) = ∫20 e|x−t|dt is (1) 2(e −1) (2) 2e −1 (3) 2 (4) e(e −1)

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.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.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 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 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.If ϕ(x) 1 π −3ϕ′(t))dt, 4 (4√2 (1) 4 (2) 8 6+√π 6+√π (3) 8 (4) 4 √π 6−√π

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 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.Let f(x) = x , x ∈R −{−1}, n ∈N, n > 2 . If f n(x) = (fofof. . . . upto n times) (x), then (1+xn) 1n lim 0 xn−2(f n(x))dx is equal to n→∞∫1