Practice Questions

Q80.Let f and g be two functions defined by f(x) = {x|x+−1|,1, xx≥0< 0 {x1, + 1, xx≥0< 0 (gof)(x) is (1) Continuous everywhere but not differentiable (2) Continuous everywhere but not differentiable at exactly at one point x = 1 (3) Differentiable everywhere (4) Not continuous at x = 1

Q80.A bag contains 6 balls. Two balls are drawn from it at random and both are found to be black. The probability that the bag contains at least 5 black balls is (1) 5 (2) 2 7 7 3 5 (3) (4) 7 6

Q80.A bag contains 6 white and 4 black balls. A die is rolled once and the number of balls equal to the number obtained on the die are drawn from the bag at random. The probability that all the balls drawn are white is (1) 1 (2) 11 4 50 (3) 1 (4) 9 5 50

Q80.The integral 16 ∫21 x3(x2+2)2dx is equal to JEE Main 2023 (25 Jan Shift 2) JEE Main Previous Year Paper (1) 11 6 + loge 4 (2) 1211 + loge 4 (3) 12 11 −loge 4 (4) 116 −loge 4 m and n are coprime natural numbers, then m2 + n2 −5 is equal to

Q80.A pair of dice is thrown 5 times. For each throw, a total of 5 is considered a success. If the probability of at 𝑘 is equal to least 4 successes is 311,𝑘 then (1) 82 (2) 75 (3) 164 (4) 123

Q80.Let 𝛺 be the sample space and 𝐴⊆𝛺 be an event. Given below are two statements: (S1): If 𝑃( 𝐴) = 0, then 𝐴= 𝜙 (S2): If 𝑃( 𝐴) = , then 𝐴= 𝛺 Then (1) only (S1) is true (2) only (S2) is true (3) both (S1) and (S2) are true (4) both (S1) and (S2) are false

Q80.Let f(x) = x + a sin x + b cos x, x ∈R be a function which satisfies π2−4 π2−4 f(x) = x + ∫π/20 sin(x + y)f(y)dy. Then (a + b) is equal to (1) −π(π + 2) (2) −2π(π + 2) (3) −2π(π −2) (4) −π(π −2)

Q80.The sum of the abosolute maximum and minimum values of the function f(x) = x2 −5x + 6 −3x + 2 in the interval [−1, 3] is equal to : (1) 10 (2) 12 (3) 13 (4) 24 π 4 x+ π4 dx is :

Q80.If ∫√sec 2x −1dx = α loge cos 2x + β + √cos 2x(1 ______.

Q80.If the equation of the normal to the curve y = (x+b)(x−2)x−a at the point (1, −3) is x −4y = 13 then the value of a + b is equal to ______

Q80.If an unbiased die, marked with -2, - 1, 0, 1, 2, 3 on its faces is thrown five times, then the probability that the product of the outcomes is positive, is : 881 521 (1) (2) 2592 2592 (3) 440 (4) 27 2592 288 1 + i ¯𝑧 12

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. 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.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.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 ∫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.Total numbers of 3-digit numbers that are divisible by 6 and can be formed by using the digits 1, 2, 3, 4, 5 with repetition, is ________

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.Let the function f : [0, 2] →R be defined as f(x) = {emin{x2,x−[x]},e[x−loge x], xx ∈[0,∈[1, 1)2] , where [t] denotes the greatest integer less than or equal to t. Then the value of the integral ∫20 xf(x)dx is (1) 1 + 3e2 (2) (e −1)(e2 + 12 ) (3) 2e −1 (4) 2e −12

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

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.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