Practice Questions

Q71.The value of 1+2−3+4+5−6+…+(3n−2)+(3n−1)−3n lim is n→∞ √2n4+4n+3−√n4+5n+4 (1) √2+1 + 2 (2) 3(√2 1) (3) 3 + 2 (√2 1) (4) 2√23

Q71.Let 𝑓𝑥= 𝑥2 - 𝑥+ -𝑥+ 𝑥, where 𝑥∈ℝ and 𝑡 denotes the greatest integer less than or equal to 𝑡. Then, 𝑓 is (1) continuous at 𝑥= 0, but not continuous at 𝑥= 1 (2) continuous at 𝑥= 1, but not continuous at 𝑥= 0 (3) continuous at 𝑥= 0 and 𝑥= 1 (4) not continuous at 𝑥= 0 and 𝑥= 1 1

Q71.Let P(x0, y0) be the point on the hyperbola 3x2 −4y2 = 36 , which is nearest to the line 3x + 2y = 1 . Then √2(y0 −x0) is equal to : (1) −3 (2) 9 (3) −9 (4) 3

Q72.If the domain of the function 𝑓𝑥= where 𝑥 is greatest integer ≤𝑥, is [2, 6 ) , then its range is 1 + 𝑥2, 5 2 9 27 18 9 5 2 (1) 26, 5 - 29, 109, 89, 53 (2) 26, 5 (3) 5 2 - 9 27 18 9 (4) 5 2 37, 5 29, 109, 89, 53 37, 5 3

Q72.Consider the following statements: P : I have fever Q : I will not take medicine R : I will take rest The statement "If I have fever, then I will take medicine and I will take rest" is equivalent to: JEE Main 2023 (30 Jan Shift 2) JEE Main Previous Year Paper (1) ((~P) ∨~Q) ∧((~P) ∨R) (2) ((~P) ∨−Q) ∧((~P) ∨~R) (3) (P ∨Q) ∧((~P) ∨R) (4) (P ∨~Q) ∧(P ∨~R)

Q72.The range of 𝑓𝑥= 4sin-1 𝑥2 is 𝑥2 + 1 (1) [0, 2𝜋] (2) [0, 𝜋] (3) [0, 2𝜋) (4) [0, 𝜋) 𝜋 4 𝑒-𝑥tan 50 𝑥𝑑𝑥 Q73. 𝑒-𝜋4 + ∫0 The value of 𝜋 ∫04 𝑒-𝑥(tan49𝑥+ tan51𝑥)𝑑𝑥 (1) 51 (2) 50 (3) 25 (4) 49 JEE Main 2023 (13 Apr Shift 2) JEE Main Previous Year Paper

Q72.The number of symmetric matrices of order 3, with all the entries from the set {0, 1, 2, 3, 4, 5, 6, 7, 8, 9} is (1) 610 (2) 106 (3) 910 (4) 109 Q73. ⎡ 1 3 α⎤ ⎡ α ⎤ Let B = 1 2 3 , α > 2 be the adjoint of a matrix A and |A| = 2. Then [α −2α α ]B −2α is equal to ⎣ α α 4 ⎦ ⎣ α ⎦ (1) 0 (2) 16 (3) −16 (4) 32

Q72.If 𝐼𝑥= ∫𝑒sin2𝑥cos𝑥 sin2𝑥- sin𝑥𝑑𝑥 and 𝐼0 = 1, then 𝐼 𝜋 is equal to 3 (1) -1 34 (2) 1 34 2𝑒 2𝑒 3 (3) -𝑒 4 (4) 𝑒 34

Q72. x→0((lim 1−cos2(3x)cos3(4x) )( (loge(2x+1))5sin3(4x) )) is equal to (1) 15 (2) 9 (3) 18 (4) 24

Q72.Let the system of linear equations 𝑥+ 𝑦+ 𝑘𝑧= 2 2𝑥+ 3𝑦- 𝑧= 1 3𝑥+ 4𝑦+ 2𝑧= 𝑘 have infinitely many solutions. Then the system 𝑘+ 1 𝑥+ 2𝑘- 1 𝑦= 7 2𝑘+ 1𝑥+ 𝑘+ 5𝑦= 10 has : (1) infinitely many solutions (2) unique solution satisfying 𝑥- 𝑦= 1 (3) no solution (4) unique solution satisfying 𝑥+ 𝑦= 1

Q72.Let 𝑆 be the set of all solutions of the equation cos-12𝑥- 2cos-1√1 - 𝑥2 = 𝜋, 𝑥∈-1 2, 12. Then ∑𝑥∈𝑆2sin-1𝑥2 is equal to -2𝜋 (1) 0 (2) 3 (3) 𝜋- sin-1√3 (4) 𝜋- 2sin-1√3 4 4

Q72.The number of values of r ∈{p, q, ~p, ~q} for which ((p ∧q) ⇒(r ∨q) ∧((p ∧r) ⇒q) is a tautology, is : (1) 1 (2) 2 (3) 4 (4) 3

Q72.If the domain of the function f(x) = loge(4x2 + 11x + 6) + sin−1(4x + 3) + cos−1( 10x+63 ) is (α, β] , then 36|α + β| is equal to (1) 54 (2) 72 (3) 63 (4) 45

Q73.Let △, ∇∈{∧, ∨} be such that (p →q) △(p∇q) is a tautology. Then (1) △= ∧, ∇= ∨ (2) △= ∨, ∇= ∧ (3) △= ∨, ∇= ∨ (4) △= ∧, ∇= ∧

Q73.The value of the integral ∫-log𝑒2log𝑒2 𝑒𝑥log𝑒𝑒𝑥+ (1) √2 ( 2 + √5 ) 2 √5 (2) ( 2 + √5 ) 2 √5 - log𝑒 √1 + √5 2 log𝑒 √1 + √5 + 2 2 ) 2 ( 2 + ( 3 √5 √2 - √5 √5 ) √5 (3) (4) - + log𝑒 2 log𝑒 + 2 √1 √5 + √1 √5

Q73.Suppose 𝑓: 𝑅→0, ∞ be a differentiable function such that 5𝑓𝑥+ 𝑦= 𝑓𝑥· 𝑓𝑦, ∀ 𝑥, 𝑦∈𝑅, If 𝑓3 = 320, then ∑𝑛=5 0 𝑓𝑛 is equal to: (1) 6875 (2) 6575 (3) 6825 (4) 6528 JEE Main 2023 (30 Jan Shift 1) JEE Main Previous Year Paper

Q73.The statement B ⇒((~A) ∨B) is not equivalent to : (1) B ⇒(A ⇒B) (2) A ⇒(A ⇔B) (3) A ⇒((~A) ⇒B) (4) B ⇒((~A) ⇒B) ¯¯

Q73.Let the six numbers a1, a2, . . . , a6 be in A. P. and a1 + a3 = 10 .If the mean of these six numbers is 192 and their variance is σ2 , then 8σ2 is equal to (1) 220 (2) 210 (3) 200 (4) 105

Q73.Let the mean of 6 observations 1, 2, 4, 5, x and y be 5 and their variance be 10 . Then their mean deviation about the mean is equal to (1) 7 (2) 3 3 (3) 8 (4) 10 3 3

Q73.Let 𝑦= 𝑓𝑥= sin3𝜋 𝜋 + 5𝑥2 + 1 2. Then, at 𝑥= 1, 3cos 3√2-4𝑥3 (1) 2𝑦' + √3𝜋2𝑦= 0 (2) 2𝑦' + 3𝜋2𝑦= 0 (3) √2𝑦' - 3𝜋2𝑦= 0 (4) 𝑦' + 3𝜋2𝑦= 0

Q73.The mean and variance of the marks obtained by the students in a test are 10 and 4 respectively. Later, the marks of one of the students is increased from 8 to 12 . If the new mean of the marks is 10. 2. then their new variance is equal to: (1) 4. 04 (2) 4. 08 (3) 3. 96 (4) 3. 92 Q74. ⎡ 1 logx y logx z ⎤ Let x, y, z > 1 and A = logy x 2 logy z . Then adj (adj A2) is equal to ⎣ logz x logz y 3 ⎦ (1) 64 (2) 28 (3) 48 (4) 24

Q73.Among the statements (S1) : (p ⇒q) ∨((~p) ∧q) is a tautology (S2) : (q ⇒p) ⇒((~p) ∧q) is a contradiction (1) Neither (S1) and (S2) is True (2) Both (S1) and (S2) are True (3) Only (S2) is True (4) Only (S1) is True

Q73.Let 9 = x1 < x2 < … < x7 be in an A.P. with common difference d. If the standard deviation of x1, x2 … , x7 ¯¯is 4 and the mean is x , then x + x6 is equal to : JEE Main 2023 (01 Feb Shift 2) JEE Main Previous Year Paper + 1 ) (2) 34 (1) 18(1 √3 + 8 ) (4) 25 (3) 2(9 √7

Q74.Among the relations S = {(a, b) : a, b ∈R −{0}, 2 + ab > 0} and T = {(a, b) : a, b ∈R, a2 −b2 ∈Z}, (1) S is transitive but T is not (2) both S and T are symmetric (3) neither S not T is transitive (4) T is symmetric but S is not ∈Z ∩[0, 4], 1 ≤i, j ≤2 . The number of matrices A such that the sum of all entries is a

Q74.The slope of tangent at any point 𝑥, 𝑦 on a curve 𝑦= 𝑦𝑥 is 𝑥2 + 𝑦2 𝑥> 0. If 𝑦2 = 0, then a value of 𝑦8 is 2𝑥𝑦, JEE Main 2023 (10 Apr Shift 1) JEE Main Previous Year Paper (1) -4√2 (2) 2√3 (3) -2√3 (4) 4√3