Practice Questions

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 𝐴= 2, 3, 4 and 𝐵= 8, 9, 12. Then the number of elements in the relation 𝑅= 𝑎1, 𝑏1, 𝑎2, 𝑏2 ∈𝐴× 𝐵, 𝐴× 𝐵: 𝑎1 divides 𝑏2 and 𝑎2 divides 𝑏1 is (1) 36 (2) 24 (3) 18 (4) 12 Q72. 5! 6! 7! 1 If 𝐴= 6! 7! 8! , then adj adj 2𝐴 is equal to 5!6!7! 7! 8! 9! (1) 220 (2) 28 (3) 212 (4) 216

Q71.tan-1 1 + √3 + sec-1√ 8 + 4√3 = 3 + √3 6 + 3√3 π π (1) (2) 4 2 (3) π (4) π 3 6

Q71.Let 𝑆 denote the set of all real values of 𝜆 such that the system of equations 𝜆𝑥+ 𝑦+ 𝑧= 1 𝑥+ 𝜆𝑦+ 𝑧= 1 𝑥+ 𝑦+ 𝜆𝑧= 1 is inconsistent, then ∑𝜆∈𝑆𝜆2 + 𝜆 is equal to (1) 2 (2) 12 (3) 4 (4) 6 - 1

Q71.Let 𝑦= 𝑓𝑥 represent a parabola with focus - 2, 0 and directrix 𝑦= - 2. Then 𝜋 𝑆= 𝑥∈ℝ: tan-1√𝑓𝑥+ sin-1√𝑓𝑥+ 1 = 2: (1) contains exactly two elements (2) contains exactly one element (3) is an infinite set (4) is an empty set 𝑥

Q71.Let the tangents at the points A(4, −11) and B(8, −5) on the circle x2 + y2 −3x + 10y −15 = 0 , intersect at the point C . Then the radius of the circle, whose centre is C and the line joining A and B is its tangent, is equal to (1) 3√3 (2) 2√13 4 (3) √13 (4) 2√13 3 Q72. 1−cos(x2−4px+q2+8q+16) ⎧ , x ≠2p Let x = 2 be a root of the equation x2 + px + q = 0 and f(x) = (x−2p)4 . Then ⎨ ⎩ 0, x = 2p x→2p+[f(x)]lim where [⋅] denotes greatest integer function, is (1) 2 (2) 1 (3) 0 (4) −1

Q71.The set of values of a for which x→a([xlim −5] −[2x + 2]) = 0 , where, [ζ] denotes the greatest integer less than or equal to ζ is equal to (1) (−7. 5, −6. 5) (2) (−7. 5, −6. 5] (3) [−7. 5, −6. 5] (4) [−7. 5, −6. 5)

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.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.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.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

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.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.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 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.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.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)

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 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 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

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.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 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.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.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