Practice Questions

Q73.Let 𝑔𝑥= 𝑓𝑥+ 𝑓1 - 𝑥 and 𝑓"𝑥> 0, 𝑥∈0, 1. If 𝑔 is decreasing in the interval 0, 𝛼 and increasing in the interval 𝛼, 1, then tan-12𝛼+ tan-1 1 tan-1𝛼+ 1 is equal to 𝛼+ 𝛼 5π (1) π (2) 4 (3) 3π (4) 3π 4 2

Q73.Let [x] denote the greatest integer function and f(x) = max{1 + x + [x], 2 + x, x + 2[x]}, 0 ≤x ≤2 , where f is not continuous and n be the number of points in (0, 2), where f is not differentiable. Then (m + n)2 + 2 is equal to (1) 2 (2) 11 (3) 6 (4) 3 α, β > 0 , then α4 −β4 is equal to dx = α1 loge( α+1β ),

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.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

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 area of the region 𝑥, 𝑦: 𝑥2 ≤𝑦≤𝑥2 - 4, 𝑦≥1 is (1) 4 + 1) 3 (4√2 - 1) (2) 43 (4√2 (3) 3 - 1) 4 (4√2 + 1) (4) 34 (4√2 2 is

Q74.Let the mean and variance of 12 observations be 29 and 4 respectively. Later on, it was observed that two observations were considered as 9 and 10 instead of 7 and 14 respectively. If the correct variance is mn , where m and n are coprime, then m + n are coprime, then m + n is equal to (1) 315 (2) 316 (3) 314 (4) 317

Q74.A wire of length 20 m is to be cut into two pieces. A piece of length ℓ1 is bent to make a square of area 𝐴1 and the other piece of length ℓ2 is made into a circle of area 𝐴2. If 2 𝐴1 + 3 𝐴2 is minimum then 𝜋ℓ1: ℓ2 is JEE Main 2023 (31 Jan Shift 1) JEE Main Previous Year Paper equal to: (1) 6: 1 (2) 3: 1 (3) 1: 6 (4) 4: 1 𝑥2 𝑥3 𝑥𝑛 𝛼𝑡50

Q74.Let α and β be real numbers. Consider a 3 × 3 matrix A such that A2 = 3A + αI . If A4 = 21A + βI , then (1) α = 1 (2) α = 4 (3) β = 8 (4) β = −8

Q74.The area enclosed between the curves 𝑦2 + 4𝑥= 4 and 𝑦- 2𝑥= 2 is 25 22 (1) (2) 3 3 (3) 9 (4) 23 3

Q74.Let the mean and variance of 8 numbers x, y, 10, 12, 6, 12, 4, 8 be 9 and 9. 25 respectively. If x > y, then 3x −2y is equal to _______

Q74.The minimum number of elements that must be added to relation R = {(a, b), (b, c), (b, d)} on the set {a, b, c, d}, so that it is an equivalence relation is

Q74.If P is a 3 × 3 real matrix such that P T = aP + (a −1)I, where a > 1, then (1) P is a singular matrix (2) |Adj P| > 1 (3) Adj P = 21 (4) |Adj P| = 1

Q74.If ∫10 (5+2x−2x2)(1+e(2−4x))1 (1) 19 (2) −21 (3) 0 (4) 21 JEE Main 2023 (15 Apr Shift 1) JEE Main Previous Year Paper

Q74.Let A, B, C be 3 × 3 matrices such that A is symmetric and B and C are skew-symmetric. Consider the statements (S1) A13 B26 −B26 A13 is symmetric (S2) A26C 13 −C 13 A26 is symmetric Then, (1) Only S2 is true (2) Only S1 is true (3) Both S1 and S2 are false (4) Both S1 and S2 are true Q75. 1 3 √10 √10 1 −i Let A = ⎡ ⎤ and B = , where i = √−1. If M = AT BA , then the inverse of the matrix −3 1 [0 1 ] ⎣ √10 √10 ⎦ AM2023 AT is (1) [10 −2023i1 ] (2) [1−2023i 01 ] (3) [12023i 10 ] (4) [10 2023i1 ] m, such that x −cos x) + m

Q74.The number of points on the curve 𝑦= 54𝑥5 - 135𝑥4 - 70𝑥3 + 180𝑥2 + 210𝑥 at which the normal lines are parallel to 𝑥+ 90𝑦+ 2 = 0 is: (1) 2 (2) 3 (3) 4 (4) 0 3𝑒- 1 2

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

Q74.Let P(S) denote the power set of S = {1, 2, 3, … , 10} . Define the relations R1 and R2 on P(S) as AR1B if (A ∩Bc) ∪(B ∩Ac) = ϕ and AR2 B if A ∪Bc = B ∪Ac, ∀A, B ∈P(S) . Then : (1) both R1 and R2 are equivalence relations (2) only R1 is an equivalence relation (3) only R2 is an equivalence relation (4) both R1 and R2 are not equivalence relations 1 1 √3 then,

Q74.The angle of elevation of the top P of a tower from the feet of one person standing due south of the tower is 45° and from the feet of another person standing due west of the tower is 30° . If the height of the tower is 5 meters, then the distance (in meters) between the two persons is equal to JEE Main 2023 (11 Apr Shift 2) JEE Main Previous Year Paper (1) 5 2 √5 (2) 10 (3) 5 (4) 5√5

Q74.If the mean and variance of the frequency distribution xi 2 4 6 8 10 12 14 16 fi 4 4 α 15 8 β 4 5 are 9 and 15. 08 respectively, then the value of α2 + β2 −αβ is _____.

Q74.The number of relations, on the set {1, 2, 3} containing (1, 2) and (2, 3) which are reflexive and transitive but not symmetric, is _________. . If B = , then the sum of all the elements of the matrix ∑50n=1 Bn is [−1 −1 ] A[ 1 1 ]

Q74.Let X = {11, 12, 13, … . , 40, 41} and Y = {61, 62, 63, . . . , 90, 91} be the two sets of observations. If x and y ¯are their respective means and σ2 is the variance of all the observations in X ∪Y, then x + y −σ2 is equal to ________

Q74.For the system of linear equations 2x + 4y + 2az = b x + 2y + 3z = 4 2x + 5y + 2z = 8 which of the following is NOT correct? (1) It has unique solution if a = b = 6 (2) It has infinitely many solutions if a = 3, b = 6 (3) It has infinitely many solutions if a = 3, b = 8 (4) It has unique solution if a = b = 8 : = π4 } then

Q74.If 2𝑥𝑦+ 3𝑦𝑥= 20, then 𝑑𝑦 at 2, 2 is equal to: 𝑑𝑥 (1) - 2 + loge8 (2) - 3 + loge16 3 + loge4 4 + loge8 (3) - 3 + loge8 (4) - 3 + loge4 2 + loge4 2 + loge8 sec2 + tan𝑥

Q74.Area of the region 𝑥, 𝑦: 𝑥2 + 𝑦- 22 ≤4, 𝑥2 ≥2𝑦 is 8 16 (1) 𝜋+ (2) 2𝜋+ 3 3 (3) 𝜋- 8 (4) 2𝜋- 16 3 3