Practice Questions

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.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.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 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.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 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.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.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 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.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.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.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.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

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.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.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

Q75.Let 𝑦= 𝑦𝑥 be a solution curve of the differential equation, 1 - 𝑥2𝑦2𝑑𝑥= 𝑦𝑑𝑥+ 𝑥𝑑𝑦, If the line 𝑥= 1 intersects the curve 𝑦= 𝑦𝑥 at 𝑦= 2 and the line 𝑥= 2 intersects the curve 𝑦= 𝑦𝑥 at 𝑦= 𝛼, then a value of 𝛼 is (1) 1 - 3𝑒2 (2) 1 + 3𝑒2 23𝑒2 + 1 23𝑒2 - 1 (3) 3𝑒2 (4) 3𝑒2 23𝑒2 - 1 23𝑒2 + 1

Q75.Let A = {1, 3, 4, 6, 9} and B = {2, 4, 5, 8, 10} . Let R be a relation defined on A × B such that R = {(a1, b1), (a2, b2) : a1 ≤b2 and b1 ≤a2}. Then the number of elements in the set R is (1) 160 (2) 52 (3) 26 (4) 180

Q75.Let A = ⌊aˆiˆj⌋⋅aij prime number p ∈(2, 13) is _____ .

Q75.Let x = x(y) be the solution of the differential equation 2(y + 2) loge(y + 2)dx + (x + 4 −2 loge(y + 2))dy = 0 , y > −1 with x(e4 −2) = 1 . Then x(e9 −2) is equal to (1) 3 (2) 49 (3) 32 (4) 10 9 3

Q75.Consider the following system of questions αx + 2y + z = 1 2αx + 3y + z = 1 3x + αy + 2z = β For some α, β ∈R . Then which of the following is NOT correct. (1) It has no solution if α = −1 and β ≠2 (2) It has no solution for α = −1 and for all β ∈R (3) It has no solution for α = 3 and for all β ≠2 (4) It has a solution for all α ≠−1 and β = 2 log(x+1)(x−2) , x ∈R is

Q75.If [ 𝑡 denotes the greatest integer ≤1, then the value of 𝑥2𝑒𝑥+ 𝑥3𝑑𝑥 is : 𝑒 ∫1 (1) 𝑒9 - 𝑒 (2) 𝑒8 - 𝑒 (3) 𝑒7 - 1 (4) 𝑒8 - 1

Q75.Let A = [10 5111 ] 1 2 −1 −2 equal to JEE Main 2023 (12 Apr Shift 1) JEE Main Previous Year Paper (1) 75 (2) 125 (3) 50 (4) 100 Q76. 1 2k 2k −1 Let Dk = n n2 + n + 2 n2 . If ∑nk=1 Dk = 96, then n is equal to _________. n n2 + n n2 + n + 2 g : D →R

Q75.Let S1 and S2 be respectively the sets of all a ∈R −{0} for which the system of linear equations ax + 2ay −3az = 1 (2a + 1) x + (2a + 3) y + (a + 1)z = 2 JEE Main 2023 (25 Jan Shift 1) JEE Main Previous Year Paper (3a + 5) x + (a + 5) y + (a + 2) z = 3 has unique solution and infinitely many solutions. Then (1) n(S1) = 2 and S2 is an infinite set (2) S1 is an infinite set an n(S2) = 2 (3) S1 = ϕ and S2 = R −{0} (4) S1 = R −{0} and S2 = ϕ