Practice Questions

Q69.If the mean of the following probability distribution of a random variable X : X 0 2 4 6 8 46 is , then the variance of the distribution is P(X) a 2a a + b 2b 3b 9 (1) 173 (2) 566 27 81 (3) 151 (4) 581 27 81

Q69.Let 𝑓: →𝑅→0, ∞ be strictly increasing function such that lim 𝑓7𝑥 1. Then, the value of lim 𝑓5𝑥 is 𝑥→∞ 𝑓𝑥= 𝑥→∞ 𝑓𝑥−1 equal to (1) 4 (2) 0 (3) 7 (4) 1 5

Q69.If R is the smallest equivalence relation on the set {1, 2, 3, 4} such that {(1, 2), (1, 3)} ⊂R, then the number of elements in R is ______. (1) 10 (2) 12 (3) 8 (4) 15 Q70. ⎡ 2 1 2 ⎤ ⎡ 1 2 0⎤ Let A = 6 2 11 and P = 5 0 2 . The sum of the prime factors of P−1AP −2I is equal to ⎣ 3 3 2 ⎦ ⎣ 7 1 5⎦ (1) 26 (2) 27 (3) 66 (4) 23

Q69.The mean and standard deviation of 20 observations are found to be 10 and 2 . respectively. On rechecking, it was found that an observation by mistake was taken 8 instead of 12. The correct standard deviation is (1) 1.8 (2) 1.94 (3) √3.96 (4) √3.86

Q70.Let A be a square matrix such that AAT = I. Then 12 A[( A + AT)2 + (A −AT)2] (1) A2 + I (2) A3 + I (3) A2 + AT (4) A3 + AT

Q70.Let A = {1, 3, 7, 9, 11} and B = {2, 4, 5, 7, 8, 10, 12}. Then the total number of one-one maps f : A →B , such that f(1) + f(3) = 14, is : (1) 480 (2) 240 (3) 120 (4) 180

Q70.Let B = [ 11 35 ] then 2β −α is equal to (1) 16 (2) 2 (3) 8 (4) 10 is equal to cot−1 √1−x1+x )dx

Q70.If the system of equations x + 4y −z = λ, 7x + 9y + μz = −3, 5x + y + 2z = −1 has infinitely many solutions, then (2μ + 3λ) is equal to : (1) 3 (2) -3 (3) -2 (4) 2 where a > 0 and g(x) = (f(x ∣) −|f(x)|)/2. Then the function

Q70.If 𝐴= √2 1 , 𝐵1 , 𝐶= 𝐴𝐵𝐴𝑇 and 𝑋= 𝐴𝑇𝐶2𝐴, then det 𝑋 is equal to: −1 √2 1 1 (1) 243 (2) 729 (3) 27 (4) 891

Q70.If the system of linear equations 𝑥- 2𝑦+ 𝑧= - 4 2𝑥+ 𝛼𝑦+ 3𝑧= 5 3𝑥- 𝑦+ 𝛽𝑧= 3 has infinitely many solutions, then 12𝛼+ 13𝛽 is equal to (1) 60 (2) 64 (3) 54 (4) 58

Q70.Consider the relations 𝑅1 and 𝑅2 defined as 𝑎𝑅1𝑏⇔𝑎2 + 𝑏2 = 1 for all 𝑎, 𝑏, ∈𝑅 and 𝑎, 𝑏𝑅2𝑐, 𝑑⇔𝑎+ 𝑑= 𝑏+ 𝑐 for all 𝑎, 𝑏, 𝑐, 𝑑∈𝑁× 𝑁. Then (1) Only 𝑅1 is an equivalence relation (2) Only 𝑅2 is an equivalence relation (3) 𝑅1 and 𝑅2 both are equivalence relation (4) Neither 𝑅1 nor 𝑅2 is an equivalence relation

Q70.For the function f(x) = (cos x) −x + 1, x ∈R, between the following two statements (S1) f(x) = 0 for only one value of x in [0, π]. (S2) f(x) is decreasing in [0, π2 ] and increasing in [ π2 , π]. (1) Both (S1) and (S2) are correct. (2) Both (S1) and (S2) are incorrect. (3) Only (S2) is correct. (4) Only (S1) is correct.

Q70. x + y + z = 4, The values of m, n, for which the system of equations 2x + 5y + 5z = 17, has infinitely many solutions, x + 2y + mz = n satisfy the equation: (1) m2 + n2 −mn = 39 (2) m2 + n2 −m −n = 46 (3) m2 + n2 + m + n = 64 (4) m2 + n2 + mn = 68

Q70.Let a relation R on N × N be defined as: (x1, y1)R (x2, y2) if and only if x1 ≤x2 or y1 ≤y2 . Consider the two statements: (I) R is reflexive but not symmetric. (II) R is transitive Then which one of the following is true? (1) Both (I) and (II) are correct. (2) Only (II) is correct. (3) Neither (I) nor (II) is correct. (4) Only (I) is correct. JEE Main 2024 (04 Apr Shift 2) JEE Main Previous Year Paper

Q70.If the domain of the function 𝑓𝑥= 2𝑥+ 3 + cos-12𝑥- 1 is ( 𝛼, 𝛽], then the value of 5𝛽- 4𝛼 is equal to log𝑒 4𝑥2 + 𝑥- 3 𝑥+ 2 (1) 10 (2) 12 (3) 11 (4) 9 𝑥2𝑔𝑥𝑑𝑥

Q70.Let the mean and the variance of 6 observation 𝑎, 𝑏, 68, 44, 48, 60 be 55 and 194, respectively if 𝑎> 𝑏, then 𝑎+ 3𝑏 is (1) 200 (2) 190 (3) 180 (4) 210 Q71. 1 1 −1 −1 0 0 Let A be a 3 × 3 real matrix such that 𝐴 0 = 2 0 , 𝐴 0 = 4 0 , 𝐴 1 = 2 1 . Then, the system 1 1 1 1 0 0 𝑥 1 𝐴−3𝐼 𝑦 = 2 has 𝑧 3 (1) unique solution (2) exactly two solutions (3) no solution (4) infinitely many solutions

Q70.Let the relations R1 and R2 on the set X = {1, 2, 3, … , 20} be given by R1 = {(x, y) : 2x −3y = 2} and R2 = {(x, y) : −5x + 4y = 0}. If M and N be the minimum number of elements required to be added in R1 and R2 , respectively, in order to make the relations symmetric, then M + N equals (1) 12 (2) 16 (3) 8 (4) 10 α

Q70.If the domain of the function f(x) = sin−1 ( 2x+3x−1 ) is R −(α, β), then 12αβ is equal to : (1) 32 (2) 40 (3) 24 (4) 36

Q70.Let a1, a2, . . . , a10 be 10 observations such that ∑10k=1 ak = 50 and ∑∀k<j ak ⋅aj = 1100. Then the standard deviation of a1, a2, … , a10 is equal to : (1) 5 (2) √5 (3) 10 (4) √115

Q70. x + (√2 sin α)y + (√2 cos α)z = 0 If the system of equations x + (cos α)y + (sin α)z = 0 has a non-trivial solution, then α ∈(0, π2 ) is x + (sin α)y −(cos α)z = 0 equal to : (1) 11π (2) 5π 24 24 (3) 7π (4) 3π 24 4 is (α, β], then 3α + 10β is equal to:

Q71.Let f(x) = { x−a+ a ifif −a0 <≤xx ≤a≤0 g : [−a, a] →[−a, a] is (1) neither one-one nor onto. (2) onto. (3) both one-one and onto. (4) one-one. Q72. , x < 0 ⎧ tan((a+1)x)+bx tan x For a, b > 0, let f(x) = be a continous function at x = 0. Then ba is equal to : ⎨ 3, x = 0 √ax+b2x2−√ax , x > 0 ⎩ b√ax√x (1) 6 (2) 4 (3) 5 (4) 8

Q71.Let f(x) = x5 + 2x3 + 3x + 1, x ∈R , and g(x) be a function such that g(f(x)) = x for all x ∈R . Then g(7) g′(7) is equal to : (1) 14 (2) 42 (3) 7 (4) 1

Q71.If f(x) = { 21 +−x2x,3 , 0−1≤x≤x≤3< 0 ; g(x) = { x,−x,0 <−3x ≤1≤x ≤0 , then range of (f ∘g(x)) is (1) (0, 1] (2) [0, 3) (3) [0, 1] (4) [0, 1)

Q71.Let x = mn (m, n are co-prime natural numbers) be a solution of the equation cos(2 sin−1 x) = 19 and let α, β(α > β) be the roots of the equation mx2 −nx −m + n = 0. Then the point (α, β) lies on the line (1) 3x + 2y = 2 (2) 5x −8y = −9 (3) 3x −2y = −2 (4) 5x + 8y = 9 −1 < x < 1. Then at x = 12 , the value of 225(y′ −y′′) is equal to

Q71.Let f(x) = 4 cos3 x + 3√3 cos2 x −10. The number of points of local maxima of f in interval (0, 2π) is (1) 3 (2) 4 (3) 1 (4) 2