Practice Questions

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 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.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.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.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.If A is a square matrix of order 3 such that det(A) = 3 and det (adj (−4 adj (−3 adj (3 adj ((2 A)−1))))) = 2m3n , then m + 2n is equal to : (1) 2 (2) 3 (3) 6 (4) 4 JEE Main 2024 (06 Apr Shift 2) JEE Main Previous Year Paper

Q71.Let f: R - -1 →R and g: R - -5 →R be defined as fx = 2x + 3 and gx = |x | + 1 . Then the domain of the function 2 2 2x + 1 2x + 5 fog is : 5 (1) R - - (2) 𝑅 2 7 5 7 (3) R - - (4) R - - - 4 2, 4

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) = 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

Q71.Let f(x) = ax3 + bx2 + cx + 41 be such that f(1) = 40, f ′(1) = 2 and f ′(1) = 4. Then a2 + b2 + c2 is equal to: (1) 73 (2) 62 (3) 51 (4) 54

Q71.For 𝛼, 𝛽, 𝛾≠0. If sin−1𝛼+ sin−1𝛽+ sin−1𝛾= 𝜋 and 𝛼+ 𝛽+ 𝛾𝛼−𝛾+ 𝛽= 3𝛼𝛽, then 𝛾 equal to √3 1 (1) (2) 2 √2 (3) √3 - 1 (4) √3 2√2

Q71.Let 𝑓: 𝑅→𝑅 be a function defined 𝑓𝑥= 𝑥 / 4 and 𝑔𝑥= 𝑓𝑓𝑓𝑓𝑥 then 18 ∫0√2√5 1 + 𝑥41 (1) 33 (2) 36 (3) 42 (4) 39

Q71.Consider the system of linear equation x + y + z = 4μ, x + 2y + 2λz = 10μ, x + 3y + 4λ2z = μ2 + 15, where λ, μ ∈R. Which one of the following statements is NOT correct? (1) The system has unique solution if λ ≠12 and (2) The system is inconsistent if λ = 12 and μ ≠1 μ ≠1, 15 (3) The system has infinite number of solutions if (4) The system is consistent if λ ≠12 λ = 21 and μ = 15 + (loge(3 −x))−1 is [−α, β) −{γ}, then α + β + γ is

Q71.Let f(x) = 7−sin1 5x be a function defined on R. Then the range of the function f(x) is equal to ; (1) [ 71 , 61 ] (2) [ 81 , 51 ] (3) [ 71 , 51 ] (4) [ 81 , 61 ]

Q71.Let the system of equations 𝑥+ 2𝑦+ 3𝑧= 5, 2𝑥+ 3𝑦+ 𝑧= 9, 4𝑥+ 3𝑦+ 𝜆𝑧= 𝜇 have infinite number of solutions. Then 𝜆+ 2𝜇 is equal to: (1) 28 (2) 17 (3) 22 (4) 15

Q71.Let f, g : R →R be defined as : f(x) = |x −1| and g(x) = {ex,x + MARA1, xx ≥0≤0 Then the function f(g(x)) is (1) neither one-one nor onto. (2) one-one but not onto. (3) onto but not one-one. (4) both one-one and onto.

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) = 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. r 1 n22 + For α, β ∈R and a natural number n, let Ar = 2r 2 n2 −β . Then n(3n−1) 3r −2 3 2 (1) 0 (2) 4α + 2β (3) 2α + 4β (4) 2n

Q71.If the system of equations 2𝑥+ 3𝑦−𝑧= 5 𝑥+ 𝛼𝑦+ 3𝑧= −4 3𝑥−𝑦+ 𝛽𝑧= 7 has infinitely many solutions, then 13𝛼𝛽 is equal to (1) 1110 (2) 1120 (3) 1210 (4) 1220

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.If the domain of the function sin−1 ( 3x−222x−19 ) + loge ( 3x2−8x+5x2−3x−10 ) (1) 100 (2) 95 (3) 97 (4) 98

Q71.Let A = [ 10 21 ] and B = I + adj(A) + (adj A)2 + … + (adj A)10 . Then, the sum of all the elements of the matrix B is: (1) -124 (2) 22 (3) -88 (4) -110