Practice Questions

Q86.Let f : [0, 3] →R be defined by f(x) = min{x −[x], 1 + [x] −x} where [x] is the greatest integer less than or equal to x. Let P denote the set containing all x ∈[0, 3] where f is discontinuous, and Q denote the set containing all x ∈(0, 3) where f is not differentiable. Then the sum of number of elements in P and Q is equal to _____.

Q86.Consider the following frequency distribution : class 10 - 20 20 - 30 30 - 40 40 - 50 50 - 60 Frequency 𝛼 110 54 30 𝛽 If the sum of all frequencies is 584 and median is 45, then |𝛼- 𝛽| is equal to . JEE Main 2021 (25 Jul Shift 1) JEE Main Previous Year Paper

Q86.Let P(a sec θ, b tan θ) and Q(a sec ϕ, b tan ϕ) where θ + ϕ = π2 , be two points on the hyperbola x2a2 −y2b2 If the ordinate of the point of intersection of normals at P and Q is −k( a2+b22b ), then k is equal to

Q86.If the system of equations kx + y + 2z = 1 3x −y −2z = 2 −2x −2y −4z = 3 has infinitely many solutions, then k is equal to ______ .

Q86.Let A = [a1a2 ] [b1b2 ] 1 1 −1 2 X = and k ∈R. If a21 + a22 = 3 (b21 + b22) and (k2 + 1)b22 ≠−2 b1b2 , then the value of k is √3 [1 k ], __________. and g(x) =

Q86.Let the mean and variance of four numbers 3, 7, x and y (x > y) be 5 and 10 respectively. Then the mean of four numbers 3 + 2x, 7 + 2y, x + y and x −y is ______.

Q86.Let X1, X2, … , X18 be eighteen observations such that ∑18i=1(Xi −α) = 36 and ∑18i=1 (Xi −β)2 = 90 , where α and β are distinct real numbers. If the standard deviation of these observations is 1 , then the value of |α −β| is _______. Q87. ⎡ 1 0 0 ⎤ ⎡1 0 0⎤ If the matrix A = 0 2 0 satisfies the equation A20 + αA19 + βA = 0 4 0 for some real numbers ⎣ 3 0 −1 ⎦ ⎣0 0 1⎦ α and β, then β −α is equal to ______.

Q86. x + a −c x + b x + a Let a, b, c, d be in arithmetic progression with common difference λ. If x −1 x + c x + b = 2 , then x −b + d x + d x + c value of λ2 is equal to________.

Q86.If the curves x = y4 and xy = k cut at right angles, then (4k)6 is equal to ___ . dx is

Q86.The maximum value of z in the following equation z = 6xy + y2, where 3x + 4y ≤100 and 4x + 3y ≤75 for x ≥0 and y ≥0 is 2 [[x2] −cos x]dx is ___________.

Q87.If f(x) = ∫ dx, (x ≥0), f(0) = 0 and f(1) = K1 , then the value of K is (x2+1+2x7)2

Q87.If the variance of 10 natural numbers 1, 1, 1, … , 1, k is less than 10, then the maximum possible value of k is ___________. 1 ) x is 1

Q87.If y = y(x) is an implicit function of x such that loge(x + y) = 4xy, then d2ydx2 at JEE Main 2021 (26 Aug Shift 1) JEE Main Previous Year Paper

Q87.Let In = ∫e1 x19(log equal to _______.

Q87.If [⋅] represents the greatest integer function, then the value of ∫ 0√π

Q87.An online exam is attempted by 50 candidates out of which 20 are boys. The average marks obtained by boys is 12 with a variance 2. The variance of marks obtained by 30 girls is also 2. The average marks of all 50 JEE Main 2021 (27 Aug Shift 2) JEE Main Previous Year Paper candidates is 15. If μ is the average marks of girls and σ2 is the variance of marks of 50 candidates, then μ + σ2 is equal to Q88. ∫2ex+3e−x4ex+7e−x dx = 141 (ux + v loge(4ex + 7e−x)) + C , where C is a constant of integration, then u + v is equal to

Q87.Let f : R →R and g : R →R be defined as f(x) = { |xx +−1|,a, xx <≥00 { (x −1)2x + 1,+ b, xx ≥0< 0 , where a, b are non-negative real numbers. If gof(x) is continuous for all x ∈R, then a + b is equal to ______ .

Q87.The value of ∫2−2 3x2 −3x −6

Q87.The minimum value of 𝛼 for which the equation sin𝑥+ 1 - sin𝑥= 𝛼 has at least one solution in 0, 2 is______.

Q87.If y1/4 + y−1/4 = 2x, and (x2 −1) dx2d2y

Q87.The area bounded by the lines y = ||x −1| −2| and y = 2 is _____.

Q87.Let A = {0, 1, 2, 3, 4, 5, 6, 7}. Then the number of bijective functions f : A →A such that f(1) + f(2) = 3 −f(3) is equal to

Q87.Let A be a 3 × 3 real matrix. If det (2 Adj (2 Adj (Adj (2A)))) = 241, then the value of det (A2) equals ______.

Q87.Let a curve y = f(x) pass through the point (2, (loge 2)2) and have slope x loge2y x for all positive real values of x. Then the value of f(e) is equal to _____. → → → → is perpendicular to and is perpendicular to + 3 −5 −4

Q87.The number of points, at which the function f(x) = |2x + 1| −3|x + 2| + x2 + x −2 , x ∈R is not differentiable, is