Practice Questions

Q87.Let 𝑀= 𝐴= 𝑎 𝑏 𝑎, 𝑏, 𝑐, 𝑑∈±3, ± 2, ± 1, 0. Define 𝑓: 𝑀→𝑍, as 𝑓𝐴= det 𝐴, for all 𝐴∈𝑀 where 𝑍 is 𝑐 𝑑: set of all integers. Then the number of 𝐴∈𝑀 such that 𝑓𝐴= 15 is equal to . 0 𝑖 𝑛𝑎 𝑏 𝑎 𝑏

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

Q88.If 𝑎𝑥+ 𝑥- 2𝑑𝑥= 22, 𝑎> 2 and 𝑥 denotes the greatest integer ≤𝑥, then -𝑎𝑥+ 𝑥𝑑𝑥 is equal to ∫-𝑎 ∫𝑎

Q88.If →a and b are unit vectors and (→a b) (7→a b) (→a b) then the angle between →a and b (in degrees) is _________. −2 (7→a → → b), y−2

Q88.Let [t] denote the greatest integer ≤t . The number of points where the function 𝑓(𝑥) = [𝑥]𝑥2 - 1 + sin 𝜋 - [𝑥+ 1], 𝑥∈( - 2, 2) is not continuous is _____ . [𝑥] + 3

Q88.If xϕ(x) = ∫x5 (3t2 −2ϕ′(t))dt, JEE Main 2021 (31 Aug Shift 1) JEE Main Previous Year Paper

Q88.Let →a, b,→cbe three mutually perpendicular vectors of the same magnitude and equally inclined at an angle θ, → with the vector →a+ b +→c. Then 36 cos2 2θ is equal to

Q88.If the normal to the curve y(x) = ∫x0 (2t2 −15t + 10)dt at a point (a, b) is parallel to the line x + 3y = −5, a > 1 , then the value of |a + 6b| is equal to ________.

Q88.Let f(x) be a polynomial of degree 6 in x, in which the coefficient of x6 is unity and it has extrema at x = −1 and x = 1 . If lim f(x) = 1, then 5 ⋅f(2) is equal to x→0 x3

Q88.Let a function g : [0, 4] →R be defined as max {t3 −6t2 + 9t −3}, 0 ≤x ≤3 ⎧ g(x) = 0≤t≤x ⎨ ⎩ 4 −x, 3 < x ≤4 then the number of points in the interval (0, 4) where g(x) is NOT differentiable, is _________. JEE Main 2021 (20 Jul Shift 2) JEE Main Previous Year Paper

Q88.Let f(x) and g(x) be two functions satisfying f(x2) + g(4 −x) = 4x3 and g(4 −x) + g(x) = 0, then the value of ∫4−4 f(x2)dx is

Q88.If →a = αˆi + βˆj + 3ˆk, →b= −βˆi −αˆj −ˆk and →c= ˆi −2ˆj −ˆk such that →a⋅→b= 1 and →b⋅→c= −3, then → 1 × is equal to _______. 3 ((→a b) ⋅→c)

Q88.Let y = y(x) be the solution of the differential equation dy = eαx+ydx; α ∈N. If y(loge 2) = loge 2 and y(0) = loge( 12 ), then the value of α is equal to ___. → →

Q88.If y = y(x), y ∈[0, π2 ) is the solution of the differential equation sec y dxdy −sin(x + y) −sin(x −y) = 0, with y(0) = 0, then 5y′( π2 ) is equal to _____. JEE Main 2021 (27 Jul Shift 1) JEE Main Previous Year Paper → → →

Q88.If ∫ sin𝑥 d𝑥= | 1 + tan𝑥| + - tan𝑥+ tan2𝑥+ 𝛾tan-1 2tan𝑥- 1 + 𝐶, when 𝐶 is constant sin3𝑥+ cos3𝑥 𝛼loge 𝛽loge1 √3 of integration, then the value of 18𝛼+ 𝛽+ 𝛾2 is 3

Q88.Let 𝑆= 𝑛∈𝑁, 𝑏, 𝑐, 𝑑∈𝑅, where 𝑖= √-1 . Then the number of 2 - digit 1 0 𝑐 𝑑= 𝑐 𝑑∀𝑎, numbers in the set 𝑆 is

Q88.If the curve, y = y(x) represented by the solution of the differential equation (2xy2 −y)dx + x dy = 0, passes through the intersection of the lines, 2x −3y = 1 and 3x + 2y = 8, then |y(1)| is equal to ___ .

Q88.For real numbers α, β, γ and δ, if (x2−1)+tan−1( x2+1x ) x2+1 γ(x2−1) x2+1 + β + δ + C where C is ∫ x2+1 dx = α loge(tan−1( x )) tan−1( x ) tan−1( x ) (x4+3x2+1) tan−1( x ) an arbitrary constant, then the value of 10(α + βγ + δ) is equal to ______ . → = 8 , then

Q88.The number of distinct real roots of the equation 3x4 + 4x3 −12x2 + 4 = 0 is _________. + C, x > 0 where C is the constant of integration, then the +

Q88. if |x| ≤2 2 ) . Let f : R →R be a function defined as f(x) = { 3(1 −|x|0 if |x| > 2 Let g : R →R be given by g(x) = f(x + 2) −f(x −2). If n and m denote the number of points in R where g is not continuous and not differentiable, respectively, then n + m is equal to ________.

Q88.Let f : [−3, 1] →R be given as f(x) = {max{√x,min{(x + 6),x2},x2}, −30 ≤x≤x≤1≤0 .If the area bounded by y = f(x) and x-axis is A sq units, then the value of 6A is equal to → → →

Q88.The difference between degree and order of a differential equation that represents the family of curves given a > 0 is _______. + √a2 ), by y2 = a(x

Q88.Let a and b respectively be the points of local maximum and local minimum of the function f(x) = 2x3 −3x2 −12x. If A is the total area of the region bounded by y = f(x), the x-axis and the lines x = a and x = b, then 4A is equal to ______.

Q88.Let y = y(x) be the solution of the differential equation xdy −ydx = √(x2 −y2)dx, x ≥1 , with y(1) = 0. If the area bounded by the line x = 1, x = eπ, y = 0 and y = y(x) is αe2π + β, then the value of 10(α + β) is equal to ___ . −b = 0 be (−3, 5, 2).

Q88.Let the normals at all the points on a given curve pass through a fixed point (a, b). If the curve passes through a −2√2 b = 3 , then (a2 + b2 + ab) is equal to _____. (3, −3) and (4, −2√2), given that