Practice Questions

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 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.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.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.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.A wire of length 36 m is cut into two pieces, one of the pieces is bent to form a square and the other is bent to form a circle. If the sum of the areas of the two figures is minimum, and the circumference of the circle is k (meter), then ( π4 + 1)k 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.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 →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.If xϕ(x) = ∫x5 (3t2 −2ϕ′(t))dt, JEE Main 2021 (31 Aug Shift 1) JEE Main Previous Year Paper

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.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.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 ∫ 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 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 →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.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.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

Q89.If ∫ b( x2+x+12x+1 ) (x2+x+1)2 dx = a tan−1( 2x+1√3 ) + value of 9(√3a b) is equal to _________. → →

Q89.Let 𝑦= 𝑦( 𝑥) be solution of the following differential equation 𝑒𝑦𝑑𝑦 2𝑒𝑦sin𝑥+ sin𝑥cos2𝑥= 0, 𝑦 𝜋 = 0. 𝑑𝑥- 2 If 𝑦0 = loge𝛼+ 𝛽e-2, then 4 ( 𝛼+ 𝛽) is equal to .

Q89.If the area of the triangle formed by the x-axis, the normal and the tangent to the circle (x −2)2 + (y −3)2 = 25 at the point (5, 7) is A, then 24A is equal to _________.

Q89.Let x be a vector in the plane containing vectors →a = 2ˆi −ˆj + ˆk and b = ˆi + 2ˆj −ˆk. If the vector x is 17√6 → 2 is 2 , then the value of x is equal to _______. perpendicular to (3ˆi + 2ˆj −ˆk) and its projection on →a

Q89.Let the plane ax + by + cz + d = 0 bisect the line joining the points (4, −3, 1) and (2, 3, −5) at the right angles. If a, b, c, d are integers, then the minimum value of (a2 + b2 + c2 + d2) is

Q89.Let a be an integer such that all the real roots of the polynomial 2x5 + 5x4 + 10x3 + 10x2 + 10x + 10 lie in the interval (a, a + 1). Then, |a| is equal to ______. dx = αIm,n, α ∈R, then α equals

Q89.If the projection of the vector ˆi + 2ˆj + ˆk on the sum of the two vectors 2ˆi + 4ˆj −5ˆk and −λˆi + 2ˆj + 3ˆk is 1, then λ is equal to _______.