Practice Questions

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

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

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.If a + α = 1, b + β = 2 and af(x) + αf( x1 ) = bx + xβ , x ≠0, then the value of the expression f(x)+f(x+ x ___________.

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 |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(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 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.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.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.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 : [−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.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.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.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

Q89.Let →a = ˆi + ˆj + ˆk, b and →c= ˆj −ˆk be three vectors such that →a× b =→cand →a⋅ b = 1. If the length of → projection vector of the vector b on the vector →a×→cis l, then the value of 3l2 is equal to _____.

Q89.The area (in sq. units) of the region bounded by the curves x2 + 2y −1 = 0, y2 + 4x −4 = 0 and y2 −4x −4 = 0 in the upper half plane is _________.

Q89.Let S be the mirror image of the point Q(1, 3, 4) with respect to the plane 2x −y + z + 3 = 0 and let R(3, 5, γ) be a point of this plane. Then the square of the length of the line segment SR is

Q89.The graphs of sine and cosine functions, intersect each other at a number of points and between two consecutive points of intersection, the two graphs enclose the same area A. Then A4 is equal to →

Q89.If y = y(x) is the solution of the equation esin y cos y dxdy + esin y cos x = cos x, y(0) = 0; then 1 + y( π6 ) + √32 y( π3 ) + √21 y( π4 ) is equal to _______.

Q89.Let →a = ˆi −αˆj + βˆk, b = 3ˆi + βˆj −αˆk and →c= −αˆi −2ˆj + ˆk, where α and β are integers. If →a⋅ b = −1 and → → is equal to ______. × b ⋅→c= 10, then (→a b) ⋅→c

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 →𝑎= 2 ^𝑖- ^𝑗+ 2 ^𝑘 and →𝑏= ^𝑖+ 2 ^𝑗- ^𝑘. Let a vector →𝑣 be in the plane containing →𝑎 and →𝑏. If →𝑣 is 2 is equal to _____. perpendicular to the vector 3 ^𝑖+ 2 ^𝑗- ^𝑘 and its projection on →𝑎 is 19 units, then |2→𝑣|