Practice Questions

Q81.If 𝑓1 = 1, 𝑓'1 = 3, then the derivative of 𝑓𝑓𝑓𝑥+ 𝑓𝑥2 at 𝑥= 1 is: JEE Main 2019 (08 Apr Shift 2) JEE Main Previous Year Paper (1) 9 (2) 12 (3) 15 (4) 33

Q81.For x > 1, if (2x)2y = 4e2x−2y , then (1 + loge 2x)2 dxdy is equal to (1) loge2x (2) xloge2x−loge2x (3) xloge2x (4) xloge2x+loge2x

Q81.Let, f : R →R be a function such that f(x) = x3 + x2f′(1) + xf′′(2) + f′′′(3), ∀x ∈R. Then f(2) equals (1) 30 (2) 8 (3) −4 (4) −2

Q81.The tangent to the curve, y = xex2 passing through the point (1, e) also passes through the point: (1) ( 34 , 2e) (2) (2, 3e) (3) ( 53 , 2e) (4) (3, 6e)

Q81.If x loge (loge x) −x2 + y2 = 4(y > 0), then dxdy at x = e is equal to : (1) (1+2e) (2) (2e−1) 2√4+e2 2√4+e2 (3) (1+2e) (4) e √4+e2 √4+e2

Q81.Let f(x) = 5 −|x −2| and g(x) = |x + 1|, x ∈ R. If f(x) attains maximum value at α and g(x) attains (x−1)(x2−5x+6) minimum value at β, then lim is equal to x→−αβ x2−6x+8 (1) 3 (2) 1 2 2 (3) −32 (4) −12

Q81.If 𝑆1 and 𝑆2 are respectively the sets of local minimum and local maximum points of the function, 𝑓𝑥= 9𝑥4 + 12𝑥3 - 36𝑥2 + 25, 𝑥∈𝑅, then (1) 𝑆1 = -2; 𝑆2 = {0,1} (2) 𝑆1 = -1; 𝑆2 = 0,2 (3) 𝑆1 = -2,0; 𝑆2 = {1} (4) 𝑆1 = -2,1; 𝑆2 = {0}

Q81.Let x, y be positive real numbers and m, n positive integers. The maximum value of the expression xmyn is : (1+x2 m)(1+y2n) (1) 1 (2) 1 2 (3) 1 (4) m+n 4 6mn

Q81.If the tangent to the curve 𝑦= 𝑥2 - 3, 𝑥∈𝑅, 𝑥≠± √3, at a point 𝛼, 𝛽≠0, 0 on it is parallel to the line 2𝑥+ 6𝑦- 11 = 0, then: (1) 2𝛼+ 6𝛽= 19 (2) 2𝛼+ 6𝛽= 11 (3) 6𝛼+ 2𝛽= 19 (4) 6𝛼+ 2𝛽= 9

Q81.If the tangent to the curve, y = x3 + ax–b at the point (1, –5) is perpendicular to the line, –x + y + 4 = 0, then which one of the following points lies on the curve? (1) (2, –2) (2) (2, –1) (3) (–2, 1) (4) (–2, 2) JEE Main 2019 (09 Apr Shift 1) JEE Main Previous Year Paper

Q81.Let f(x) = ex −x and g(x) = x2 −x, ∀ x ϵ R . Then the set of all x ϵ R , where the function h(x) = (fog)(x) is increasing, is: (1) [−1, −12 ] ⋃[ 21 , ∞) (2) [0, ∞) (3) [0, 12 ] ∪[1, ∞) (4) [−12 , 0] ∪[1, ∞) + C , then (where C is a constant of integration)

Q81.If 𝑚 is the minimum value of 𝑘 for which the function 𝑓𝑥= 𝑥√𝑘𝑥- 𝑥2 is increasing in the interval [0, 3] and 𝑀 is the maximum value of 𝑓 in [0, 3] when 𝑘= 𝑚, then the ordered pair ( 𝑚, 𝑀) is equal to: (1) 4, 3√3 (2) 5, 3√6 (3) 3, 3√3 (4) 4, 3√2

Q81.Let A = { x ∈R : x is not a positive integer} . Define a function f : A →R as f(x) = x−12x , then f is: (1) Injective but not surjective (2) Not injective (3) Surjective but not injective (4) Neither injective nor surjective

Q81.If the function f given by f(x) = x3 −3(a −2)x2 + 3ax + 7, for some a ∈R is increasing in (0, 1] and decreasing in [1, 5), then a root of the equation, f(x)−14 = 0, (x ≠1) is : (x−1)2 (1) 7 (2) −7 (3) 6 (4) 5

Q82.The shortest distance between the point ( 23 , 0) and the curve y = √x, (x > 0) , is (1) √3 (2) 5 2 4 (3) 3 (4) √5 2 2 π

Q82.The integral ∫ 3x13+2x11 dx, is equal to (2x4+3x2+1)4 (1) x4 + C (2) x4 + C 6(2x4+3x2+1)3 (2x4+3x2+1)3 (3) x12 + C (4) x12 + C (2x4+3x2+1)3 6(2x4+3x2+1)3 e x e x dx is equal to

Q82.If ∫ x+1 dx = f(x)√2x −1 + C, where C is a constant of integration, then f(x) is equal to: √2x−1 (1) 3 1 (x + 1) (2) 32 (x + 2) (3) 3 2 (x −4) (4) 31 (x + 4)

Q82.If 𝜃 denotes the acute angle between the curves, 𝑦= 10 - 𝑥2 and 𝑦= 2 + 𝑥2 at a point of their intersection, then tan⁡𝜃 is equal to: (1) 4 (2) 8 9 17 7 8 (3) (4) 17 15

Q82.The maximum area (in sq. units) of a rectangle having its base on the x− axis and its other two vertices on the parabola, y = 12 −x2 such that the rectangle lies inside the parabola, is : (1) 20√2 (2) 32 (3) 36 (4) 18√3

Q82.The height of a right circular cylinder of maximum volume inscribed in a sphere of radius 3 is: 2 (1) √3 (2) 3√3 (3) √6 (4) 2 √3

Q82.If ∫x5e−4x3dx = 481 e−4x3f(x) + C , where C is a constant of integration, then f(x) is equal to (1) −4x3 −1 (2) −2x3 + 1 (3) −2x3 −1 (4) 4x3 + 1 π/2 dx where [t] denotes the greatest integer less than or equal to t, is

Q82.If ∫ dx = A(tan−1( x−13 ) + x2−2x+10f(x) ) (x2−2x+10)2 (1) A = 271 and f(x) = 9(x −1) (2) A = 811 and f(x) = 3(x −1) (3) A = 541 and f(x) = 9(x −1)2 (4) A = 541 and f(x) = 3(x −1) JEE Main 2019 (10 Apr Shift 1) JEE Main Previous Year Paper

Q82.Let S be the set of all values of x for which the tangent to the curve y = f(x) = x3 −x2 −2x at (x, y) is parallel to the line segment joining the points (1, f(1)) and (−1, f(−1)), then S is equal to (1) {−13 , −1} (2) {−13 , 1} (3) { 31 , 1} (4) { 13 , −1} 3 xdx is equal to Q83. ∫sec2x ⋅cot 4 3 x + C (1) 3tan−13 x + C (2) −34 tan−4 (3) −3tan−13 x + C (4) −3cot−13 x + C π/2 sin3x dx is:

Q82.Let f be a differentiable function from R to R such that |f(x) −f(y)| ≤2|x −y|3/2, for all x, y ∈R. If 1 f(0) = 1 then ∫ f 2(x)dx is equal to 0 (1) 0 (2) 1 (3) 2 (4) 21

Q82.If ∫esecx(secx tan xf(x) + (secx tan x + sec2x))dx = esecxf(x) + C, then a possible choice of f(x) is: (1) secx −tanx −12 (2) secx + tanx + 12 (3) xsecx + tanx + 12 (4) secx + xtanx −12