Practice Questions

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.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 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.A water tank has the shape of an inverted right circular cone, whose semi-vertical angle is tan−1( 21 ). Water is poured into it at a constant rate of 5 cubic m/min. Then the rate (in m/min), at which the level of water is rising at the instant when the depth of water in the tank is 10 m; is: (1) 1 (2) 1 10π 15π (3) 1 (4) 2 5π π

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.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.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.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.The maximum volume in 𝑐𝑢. 𝑚 of the right circular cone having slant height 3 𝑚 is: JEE Main 2019 (09 Jan Shift 1) JEE Main Previous Year Paper (1) 2√3 𝜋 (2) 3√3 𝜋 4 (3) 6 𝜋 (4) 3𝜋

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 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 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 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

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

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.The integral ∫2𝑥3 - 1 is equal to 𝑥4 + 𝑥𝑑𝑥, (1) 2 (2) |𝑥3 + 1| 1 (𝑥3 + 1) + 𝐶 + 𝐶 log𝑒 𝑥2 2log𝑒 |𝑥3| (3) 𝑥3 + 1 (4) 1 |𝑥3 + 1| log𝑒 𝑥 + 𝐶 2log𝑒 𝑥2 + 𝐶

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.Themaximum value of the finction f(x) = 3x3 −18x2 + 27x −40 on the set S = {x ∈R : x2 + 30 ≤11x} is : (1) -122 (2) -222 (3) 122 (4) 222 JEE Main 2019 (11 Jan Shift 1) JEE Main Previous Year Paper + C, for a suitable chosen integer m and a function A(x), where C is a

Q82.Let 𝑓: 0, 2 →𝑅 be a twice differentiable function such that 𝑓''𝑥> 0, for all 𝑥∈0, 2 . If 𝜙𝑥= 𝑓𝑥+ 𝑓2 – 𝑥, then 𝜙 is (1) decreasing on 0,2 (2) increasing on 0,2 (3) increasing on ( 0,1 ) (4) decreasing on 0,1 and and decreasing on 1,2 increasing on ( 1,2 )

Q82.Let α ∈(0, π2 ) , be constant.If the integral ∫ tanx−tantanx+tanαα dx = A(x)cos2α + B(x)sin2α + C , where C is a constant of integration, then the functions A(x) and B(x) are respectively (1) x −α and loge|sin(x −α)| (2) x + α and loge|cos(x −α)| (3) x + α and loge|sin(x + α)| (4) x −α and loge|cos(x −α)| JEE Main 2019 (12 Apr Shift 2) JEE Main Previous Year Paper α+1 dx 9 = loge( 8 ) is

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.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 ∫ 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