Practice Questions

Q80.If the function f(x) = {a|πb|x −π|−x| ++ 3,1, xx >≤55 is continuous at x = 5, then the value of a −b is: (1) 2 (2) −2 5−π π+5 (3) 2 (4) 2 π+5 π−5

Q80.Let S be the set of all points in (−π, π) at which the function, f(x) = min{sin x, cos x} is not differentiable. Then S is a subset of which of the following? (1) {−3π4 , −π2 , π2 , 3π4 } (2) {−3π4 , −π4 , 3π4 , π4 } (3) {−π4 , 0, π4 } (4) {−π2 , −π4 , π4 , π2 }

Q80.Let 𝑓𝑥= log𝑒sin𝑥, 0 < 𝑥< 𝜋 and 𝑔𝑥= sin-1 ( 𝑒-𝑥) , (𝑥≥0) . If 𝛼 is a positive real number such that 𝑎= 𝑓𝑜𝑔' (𝛼) and 𝑏= 𝑓𝑜𝑔( 𝛼) , then (1) 𝑎𝛼2 + 𝑏𝛼+ 𝑎= 0 (2) 𝑎𝛼2 + 𝑏𝛼- 𝑎= - 2𝛼 (3) 𝑎𝛼2 - 𝑏𝛼- 𝑎= 0 (4) 𝑎𝛼2 - 𝑏𝛼- 𝑎= 1 𝑥

Q80.Let f(x) = { max(|x|,8 −2|x|,x2), 2 <|x||x|≤2≤4 differentiable. Then S (1) equals {−2, −1, 0, 1, 2} (2) equals {−2, 2} (3) is an empty set (4) equal {−2, −1, 1, 2}

Q80.The derivative of tan−1( sinx+cosxsinx−cosx ) with respect to x2 , where x ∈(0, π2 ), is (1) 2 (2) 21 (3) 2 (4) 1 3

Q80.A 2m ladder leans against a vertical wall. If the top of the ladder begins to slide down the wall at the rate 25cm / sec , then the rate (in cm/sec.) at which the bottom of the ladder slides away from the wall on the horizontal ground when the top of the ladder is 1 m above the ground is: (1) 25 (2) 25√3 25 25 (3) (4) 3 √3 JEE Main 2019 (12 Apr Shift 1) JEE Main Previous Year Paper

Q80.Let f : [0,1] →R be such that f(xy) = f(x). f(y), for all x, y ∈[0,1], and f(0) ≠0. If y = y(x) satisfies the differential equation, dx dy = f(x) with y(0) = 1 then y( 41 ) + y( 34 ) is equal to: (1) 5 (2) 2 (3) 3 (4) 4

Q80.The tangent to the curve y = x2 −5x + 5, parallel to the line 2y = 4x + 1, also passes through the point : (1) ( 14 , 27 ) (2) ( 27 , 41 ) (3) (−18 , 7) (4) ( 81 , −7)

Q80.The shortest distance between the line 𝑦= 𝑥 and the curve 𝑦2 = 𝑥– 2 is (1) 7 (2) 7 (3) 11 (4) 2 4√2 8 4√2

Q80.If f(x) is a non-zero polynomial of degree four, having local extreme points at x = –1, 0, 1; then the set S = {x ∈R : f(x) = f(0)} contains exactly (1) Two irrational and two rational numbers (2) Four rational numbers (3) Two irrational and one rational number (4) Four irrational numbers

Q80.Let f(x) = x − d−x , x ∈R wherea, b and d are non-zero real constants. Then : √a2+x2 √b2+(d−x)2 JEE Main 2019 (11 Jan Shift 2) JEE Main Previous Year Paper (1) f is an increasing function of x (2) f is a decreasing function of x (3) f ′ is not a continuous function of x (4) f is neither increasing nor decreasing function of x

Q80.Let f(x) = { x2−1,−1, 0−2≤x≤x≤2< 0 (1) differentiable at all points (2) not continuous (3) not differentiable at two points (4) not differentiable at one point

Q80.Let 𝑓: -1,3 →R be defined as 𝑥+ 𝑥, -1 ≤𝑥< 1 𝑓𝑥= 𝑥+ 𝑥, 1 ≤𝑥< 2 𝑥+ 𝑥, 2 ≤𝑥≤3, Where t denotes the greatest integer less than or equal to 𝑡. Then, 𝑓 is discontinuous at: (1) Only one point (2) Only two points (3) Four or more points (4) Only three points

Q80.Let 𝑓: 𝑅→𝑅 be a function defined as 5, 𝑖𝑓 𝑥≤1 𝑎+ 𝑏𝑥, 𝑖𝑓 1 < 𝑥< 3 𝑓𝑥= 𝑏+ 5𝑥, 𝑖𝑓 3 ≤𝑥< 5 30, 𝑖𝑓 𝑥≥5 Then 𝑓 is: (1) continuous if 𝑎= - 5 and 𝑏= 10 (2) continuous if 𝑎= 0 and 𝑏= 5 (3) not continuous for any values of 𝑎 and 𝑏 (4) continuous if 𝑎= 5 and 𝑏= 5

Q80.A helicopter is flying along the curve given by y −x 32 = 7, (x ≥0). A soldier positioned at the point ( 12 , 7) , who wants to shoot down the helicopter when it is nearest to him. Then this nearest distance is: (1) 1 (2) 1 2 6 √73 (3) 1 (4) √5 6 3 √73

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.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.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 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 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.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.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(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 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