Practice Questions

Q79.If x = sin−1(sin 10) and y = cos−1 (cos 10), then y −x is equal to: (1) 10 (2) π (3) 0 (4) 7π

Q79.Considering only the principal values of inverse functions, the set A = {x ≥0 : tan−1(2x) + tan−1(3x) = π4 } (1) Is an empty set (2) Contains more than two elements (3) Contains two elements (4) Is a singleton

Q79.Let 𝑓𝑥= a𝑥 ( a > 0 ) be written as 𝑓𝑥= 𝑓1𝑥+ 𝑓2𝑥, where 𝑓1 ( 𝑥) is an even function and 𝑓2 ( 𝑥) is an odd function. Then 𝑓1𝑥+ 𝑦+ 𝑓1 ( 𝑥- 𝑦) equals: (1) 2𝑓1𝑥𝑓1𝑦 (2) 2𝑓1𝑥+ 𝑦𝑓1𝑥- 𝑦 (3) 2𝑓1𝑥𝑓2𝑦 (4) 2𝑓1𝑥+ 𝑦𝑓2𝑥- 𝑦

Q79.Let f : (−1, 1) →R be a function defined by f(x) = max{−|x|, −√1 −x2}. If at which f is not differentiable, then K has exactly (1) two elements (2) one element (3) three elements (4) five elements

Q79.If 2𝑦= cot-1√3cos𝑥+ 2 ∀𝑥∈0, cos𝑥- √3sin𝑥 2, 𝑑𝑥 (1) 𝜋 - 𝑥 (2) 2𝑥- 𝜋 (3) 𝑥- 𝜋 (4) None of these 6 3 6

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 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 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.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

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.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.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.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 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.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.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.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 : [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

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.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 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.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) = 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 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.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