Practice Questions

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.The domain of the definition of the function f(x) = 1 + log10(x3 −x) is: 4−x2 (1) (−1, 0) ∪(1, 2) ∪(2, ∞) (2) (1, 2) ∪(2, ∞) (3) (−2, −1) ∪(−1, 0) ∪(2, ∞) (4) (−1, 0) ∪(1, 2) ∪(3, ∞) JEE Main 2019 (09 Apr Shift 2) JEE Main Previous Year Paper

Q79.Let K be the set of all real values of x where the function f(x) = sin |x| −|x| + 2(x −π) cos |x| is not differentiable. Then the set K is equal to : (1) ϕ (an empty set) (2) (π} (3) {0} (4) {0, π}

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.Let f : R →R be differentiable at c ∈R and f(c) = 0. If g(x) = |f(x)|, then at x = c, g is: (1) not differentiable (2) not differentiable if f '(c) = 0 (3) differentiable if f '(c) = 0 (4) differentiable if f '(c) ≠0 Q80. , x < 0 ⎧ sin(p+1)x+sinxx If f(x) = is continuous at x = 0 , then the ordered pair (p, q) is equal to: ⎨ q , x = 0 √x+x2−√x , x > 0 ⎩ x3/2 (1) (−32 , −12 ) (2) (−12 , 32 ) (3) ( 52 , 12 ) (4) (−32 , 12 )

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.If cos-1𝑥- cos = 𝛼, where -1 ≤𝑥≤1, - 2 ≤𝑦≤2, 𝑥≤ then for all 𝑥, 𝑦, 4𝑥2 - 4𝑥𝑦cos𝛼+ 𝑦2 is 2 2, equal to : (1) 4cos2𝛼+ 2𝑥2𝑦2 (2) 4sin2𝛼- 2𝑥2𝑦2 (3) 2sin2α (4) 4sin2α

Q79.Let f be a differentiable function such that f(1) = 2 and f ′(x) = f(x) for all x ∈R. If h(x) = f(f(x)), then h′(1) is equal to : (1) 4e2 (2) 2e (3) 4e (4) 2e2

Q79.If [x] denotes the greatest integer ≤x, then the system of linear equations [sinθ]x + [−cosθ]y = 0, [cotθ]x + y = 0 (1) has a unique solution if θ ∈( π2 , 2π3 ) ∪(π, 7π6 ) (2) have infinitely many solution if θ ∈( π2 , 2π3 ) ∪(π, 7π6 ) (3) has a unique if θ ∈( π2 , 2π3 ) and have infinitely (4) have infinitely many solutions if θ ∈( π2 , 2π3 ) many solutions if θ ∈(π, 7π6 ) and has a unique solution if θ ∈(π, 7π6 )

Q79.If 𝑒𝑦+ 𝑥𝑦= 𝑒, the ordered pair 𝑑𝑦 𝑑2𝑦 at 𝑥= 0 is equal to 𝑑𝑥, 𝑑𝑥2 1 1 1 1 (1) - 𝑒, - 𝑒2 (2) - 𝑒, 𝑒2 (3) 1 - 1 (4) 1 1 𝑒, 𝑒2 𝑒, 𝑒2

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 : [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 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.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.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.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.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: 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.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 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.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.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