Practice Questions

Q72.Let 𝑓: [0, ∞) →[0, ∞) be defined as 𝑓𝑥= 𝑥𝑦𝑑𝑦 where [𝑥] is the greatest integer less than or equal to 𝑥. ∫0 Which of the following is true? (1) 𝑓 is continuous at every point in [0, ∞) and (2) 𝑓 is both continuous and differentiable except at differentiable except at the integer points. the integer points in [0, ∞) . (3) 𝑓 is continuous everywhere except at the integer (4) 𝑓 is differentiable at every point in [0, ∞) . points in [0, ∞) . 𝜋 𝜋

Q72.Let f, g : N →N such that f(n + 1) = f(n) + f(1) ∀ n ∈N and g be any arbitrary function. Which of the following statements is NOT true? (1) If f is onto, then f(n) = n∀n ∈N (2) If g is onto, then fog is one-one (3) f is one-one (4) If fog is one-one, then g is one-one

Q72.Let A and B be two 3 × 3 real matrices such that (A2 −B2) is invertible matrix. If A5 = B5 and A3 B2 = A2 B3, then the value of the determinant of the matrix A3 + B3 is equal to : (1) 2 (2) 4 (3) 1 (4) 0

Q72.Let f be any function defined on R and let it satisfy the condition: |f(x) −f(y)| ≤(x −y)2 , ∀(x, y) ∈R. If f(0) = 1, then : (1) f(x) = 0, ∀x ∈R (2) f(x) can take any value in R (3) f(x) < 0, ∀x ∈R (4) f(x) > 0, ∀x ∈R

Q72.The sum of all the local minimum values of the twice differentiable function f : R →R defined by ′′(2) x + f ′′(1) is: f(x) = x3 −3x2 −3f 2 (1) −22 (2) 5 (3) −27 (4) 0

Q72.The function 𝑓𝑥= 𝑥3 - 6𝑥2 + 𝑎𝑥+ 𝑏 is such that 𝑓2 = 𝑓4 = 0. Consider two statements: 𝑆1 there exists 𝑥1, 𝑥2 ∈2, 4, 𝑥1 < 𝑥2, such that 𝑓'𝑥1 = - 1 and 𝑓'𝑥2 = 0 . 𝑆2 there exists 𝑥3, 𝑥4 ∈2, 4, 𝑥3 < 𝑥4, such that 𝑓 is decreasing in 2, 𝑥4, increasing in 𝑥4, 4 and 2𝑓'𝑥3 = √3𝑓𝑥4 then (1) 𝑆1 is true and 𝑆2 is false (2) both 𝑆1 and 𝑆2 are false (3) both 𝑆1 and 𝑆2 are true (4) 𝑆1 is false and 𝑆2 is true JEE Main 2021 (01 Sep Shift 2) JEE Main Previous Year Paper Q73. 𝜋 sec2𝑥𝑓(𝑥)d𝑥 4 ∫2 Let f : R →R be a continuous function. Then lim 𝜋2 is equal to: 𝑥→𝜋/ 4 𝑥2 - 16 (1) 𝑓( 2 ) (2) 2𝑓( √2 ) (3) 2𝑓( 2 ) (4) 4𝑓( 2 )

Q72.The number of solutions of the equation sin−1[x2 + 13 ] + cos−1[x2 −23 ] = x2 for x ∈[−1, 1], and [x] denotes the greatest integer less than or equal to x, is : (1) 2 (2) 0 (3) 4 (4) Infinite . Then f is:

Q73.Let f : R →R be defined as f(x + y) + f(x −y) = 2f(x)f(y), f( 21 ) = −1. Then the value of ∑20k=1 sin(k) sin(k+f(k))1 is equal to : (1) cosec2 (21) cos(20) cos(2) (2) sec2(1) sec(21) cos(20) (3) cosec2 (1) cosec (21) sin(20) (4) sec2(21) sin(20) sin(2) . Then which of

Q73.Let [t] denote the greatest integer less than or equal to t. Let f(x) = x −[x], g(x) = 1 −x + [x], and h(x) = min{f(x), g(x)}, x ∈[−2, 2]. Then h is : (1) continuous in [−2, 2] but not differentiable at (2) Continous in [−2, 2] but not differentiable at more than four points in (−2, 2) exactly three poionts in (−2, 2) (3) not continuous at exactly four points in [−2, 2] (4) not continuous at exactly three points in [−2, 2] is

Q73.If cot−1(α) = cot−1 2 + cot−1 8 + cot−1 18 + cot−1 32 + … . upto 100 terms, then α is: JEE Main 2021 (17 Mar Shift 1) JEE Main Previous Year Paper (1) 1. 01 (2) 1. 00 (3) 1. 02 (4) 1. 03

Q73.A wire of length 20 m is to be cut into two pieces. One of the pieces is to be made into a square and the other into a regular hexagon. Then the length of the side (in meters) of the hexagon, so that the combined area of the square and the hexagon is minimum, is (1) 10 (2) 5 2+3√3 3+√3 (3) 10 (4) 5 3+2√3 2+√3 + … + n2

Q73.If the tangent to the curve 𝑦= 𝑥3 at the point 𝑃𝑡, 𝑡3 meets the curve again at 𝑄, then the ordinate of the point which divides 𝑃𝑄 internally in the ratio 1: 2 is: (1) 0 (2) -2𝑡3 (3) -𝑡3 (4) 2𝑡3

Q74.Let 𝑓 be any continuous function on 0, 2 and twice differentiable on 0, 2 . If 𝑓0 = 0, 𝑓1 = 1 and 𝑓2 = 2, then : (1) 𝑓"𝑥> 0 for all 𝑥∈0, 2 (2) 𝑓'𝑥= 0 for some 𝑥∈0, 2 (3) 𝑓"𝑥= 0 for some 𝑥∈0, 2 (4) 𝑓"𝑥= 0 for all 𝑥∈0, 2 2 𝜋𝑥

Q74.Let f : [0, ∞) →[0, 3] be a function defined by f(x) = {max{sin2 + cos x,t :x0>≤tπ ≤π}, x ∈[0, π] the following is true ? (1) f is continuous everywhere but not differentiable (2) f is differentiable everywhere in (0, ∞) exactly at one point in (0, ∞) (3) f is not continuous exactly at two points in (4) f is continuous everywhere but not differentiable (0, ∞) exactly at two points in (0, ∞)

Q74.The value of the integral ∫10 (1+x)(1+3x)(3+x)√xdx is: (1) π 4 (1 −√32 ) (2) π8 (1 −√36 ) (3) π 8 (1 −√32 ) (4) π4 (1 −√36 )

Q74.The value of the integral ∫ sin θ⋅sin 2θ(sin6 θ+sin41−cosθ+sin22θθ)√2 sin4 θ+3 sin2 θ+6 (1) 1 32 (2) 1 32 18 [11 −18 sin2 θ + 9 sin4 θ −2 sin6 θ] + c 18 [9 −2 sin6 θ −3 sin4 θ −6 sin2 θ] + c (3) 1 32 (4) 1 −32 18 [11 −18 cos2 θ + 9 cos4 θ −2 cos6 θ] + c 18 [9 −2 cos6 θ −3 cos4 θ −6 cos2 θ] + c

Q74.Let f(x) cos(2 sin(cot−1 √1−x )), (1) (1 −x)2f ′(x) + 2(f(x))2 = 0 (2) (1 + x)2f ′(x) + 2(f(x))2 = 0 (3) (1 −x)2f ′(x) −2(f(x))2 = 0 (4) (1 + x)2f ′(x) −2(f(x))2 = 0

Q74.Let a be a real number such that the function f(x) = ax2 + 6x −15, x ∈R is increasing in (−∞, 43 ) and decreasing in ( 34 , ∞) . Then the function g(x) = ax2 −6x + 15, x ∈R has a (1) local maximum at x = −34 (2) local minimum at x = −34 (3) local maximum at x = 34 (4) local minimum at x = 34

Q74.If ∫100π0 sin2x xx dx = 1+4π2απ3 π −[ π ]) e ( α is: (1) 200(1 −e−1) (2) 100(1 −e) (3) 50(e −1) (4) 150(e−1 −1)

Q74.Let α, β, γ be the real roots of the equation, x3 + ax2 + bx + c = 0, ( a, b, c ∈R and a, b ≠0). If the system of equations (in, u, v, w) given by αu + βv + γw = 0, βu + γv + αw = 0, γu + αv + βw = 0 has non-trivial solution, then the value of a2 is b (1) 5 (2) 3 (3) 1 (4) 0

Q74.Let f : R →R be defined as f(x) = e−x sin x. If F : [0, 1] →R is a differentiable function such that F(x) = ∫x0 f(t)dt, then the value of ∫10 (F ′(x) + f(x))exdx lies in the interval (1) [ 327360 , 360329 ] (2) [ 360330 , 360331 ] (3) [ 331360 , 360334 ] (4) [ 360335 , 360336 ] dx = αe−1 + βe−12 + γ, where α, β, γ are integers and [x] denotes the greatest

Q74.Let 1 / 2 𝑥𝑛 ∀𝑛> 𝑚 and 𝑛, 𝑚∈𝑁. Consider a matrix 𝐴= where 𝐽𝑛, 𝑚= ∫0 𝑥𝑚- 1𝑑𝑥, 𝑎𝑖𝑗3 × 3 J6 + 𝑖, 3 - J𝑖+ 3, 3 , 𝑖≤𝑗 a𝑖𝑗= Then adj A-1 is : 0 , 𝑖> 𝑗. (1) (15 ) 2 × 234 (2) (15 ) 2 × 242 (3) (105 ) 2 × 236 (4) (105 ) 2 × 238

Q75.The number of real roots of the equation e4x + 2e3x −ex −6 = 0 is : (1) 0 (2) 1 (3) 4 (4) 2

Q75.Let f be a twice differentiable function defined on R such that f(0) = 1, f ′(0) = 2 and f ′(x) ≠0 for all f(x) f ′(x) x ∈R. If = 0, for all x ∈R, then the value of f(1) lies in the interval f ′(x) f ′′(x) JEE Main 2021 (24 Feb Shift 2) JEE Main Previous Year Paper (1) (9, 12) (2) (3, 6) (3) (0, 3) (4) (6, 9)

Q75.Let f : (a, b) →R be twice differentiable function such that f(x) = ∫xa g(t)dt for a differentiable function g(x). If f(x) = 0 has exactly five distinct roots in (a, b), then g(x)g′(x) = 0 has at least : (1) twelve roots in (a, b) (2) five roots in (a, b) (3) seven roots in (a, b) (4) three roots in (a, b)