Practice Questions

Q68.A ray of light through (2, 1) is reflected at a point P on the y− axis and then passes through the point (5, 3). If this reflected ray is the directrix of an ellipse with eccentricity 1 and the distance of the nearer focus from this 3 directrix is 8 , then the equation of the other directrix can be: √53 JEE Main 2021 (27 Jul Shift 1) JEE Main Previous Year Paper (1) 11x + 7y + 8 = 0 or 11x + 7y −15 = 0 (2) 11x −7y −8 = 0 or 11x + 7y + 15 = 0 (3) 2x −7y + 29 = 0 or 2x −7y −7 = 0 (4) 2x −7y −39 = 0 or 2x −7y −7 = 0 x2f(2)−4f(x) is equal to:

Q68.A spherical gas balloon of radius 16 meter subtends an angle 60° at the eye of the observer 𝐴 while the angle of elevation of its center from the eye of 𝐴 is 75°. Then the height (in meter) of the top most point of the balloon from the level of the observer's eye is : (1) 8 ( 2 + 2√3 + √2 ) (2) 8 ( √6 + √2 + 2 ) (3) 8 ( √2 + 2 + √3 ) (4) 8 ( √6 - √2 + 2 )

Q68.Let A = [aij] be a real matrix of order 3 × 3, such that ai1 + ai2 + ai3 = 1, for i = 1, 2, 3 . Then, the sum of all the entries of the matrix A3 is equal to: (1) 2 (2) 1 (3) 3 (4) 9 JEE Main 2021 (22 Jul Shift 1) JEE Main Previous Year Paper

Q68.Let Z be the set of all integers, A = {(x, y) ∈Z × Z : (x −2)2 + y2 ≤4} B = {(x, y) ∈Z × Z : x2 + y2 ≤4} and C = {(x, y) ∈Z × Z : (x −2)2 + (y −2)2 ≤4} If the total number of relations from A ∩B to A ∩C is 2p , then the value of p is: (1) 25 (2) 9 (3) 16 (4) 49

Q68.Let A and B be 3 × 3 real matrices such that A is a symmetric matrix and B is a skew-symmetric matrix. Then the system of linear equations (A2 B2 −B2 A2)X = O, where X is a 3 × 1 column matrix of unknown variables and O is a 3 × 1 null matrix, has (1) exactly two solutions (2) infinitely many solutions (3) a unique solution (4) no solution is:

Q69.If a tangent to the ellipse x2 + 4y2 = 4 meets the tangents at the extremities of its major axis at B and C, then the circle with BC as diameter passes through the point. (1) (√3, 0) (2) (√2, 0) (3) (1, 1) (4) (−1, 1)

Q69.Let A = {1, 2, 3, … , 10} and f : A →A be defined as + 1 if k is odd f(k) = {k k if k is even JEE Main 2021 (26 Feb Shift 2) JEE Main Previous Year Paper Then the number of possible functions g : A →A such that gof = f is: (1) 10C5 (2) 55 (3) 5! (4) 105

Q69.Let A = [2a 30 ], If det (Q) = 9 , then the modulus of the sum of all possible values of determinant of P is equal to: (1) 36 (2) 24 (3) 45 (4) 18

Q69.Let [λ] be the greatest integer less than or equal to λ. The set of all values of λ for which the system of linear equations x + y + z = 4, 3x + 2y + 5z = 3, 9x + 4y + (28 + [λ])z = [λ] has a solution is: (1) R (2) (−∞, −9) ∪[−8, ∞) (3) (−∞, −9) ∪(−9, ∞) (4) [−9, −8) Q70. ⎡[x + 1] [x + 2] [x + 3]⎤ Let A = [x] [x + 3] [x + 3] , where [x] denotes the greatest integer less than or equal to x. If ⎣ [x] [x + 2] [x + 4] ⎦ det (A)= 192 , then the set of values of x is in the interval: (1) [62, 63) (2) [65, 66) (3) [60, 61) (4) [68, 69) = x ∈( π2 , π), then dxdy at x = 5π6 is:

Q70.Let A = [ −ii −ii ], [ 648 ] (1) A unique solution (2) Infinitely many solutions (3) No solution (4) Exactly two solutions lim is equal to :

Q70. cos−1(1−{x}2) sin−1(1−{x}) ⎧ , x ≠0 Let α ∈R be such that the function f(x) = {x}−{x}3 is continuous at x = 0, where ⎨ ⎩α, x = 0 {x} = x −[x], [x] is the greatest integer less than or equal to x. Then : (1) α = π (2) α = 0 √2 (3) no such α exists (4) α = π4

Q71. a1 a2 a3 If ar = cos 2rπ9 + i sin 2rπ9 , r = 1, 2, 3, … , i = √−1, then the determinant a4 a5 a6 is equal to : a7 a8 a9 (1) a9 (2) a1a9 −a3a7 (3) a5 (4) a2a6 −a4a8

Q71.If the domain of the function f(x) = cos−1 √x2−x+1 is the interval (α, β], then α + β is equal to: √sin−1( 2x−12 ) (1) 3 (2) 2 2 (3) 1 (4) 1 2

Q71.Let f : S →S where S = (0, ∞) be a twice differentiable function such that f(x + 1) = xf(x). If g : S →R be defined as g(x) = loge f(x), then the value of |g′′(5) −g′′(1)| is equal to : (1) 205 (2) 197 144 144 (3) 187 (4) 1 144 JEE Main 2021 (16 Mar Shift 2) JEE Main Previous Year Paper

Q71.Let 𝑓: 𝑅→𝑅 be defined as 𝜆𝑥2 - 5𝑥+ 6 𝑥< 2 𝜇5𝑥- 𝑥2 - 6 𝑓𝑥= tan ( 𝑥- 2 ) 𝑒 𝑥- [𝑥] 𝑥> 2 𝜇 𝑥= 2 where 𝑥 is the greatest integer less than or equal to 𝑥. If 𝑓 is continuous at 𝑥= 2, then 𝜆+ 𝜇 is equal to : (1) 𝑒( - 𝑒+ 1 ) (2) 𝑒( 𝑒- 2 ) (3) 1 (4) 2𝑒- 1

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:

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.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.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.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.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.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

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.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 the functions f : R →R and g : R →R be defined as : + 2, x < 0 x < 1 f(x) = and g(x) = {xx2, x ≥0 {x3,3x −2, x ≥1 Then, the number of points in R where (fog)(x) is NOT differentiable is equal to : (1) 3 (2) 1 (3) 0 (4) 2