Practice Questions

Q65.Two tangents are drawn from the point P(−1, 1) to the circle x2 + y2 −2x −6y + 6 = 0. If these tangents touch the circle at points A and B, and if D is a point on the circle such that length of the segments AB and AD are equal, then the area of the triangle ABD is equal to: + (1) 2 (2) (3√2 2) (3) 4 (4) 3(√2 −1)

Q66.Let a tangent be drawn to the ellipse x2 cos θ, sin θ ∈(0, π2 ). Then the value of θ 27 + y2 = 1 at (3√3 θ) where such that the sum of intercepts on axes made by this tangent is minimum is equal to : (1) π (2) π 8 4 (3) π (4) π 6 3 x-axis at Q and latus

Q66.The locus of the mid points of the chords of the hyperbola x2 −y2 = 4, which touch the parabola y2 = 8x, is : (1) y2(x −2) = x3 (2) x3(x −2) = y2 (3) x2(x −2) = y3 (4) y3(x −2) = x2 lim n=1 n(n+1)x2+2(2n+1)x+4x ) is equal to :

Q66.Consider the parabola with vertex 2, 4 and the directrix 𝑦= 2 . Let P be the point where the parabola meets the line 𝑥= - 12. If the normal to the parabola at P intersects the parabola again at the point Q . then ( PQ ) 2 is equal to : 25 75 (1) (2) 2 8 (3) 125 (4) 15 16 2

Q66.Let ABC be a triangle with A(−3, 1) and ∠ACB = θ, 0 < θ < π2 . If the equation of the median through B is 2x + y −3 = 0 and the equation of angle bisector of C is 7x −4y −1 = 0, then tan θ is equal to: (1) 3 (2) 4 4 3 (3) 2 (4) 12

Q67.Let A = {(x, y) ∈R × R ∣2x2 + 2y2 −2x −2y = 1} B = {(x, y) ∈R × R ∣4x2 + 4y2 −16y + 7 = 0} and C = {(x, y) ∈R × R ∣x2 + y2 −4x −2y + 5 ≤r2}. Then the minimum value of |r| such that A ∪B ⊆C is equal to (1) 3+√10 (2) 2+√10 2 2 (3) 3+2√5 (4) 1 + √5 2

Q67.If a line along a chord of the circle 4x2 + 4y2 + 120x + 675 = 0, passes through the point (−30, 0) and is tangent to the parabola y2 = 30x, then the length of this chord is: (1) 5 (2) 7 (3) 3√5 (4) 5√3 y2

Q67.A tangent and a normal are drawn at the point P(2, −4) on the parabola y2 = 8x, which meet the directrix of the parabola at the points A and B respectively. If Q(a, b) is a point such that AQBP is a square, then 2a + b is equal to (1) −12 (2) −20 (3) −16 (4) −18

Q67.The locus of the midpoints of the chord of the circle, x2 + y2 = 25 which is tangent to the hyperbola, x2 is : 9 −y216 = 1 (1) (x2 + y2)2 −16x2 + 9y2 = 0 (2) (x2 + y2)2 −9x2 + 144y2 = 0 2 2 (3) (x2 + y2) −9x2 −16y2 = 0 (4) (x2 + y2) −9x2 + 16y2 = 0

Q67.Let 𝜃 be the acute angle between the tangents to the ellipse 𝑥2 + 𝑦2 = 1 and the circle 𝑥2 + 𝑦2 = 3 at their 9 1 point of intersection in the first quadrant. Then tan𝜃 is equal to : (1) 5 (2) 4 2√3 √3 (3) 2 (4) 2 √3

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. sin2 x 1 + cos2 x cos 2x The maximum value of f(x) = 1 + sin2 x cos2 x cos 2x , x ∈R is sin2 x cos2 x sin 2x (1) √7 (2) 34 (3) √5 (4) 5

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:

Q68.If the curves, x2 intersect each other at an angle of 90°, then which of the a + b = 1 and x2c + y2d = 1 following relations is TRUE? (1) a −c = b + d (2) a −b = c −d (3) a + b = c + d (4) ab = a+bc+d 1 1 n 1+ 2 +……+ n

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.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.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

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:

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

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.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. 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: 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