Practice Questions

Q64.Let S = {θ ∈(0, π2 ) : ∑9m=1 sec(θ + (m −1) π6 ) sec(θ + mπ6 ) = −8√3 }. Then (1) S = { 12π } (2) S = { 2π3 } (3) ∑θ∈S θ = π2 (4) ∑θ∈S θ = 3π4

Q65.Let the focal chord of the parabola P : y2 = 4x along the line L : y = mx + c, m > 0 meet the parabola at the points M and N . Let the line L be a tangent to the hyperbola H : x2 −y2 = 4. If O is the vertex of P and F is the focus of H on the positive x-axis, then the area of the quadrilateral OMFN is (1) 2√6 (2) 2√14 (3) 4√6 (4) 4√14 αex+βe−x+γ sin x 2

Q65.Let the tangent to the circle C1 : x2 + y2 = 2 at the point M(−1, 1) intersect the circle C2 : (x −3)2 + (y −2)2 = 5 , at two distinct points A and B. If the tangents to C2 at the points A and B intersect at N , then the area of the triangle ANB is equal to (1) 12 (2) 23 (3) 1 (4) 5 6 3

Q65.Let 𝑥= 2𝑡, 𝑦= 𝑡2 be a conic. Let 𝑆 be the focus and 𝐵 be the point on the axis of the conic such that 𝑆𝐴⊥𝐵𝐴, 3 where 𝐴 is any point on the conic. If 𝑘 is the ordinate of the centroid of the 𝛥𝑆𝐴𝐵, then 𝑡→1𝑘lim is equal to (1) 17 (2) 19 18 18 (3) 11 (4) 13 18 18

Q65.For 𝑡∈0, 2𝜋, if 𝐴𝐵𝐶 is an equilateral triangle with vertices 𝐴sin𝑡, - cos𝑡, 𝐵cos𝑡, sin𝑡 and 𝐶𝑎, 𝑏 such that its 1 orthocentre lies on a circle with centre 1, 3, then 𝑎2 - 𝑏2 is equal to (1) 8 (2) 8 3 77 80 (3) (4) 9 9 11

Q65.The distance of the origin from the centroid of the triangle whose two sides have the equations x −2y + 1 = 0 and 2x −y −1 = 0 and whose orthocenter is ( 73 , 37 ) is: (1) √2 (2) 2 (3) 2√2 (4) 4 JEE Main 2022 (29 Jun Shift 2) JEE Main Previous Year Paper

Q65.Let the locus of the centre 𝛼, 𝛽, 𝛽> 0, of the circle which touches the circle 𝑥2 + 𝑦- 12 = 1 externally and also touches the 𝑥-axis be 𝐿. Then the area bounded by 𝐿 and the line 𝑦= 4 is (1) 32√2 (2) 40√2 3 3 64 32 (3) (4) 3 3

Q65.A particle is moving in the xy-plane along a curve C passing through the point (3, 3). The tangent to the curve C at the point P meets the x-axis at Q . If the y-axis bisects the segment PQ , then C is a parabola with (1) length of latus rectum 3 (2) length of latus rectum 6 (3) focus ( 34 , 0) (4) focus (0, 33 ) y2

Q66.Let a be an integer such that lim 18−[1−x][x−3a] exists, where [t] is greatest integer ≤t . Then x→7 (1) −2 (2) 6 (3) −6 (4) −7

Q66.If lim = 3 , where α, β, γ ∈R, then which of the following is NOT correct? x sin2 x x→0 (1) α2 + β2 + γ 2 = 6 (2) αβ + βγ + γα + 1 = 0 (3) αβ2 + βγ 2 + γα2 + 3 = 0 (4) α2 −β2 + γ 2 = 4

Q66.The acute angle between the pair of tangents drawn to the ellipse 2𝑥2 + 3𝑦2 = 5 from the point 1, 3 is 16 24 (1) tan-1 (2) tan-1 7√5 7√5 32 + 8√5 (3) tan-1 (4) tan-13 7√5 35

Q66.Let PQ be a focal chord of the parabola y2 = 4x such that it subtends an angle of π2 at the point (3, 0). Let the x2 y2 line segment PQ be also a focal chord of the ellipse E : + = 1, a2 > b2 . If e is the eccentricity of the a2 b2 ellipse E , then the value of 1 is equal to e2 (1) 1 + √2 (2) 3 + 2√2 (3) 1 + 2√3 (4) 4 + 5√3

Q66. lim cos(sin x)−cos x is equal to x→0 x4 (1) 1 (2) 1 3 6 (3) 1 (4) 1 4 12

Q66.Let 𝑚1, 𝑚2 be the slopes of two adjacent sides of a square of side 𝑎 such that 𝑎2 + 11𝑎+ 3 𝑚12 + 𝑚22 = 220. 𝜋 If one vertex of the square is 10cos𝛼- sin𝛼, 10sin𝛼+ cos𝛼, where 𝛼∈0, and the equation of one diagonal is 2 cosα - sinα𝑥+ sin𝛼+ cos𝛼𝑦= 10, then 72sin4𝛼+ cos4𝛼+ 𝑎2 - 3𝑎+ 13 is equal to (1) 119 (2) 128 (3) 145 (4) 155

Q66.Let x2 + y2 + Ax + By + C = 0 be a circle passing through (0, 6) and touching the parabola y = x2 at (2, 4). Then A + C is equal to _____ (1) 16 (2) 885 (3) 72 (4) −8 is equal to

Q66.Let the maximum area of the triangle that can be inscribed in the ellipse x2 + 4 = 1, a > 2, having one of its a2 vertices at one end of the major axis of the ellipse and one of its sides parallel to the y-axis, be 6√3. Then the eccentricity of the ellipse is: (1) √3 (2) 1 2 2 (3) 1 (4) √3 √2 4

Q67.Let f : R →R be a function defined as f(x) = a sin( π[x]2 ) less than or equal to t. If lim f(x) exists, then the value of ∫40 f(x)dx is equal to x→−1 (1) −1 (2) −2 (3) 1 (4) 2

Q67.If the tangents drawn at the points 𝑃 and 𝑄 on the parabola 𝑦2 = 2𝑥- 3 intersect at the point 𝑅0, 1, then the orthocentre of the triangle 𝑃𝑄𝑅 is (1) 0, 1 (2) 2, - 1 (3) 6, 3 (4) 2, 1

Q67.If the equation of the parabola, whose vertex is at (5, 4) and the directrix is 3x + y −29 = 0, is x2 + ay2 + bxy + cx + dy + k = 0, then a + b + c + d + k is equal to (1) 575 (2) −575 (3) 576 (4) −576

Q67.Let A be a 2 × 2 matrix with det(A) = −1 and det((A + I)(Adj(A) + I)) = 4 . Then the sum of the diagonal elements of A can be: (1) −1 (2) 2 (3) 1 (4) −√2

Q67.Let P : y2 = 4ax, a > 0 be a parabola with focus S .Let the tangents to the parabola P make an angle of π4 with the line y = 3x + 5 touch the parabola P at A and B . Then the value of a for which A, B and S are collinear is: (1) 8 only (2) 2 only (3) 1 only (4) any a > 0 4

Q68.Let the system of linear equations x + y + az = 2 3x + y + z = 4 x + 2z = 1 have a unique solution ( x∗, y∗, z∗). If ( (a, x∗), (y∗, α) and ( x∗, −y∗) are collinear points, then the sum of absolute values of all possible values of α is: (1) 4 (2) 3 (3) 2 (4) 1

Q68.Let A be a matrix of order 3 × 3 and det(A) = 2 . Then det(det (A) adj (5 adj (A3)) is equal to _____. (1) 256 × 106 (2) 1024 × 106 (3) 512 × 106 (4) 256 × 1011

Q69.The number of 𝜃∈0, 4𝜋 for which the system of linear equations 3sin3𝜃𝑥- 𝑦+ 𝑧= 2 3cos2𝜃𝑥+ 4𝑦+ 3𝑧= 3 6𝑥+ 7𝑦+ 7𝑧= 9 has no solution is (1) 6 (2) 7 (3) 8 (4) 9

Q69.For 𝛼∈𝑁, consider a relation 𝑅 on 𝑁 given by 𝑅= {𝑥, 𝑦: 3𝑥+ 𝛼𝑦 is a multiple of 7}. The relation 𝑅 is an equivalence relation if and only if (1) 𝛼= 14 (2) 𝛼 is a multiple of 4 (3) 4is the remainder when 𝛼 is divided by 10 (4) 4 is the remainder when 𝛼 is divided by 7 Q70. 0 1 0 Let the matrix 𝐴= 1 0 0 and the matrix 𝐵0 = 𝐴49 + 2𝐴98. If 𝐵𝑛= Adj𝐵𝑛- 1 for all 𝑛≥1, then det 𝐵4 is 0 0 1 equal to (1) 328 (2) 330 (3) 332 (4) 336