Practice Questions

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.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 p : Ramesh listens to music. q : Ramesh is out of his village r : It is Sunday s : It is Saturday Then the statement "Ramesh listens to music only if he is in his village and it is Sunday or Saturday" can be expressed as (1) ((~q) ∧(r ∨s)) ⇒p (2) (q ∧(r ∨s)) ⇒p (3) p ⇒(q ∧(r ∨s)) (4) p ⇒((~q) ∧(r ∨s))

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

Q65.If cot α = 1 and sec β = −53 , where π < α < 3π2 and π2 < β < π, then the value of tan(α + β) and the quadrant in which α + β lies, respectively are (1) −17 and IVth quadrant (2) 7 and Ist quadrant (3) −7 and IVth quadrant (4) 71 and Ist quadrant

Q65.Let the normal at the point P on the parabola y2 = 6x pass through the point (5, −8). If the tangent at P to the parabola intersects its directrix at the point Q, then the ordinate of the point Q is (1) −9 (2) 9 4 4 (3) −5 (4) −3 2

Q65.Let the eccentricity of an ellipse x2 + = 1, a > b, be 14 . If this ellipse passes through the point a2 b2 5 , , then a2 + b2 is equal to (−4√2 3) (1) 29 (2) 31 (3) 32 (4) 34 a is equal to

Q65.Let the tangent drawn to the parabola y2 = 24x at the point (α, β) is perpendicular to the line 2x + 2y = 5 . Then the normal to the hyperbola x2 −y2 = 1 at the point (α + 4, β + 4) does NOT pass through the point: α2 β2 (1) (25, 10) (2) (20, 12) (3) (30, 8) (4) (15, 13)

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.Let the point P(α, β) be at a unit distance from each of the two lines L1 : 3x −4y + 12 = 0 , and L2 : 8x + 6y + 11 = 0 . If P lies below L1 and above L2 , then 100(α + β) is equal to (1) −14 (2) 42 (3) −22 (4) 14

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.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.The normal to the hyperbola x2 −y29 = 1 a2 at the point (8, 3√3) on it passes through the point (1) (15, −2√3) (2) (9, 2√3) (3) (−1, 9√3) (4) (−1, 6√3)

Q65.If the circle x2 + y2 −2gx + 6y −19c = 0, g, c ∈R passes through the point (6, 1) and its centre lies on the line x −2cy = 8 , then the length of intercept made by the circle on x-axis is (1) √11 (2) 4 (3) 3 (4) 2√23

Q65.The equations of the sides AB, BC and CA of a triangle ABC are 2x + y = 0, x + py = 39 and x −y = 3 respectively and P(2, 3) is its circumcentre. Then which of the following is NOT true (1) (AC)2 = 9p (2) (AC)2 + p2 = 136 (3) 32 <area (ΔABC) < 36 (4) 34 < area (ΔABC) < 38

Q65.The number of elements in the set 𝑥2 + 𝑥 𝑆= 𝑥∈ℝ: 2cos = 4𝑥+ 4-𝑥 is 6 (1) 1 (2) 3 (3) 0 (4) infinite JEE Main 2022 (29 Jul Shift 2) JEE Main Previous Year Paper

Q65.Let S = {θ ∈[−π, π] −{± π2 } : sin θ tan θ + tan θ = sin 2θ}. If T = ∑θ∈S cos 2θ, then T + n(S) is equal to (1) 7 + √3 (2) 5 (3) 8 + √3 (4) 9

Q65.The coefficient of 𝑥101 in the expression 5 + 𝑥500 + 𝑥5 + 𝑥499 + 𝑥25 + 𝑥498 + … … + 𝑥500, 𝑥> 0 is (1) 501 𝐶101 × 5399 (2) 501𝐶101 × 5400 (3) 501𝐶100 × 5400 (4) 500𝐶101 × 5399

Q65.The equation of a common tangent to the parabolas 𝑦= 𝑥2 and 𝑦= - 𝑥- 22 is (1) 𝑦= 4𝑥- 2 (2) 𝑦= 4𝑥- 1 (3) 𝑦= 4𝑥+ 1 (4) 𝑦= 4𝑥+ 2

Q66.A horizontal park is in the shape of a triangle OAB with AB = 16 . A vertical lamp post OP is erected at the point O such that ∠PAO = ∠PBO = 15° and ∠PCO = 45° , where C is the midpoint of AB. Then (OP)2 is equal to (1) √3 32 (√3 −1) (2) √332 (2 −√3) (3) 16 (4) 16 √3 (√3 −1) √3 (2 −√3)

Q66.Let a triangle ABC be inscribed in the circle x2 −√2(x + y) + y2 = 0 such that ∠BAC = π2 . If the length of side AB is √2 , then the area of the △ABC is equal to: (1) 1 (2) (√6+√3) 2 (3) (√3+√3) (4) (√6+2√3) 2 4

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

Q66.Let 𝑓𝑥 be a polynomial function such that 𝑓𝑥+ 𝑓'𝑥+ 𝑓''𝑥= 𝑥5 + 64. Then, the value of lim 𝑓𝑥 is equal to 𝑥→1 𝑥- 1 (1) -15 (2) 15 (3) -60 (4) 60

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