Practice Questions

Q66.If 𝑛→∞√𝑛2lim - 𝑛- 1 + 𝑛𝛼+ 𝛽= 0 then 8𝛼+ 𝛽 is equal to JEE Main 2022 (25 Jul Shift 1) JEE Main Previous Year Paper (1) 4 (2) -8 (3) -4 (4) 8

Q66.The tangents at the points A(1, 3) and B(1, −1) on the parabola y2 −2x −2y = 1 meet at the point P . Then the area (in unit2 ) of the triangle PAB is: (1) 4 (2) 6 (3) 7 (4) 8 y2

Q66.Let p, q, r be three logical statements. Consider the compound statements S1 : ((~p) ∨q) ∨((~p) ∨r) and S2 : p →(q ∨r) Then, which of the following is NOT true? (1) If S2 is True, then S1 is True (2) If S2 is False, then S1 is False (3) If S2 is False, then S1 is True (4) If S1 is False, then S2 is False

Q66.The set of values of k for which the circle C : 4x2 + 4y2 −12x + 8y + k = 0 lies inside the fourth quadrant and the point (1, −13 ) lies on or inside the circle C is (1) An empty set (2) (6, 959 ] (3) [ 809 , 10) (4) (9, 929 ]

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. 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 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.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.The statement (~(p ⇔~q)) ∧q is: (1) a tautology (2) a contradiction (3) equivalent to (p ⇒q) ∧q (4) equivalent to (p ⇒q) ∧p

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.A circle C1 passes through the origin O and has diameter 4 on the positive x-axis. The line y = 2x gives a chord OA of a circle C1 . Let C2 be the circle with OA as a diameter. If the tangent to C2 at the point A meets the x-axis at P and y-axis at Q, then QA : AP is equal to (1) 1 : 4 (2) 1 : 5 (3) 2 : 5 (4) 1 : 3

Q66.Let 𝐶 be the centre of the circle 𝑥2 + 𝑦2 - 𝑥+ 2𝑦= and 𝑃 be a point on the circle. A line passes through the 4 point 𝐶, makes an angle of 𝜋 with the line 𝐶𝑃 and intersects the circle at the points 𝑄 and 𝑅. Then the area of 4 the triangle 𝑃𝑄𝑅 (in unit2) is (1) 2 (2) 2√2 𝜋 𝜋 (3) 8sin (4) 8cos 8 8

Q66.Let P(a, b) be a point on the parabola y2 = 8x such that the tangent at P passes through the centre of the circle x2 + y2 −10x −14y + 65 = 0 . Let A be the product of all possible values of a and B be the product of all possible values of b. Then the value of A + B is equal to (1) 0 (2) 25 (3) 40 (4) 65 + [2 −x], a ∈R, where [t] is the greatest integer

Q66.The value of 2sin12° - sin72° is (1) √51 - √3 (2) 1 - √5 4 8 (3) √31 - √5 (4) √31 - √5 2 4

Q66.Let a triangle be bounded by the lines L1 : 2x + 5y = 10 ; L2 : −4x + 3y = 12 and the line L3 , which passes through the point P(2, 3), intersect L2 at A and L1 at B. If the point P divides the line-segment AB, internally in the ratio 1 : 3, then the area of the triangle is equal to (1) 110 (2) 132 13 13 (3) 142 (4) 151 13 13

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.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.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.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.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)

Q67.Let A and B be any two 3 × 3 symmetric and skew symmetric matrices respectively. Then which of the following is NOT true? (1) A4 −B4 is a symmetric matrix (2) AB −BA is a symmetric matrix (3) B5 −A5 is a skew-symmetric matrix (4) AB + BA is a skew-symmetric matrix

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

Q67.A circle touches both the 𝑦-axis and the line 𝑥+ 𝑦= 0. Then the locus of its center (1) 𝑦= √2𝑥 (2) 𝑥= √2𝑦.. (3) 𝑦2 - 𝑥2 = 2𝑥𝑦 (4) 𝑥2 −𝑦2 = 2𝑥𝑦

Q67.Consider the following two propositions : 𝑃1: ~𝑝→~𝑞 𝑃2: 𝑝∧~𝑞∧~𝑝∨𝑞 If the proposition 𝑝→~𝑝∨𝑞 is evaluated as FALSE, then (1) 𝑃1 is TRUE and 𝑃2 is FALSE (2) 𝑃1 is FALSE and 𝑃2 is TRUE (3) Both 𝑃1 and 𝑃2 are FALSE (4) Both 𝑃1 and 𝑃2 are TRUE