Practice Questions

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.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

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.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 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.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.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 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

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

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

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.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.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.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 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. lim sin(cos−1 x)−x is equal to 1−tan(cos−1 x) x→1 √2 (1) 1 (2) −1 √2 √2 (3) √2 (4) −1

Q67.If the length of the latus rectum of a parabola, whose focus is (a, a) and the tangent at its vertex is x + y = a, is 16 , then |a| is equal to (1) 2√2 (2) 2√3 (3) 4√2 (4) 4 JEE Main 2022 (27 Jul Shift 2) JEE Main Previous Year Paper

Q67.If the line 𝑥- 1 = 0, is a directrix of the hyperbola 𝑘𝑥2 - 𝑦2 = 6, then the hyperbola passes through the point (1) -2√5, 6 (2) -√5, 3 (3) √5, - 2 (4) 2√5, 3√6

Q67.Let Δ, ∇∈{∧, ∨} be such that p∇q →((pΔq)∇r) is a tautology. Then (p∇q) Δ r is logically equivalent to (1) (pΔr) ∨q (2) (pΔr) ∧q (3) (p ∧r)Δq (4) (p∇r) ∧q