Practice Questions

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 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 eccentricity of the hyperbola H : x2 −y2 and length of its latus rectum be 6√2 . If = 1 be √52 a2 b2 y = 2x + c is a tangent to the hyperbola H , then the value of c2 is equal to (1) 18 (2) 20 (3) 24 (4) 32

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.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.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 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.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 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.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 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.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 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.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.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

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.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 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. 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 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, 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.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 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