Practice Questions

Q70.The common tangent to the circles x2 + y2 = 4 and x2 + y2 + 6x + 8y −24 = 0 also passes through the point: (1) (4, −2) (2) (−4, 6) (3) (6, −2) (4) (−6, 4) JEE Main 2019 (09 Apr Shift 2) JEE Main Previous Year Paper

Q71.If the line y = mx + 7√3 is normal to the hyperbola x224 −y218 = 1 (1) √5 (2) 3 2 √5 (3) √15 (4) 2 2 √5

Q71.If the eccentricity of the standard hyperbola passing through the point ( 4,6 ) is 2, then the equation of the tangent to the hyperbola at ( 4,6 ) is: (1) 2𝑥- 3𝑦+ 10 = 0 (2) 𝑥- 2𝑦+ 8 = 0 (3) 3𝑥- 2𝑦= 0 (4) 2𝑥- 𝑦- 2 = 0 1 1 + 𝑓3 + 𝑥- 𝑓3 𝑥

Q71.The area (in sq. units) of the smaller of the two circles that touch the parabola, y2 = 4x at the point (1, 2) and the x -axis is (1) 8π(3 −2√2) (2) 8π(2 −√2) + (3) 4π(3 √2) (4) 4π(2 −√2)

Q71.The straight line x + 2y = 1 meets the coordinate axes at A and B. A circle is drawn through A, B and the origin. Then the sum of perpendicular distances from A and B on the tangent to the circle at the origin is: (1) √5 (2) 2√5 2 (3) √5 (4) 4√5 4

Q71.If the parabolas y2 = 4b(x −c) and y2 = 8ax have a common normal, then which one of the following is a valid choice for the ordered triad (a, b, c) (1) (1, 1, 3) (2) ( 12 , 2, 0) (3) ( 12 , 2, 3) (4) All of above

Q71.If a variable line 3x + 4y −λ = 0 is such that the two circles x2 + y2 −2x −2y + 1 = 0 and x2 + y2 −18x −2y + 78 = 0 are on its opposite sides, then the set of all values of λ is the interval : (1) [13, 23] (2) (23, 31) (3) [12, 21] (4) (2, 17)

Q71.Equation of a common tangent to the circle, 𝑥2 + 𝑦2 - 6𝑥= 0 and the parabola, 𝑦2 = 4𝑥 is: (1) 2√3𝑦= - 𝑥- 12 (2) √3𝑦= 𝑥+ 3 (3) √3𝑦= 3𝑥+ 1 (4) 2√3𝑦= 12𝑥+ 1

Q71.If a directrix of a hyperbola centered at the origin and passing through the point (4, −2√3) is and its eccentricity is e, then: (1) 4e4 + 8e2 −35 = 0 (2) 4e4 −24e2 + 35 = 0 (3) 4e4 −24e2 + 27 = 0 (4) 4e4 −12e2 −27 = 0 x4−1

Q71.The tangent and normal to the ellipse 3𝑥2 + 5𝑦2 = 32 at the point 𝑃2, 2 meet the 𝑥-axis at 𝑄 and 𝑅, respectively. Then the area (in sq. units) of the triangle 𝑃𝑄𝑅 is: 68 16 (1) (2) 15 3 (3) 14 (4) 34 3 15

Q71.Let S and S ′ be the foci of an ellipse and B be any one of the extremities of its minor axis. If ΔS ′BS is a right angled triangle with right angle at B and area (ΔS ′BS) = 8 sq. units, then the length of a latus rectum of the ellipse is : (1) 2√2 (2) 2 (3) 4 (4) 4√2 Q72. √π−√2 sin−1 x lim is equal to x→1− √1−x (1) √π (2) √2π (3) 1 (4) √π2 √2π

Q71.If the circles x2 + y2 −16x −20y + 164 = r2 and (x −4)2 + (y −7)2 = 36 intersect at two distinct points, then: (1) r > 11 (2) 0 < r < 1 (3) 1 < r < 11 (4) r = 11

Q71.Let 𝑃 be the point of intersection of the common tangents to the parabola 𝑦2 = 12𝑥 and the hyperbola 8𝑥2 - 𝑦2 = 8. If 𝑆 and 𝑆' denote the foci of the hyperbola where 𝑆 lies on the positive 𝑥-axis then 𝑃 divides 𝑆𝑆' in a ratio: (1) 5: 4 (2) 2: 1 (3) 13: 11 (4) 14: 13

Q71.Consider the following three statements: P : 5 is a prime number Q : 7 is a factor of 192 R : LCM of 5 and 7 is 35 Then the truth value of which one of the following statements is true? (1) P ∨(~Q ∧R) (2) (P ∧Q) ∨(~R) (3) (~P) ∨(Q ∧R) (4) (~P) ∧(~Q ∧R)

Q71.If the tangents on the ellipse 4𝑥2 + 𝑦2 = 8 at the points 1, 2 and ( 𝑎, 𝑏) are perpendicular to each other, then 𝑎2 is equal to (1) 2 (2) 4 (3) 64 (4) 128 17 17 17 17

Q71.The tangents to the curve y = (x −2)2 −1 at its points of intersection with the line x −y = 3, intersect at the point: (1) ( 25 , 1) (2) ( 52 , −1) (3) (−52 , −1) (4) (−52 , 1)

Q71. limx→0 x cot(4x) is equal to: sin2 x cot2(2x) JEE Main 2019 (11 Jan Shift 2) JEE Main Previous Year Paper (1) 0 (2) 2 (3) 4 (4) 1

Q72.If x3−k3 , then k is lim lim x−1 = x2−k2 x→1 x→k (1) 3 (2) 4 2 3 (3) 3 (4) 8 8 3

Q72.For any two statement p and q, the negative of the expression p ∨(~p ∧q) is (1) ~p ∨~q (2) p ∧q (3) ~p ∧~q (4) p ↔q

Q72. lim sin2𝑥 equals 𝑥→0 √2 - √1 + cos𝑥 (1) 4√2 (2) 2√2 (3) √2 (4) 4

Q72.If tangents are drawn to the ellipse x2 + 2y2 = 2 at all points on the ellipse other than its four vertices then the mid points of the tangents intercepted between the coordinate axes lie on the curve : y2 (1) 1 + 1 = 1 (2) x2 4x2 2y2 4 + 2 = 1 y2 (3) 1 + 1 = 1 (4) x2 2x2 4y2 2 + 4 = 1

Q72.The equation of a tangent to the hyperbola, 4x2 −5y2 = 20, parallel to the line x −y = 2, is (1) x −y + 7 = 0 (2) x −y −3 = 0 (3) x −y + 1 = 0 (4) x −y + 9 = 0 (1−|x|+sin|1−x|)sin([1−x] π2 )

Q72.An ellipse, with foci at (0,2) and (0, −2) and minor axis of length 4 , passes through which of the following points? (1) (1, 2√2) (2) (2, √2) (3) (√2, 2) (4) (2, 2√2)

Q72.Let P(4, −4) and Q(9, 6) be two points on the parabola, y2 = 4x and let X be any point on the arc POQ of this parabola, where O is the vertex of this parabola, such that the area of ΔPXQ is maximum. Then this maximum area (in sq. units) is : (1) 625 (2) 75 4 2 (3) 125 (4) 125 4 2

Q72.Let 0 < 𝜃< 𝜋 . If the eccentricity of the hyperbola 𝑥2 𝑦2 1 is greater than 2, then the length of its 2 cos2⁡𝜃- sin2⁡𝜃= latus rectum lies in the interval: (1) 3, ∞ (2) 1, 3 2 3 (3) 2, 3 (4) 2, 2 Q73. √1 + √1 + 𝑦4 - √2 The value of lim 𝑦→0 𝑦4 JEE Main 2019 (09 Jan Shift 1) JEE Main Previous Year Paper 1 1 (1) exists and equals (2) exists and equals 2√2 4√2 1 (3) does not exist (4) exists and equals 2√2√2 + 1