Practice Questions

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.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 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.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.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 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.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)

Q72.Let 𝑓: 𝑅→𝑅 be a differentiable function satisfying 𝑓'3 + 𝑓'2 = 0 . Then lim is equal to 𝑥→0 1 + 𝑓2 - 𝑥- 𝑓2 (1) 1 (2) e (3) 𝑒2 (4) e-1

Q72.Contrapositive of the statement "If two numbers are not equal, then their squares are not equal". is : (1) If the squares of two numbers are not equal, then (2) If the squares of two numbers are equal, then the the numbers are equal. numbers are not equal. (3) If the squares of two numbers are equal, then the (4) If the squares of two numbers are not equal, then numbers are equal. the numbers are not equal.

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.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.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.If the tangent to the parabola y2 = x at a point (α, β), (β > 0) is also a tangent to the ellipse, x2 + 2y2 = 1 then α is equal to: (1) √2 −1 (2) 2√2 + 1 (3) √2 + 1 (4) 2√2 −1

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

Q72.If the mean and standard deviation of 5 observations x1, x2, x3, x4, x5 are 10 and 3, respectively, then the variance of 6 observations x1, x2, … , x5 and −50 is equal to (1) 582.5 (2) 507.5 (3) 509.5 (4) 586.5

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 A(4, −4) and B(9, 6) be points on the parabola, y2 = 4x. Let C be chosen on the arc AOB of the parabola, where O is the origin, such that the area of ΔACB is maximum. Then, the area (in sq. units) of ΔACB , is: (1) 32 (2) 31 34 (3) 30 12 (4) 31 14

Q72.If 5𝑥+ 9 = 0 is the directrix of the hyperbola 16𝑥2 - 9𝑦2 = 144, then its corresponding focus is: (1) -5, 0 (2) 5, 0 5 5 (3) - 3, 0 (4) 3, 0

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.If the truth value of the statement 𝑝→~𝑞∨𝑟 is false 𝐹, then the truth values of the statements 𝑝, 𝑞, 𝑟 are respectively JEE Main 2019 (12 Apr Shift 1) JEE Main Previous Year Paper (1) 𝑇, 𝐹, 𝑇 (2) 𝑇, 𝐹, 𝐹 (3) 𝑇, 𝑇, 𝐹 (4) 𝐹, 𝑇, 𝑇

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

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

Q73.If the vertices of a hyperbola be at (−2, 0) and (2, 0) and one of its foci be at (−3, 0), then which one of the following points does not lie on this hyperbola ? (1) (6, 5√2) (2) (−6, 2√10) (3) (2√6, 5) (4) (4, √15)