Practice Questions

Q70.Axis of a parabola lies along 𝑥-axis. If its vertex and focus are at distances 2 and 4 respectively from the origin, on the positive 𝑥-axis then which of following points does not lie on it? (1) 6, 4√2 (2) 5, 2√6 (3) 8, 6 (4) 4, - 4

Q70.If the line 𝑎𝑥+ 𝑦= 𝑐, touches both the curves 𝑥2 + 𝑦2 = 1 and 𝑦2 = 4√2𝑥, then 𝑐 is equal to: 1 (1) (2) √2 2 (3) 1 (4) 2 √2

Q70.A circle touching the x− axis at (3, 0) and making an intercept of length 8 on the y− axis passes through the point: (1) (3, 10) (2) (2, 3) (3) (3, 5) (4) (1, 5)

Q70.Let 𝑂0,0 and 𝐴0,1 be two fixed points. Then, the locus of a point 𝑃 such that the perimeter of 𝛥𝐴𝑂𝑃 is 4 is (1) 8𝑥2 + 9𝑦2 - 9𝑦= 18 (2) 9𝑥2 - 8𝑦2 + 8𝑦= 16 (3) 8𝑥2 - 9𝑦2 + 9𝑦= 18 (4) 9𝑥2 + 8𝑦2 - 8𝑦= 16

Q70.In an ellipse, with centre at the origin, if the difference of the lengths of major axis and minor axis is 10 and one of the foci is at 0,5√3, then the length of its latus rectum is: (1) 6 (2) 10 (3) 8 (4) 5

Q70.If one end of a focal chord of the parabola, y2 = 16x is at (1, 4), then the length of this focal chord is (1) 24 (2) 25 (3) 22 (4) 20 , then a value of m is:

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

Q70.If the normal to the ellipse 3𝑥2 + 4𝑦2 = 12 at a point 𝑃 on it is parallel to the line, 2𝑥+ 𝑦= 4 and the tangent to the ellipse at 𝑃 passes through 𝑄( 4,4 ) then 𝑃𝑄 is equal to: (1) √61 (2) 5√5 2 2 (3) √157 (4) √221 2 2

Q70.The equation of a tangent to the parabola, x2 = 8y, which makes an angle θ with the positive direction of x− axis, is (1) y = xtanθ + 2cotθ (2) y = xtanθ −2cotθ (3) x = ycotθ + 2tanθ (4) x = ycotθ −2tanθ

Q70.Let S = {(x, 1}, where (1) An ellipse whose eccentricity is 1 , when (2) A hyperbola whose eccentricity is 2 , when √r+1 √r+1 r > 1. 0 < r < 1. (3) (4) A hyperbola whose eccentricity is 2 , when An ellipse whose eccentricity is , when √1−r √ r+12 r > 1 0 < r < 1

Q70.If a circle C passing through the point (4, 0) touches the circle x2 + y2 + 4x −6y = 12 externally at the point (1, −1), then the radius of C is: (1) 4 units (2) 5 units (3) 2√5 units (4) √57 units

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 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. 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

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.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 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.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.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.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.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.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 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.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