Practice Questions

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 the line x −2y = 12 is a tangent to the ellipse x2 + = 1 at the point (3, −92 ), then the length of the a2 b2 latus rectum of the ellipse is (1) 5 units (2) 12√2 units (3) 9 units (4) 8√3 units 5x = 4√5

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

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.Let C1 and C2 be the centres of the circles x2 + y2 −2x −2y −2 = 0 and x2 + y2 −6x −6y + 14 = 0 respectively. If P and Q are the points of intersection of these circles, then the area (in sq. units) of the quadrilateral PC1 QC2 is : JEE Main 2019 (12 Jan Shift 1) JEE Main Previous Year Paper (1) 6 (2) 4 (3) 8 (4) 9

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.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.Two circles with equal radii are intersecting at the points (0,1) and (0,-1) . The tangent at the point (0,1) to one of the circles passes through the centre of the other circle. Then the distance between the centres of these circles is: (1) 1 (2) 2 (3) 2√2 (4) √2

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 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.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 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.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.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 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

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

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

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