Practice Questions

Q56.Let P be a point on the parabola, y2 = 12x and N be the foot of the perpendicular drawn from P , on the axis of the parabola. A line is now drawn through the mid-point M of PN , parallel to its axis which meets the parabola at Q . If the y−intercept of the line NQ is 43 , then : JEE Main 2020 (03 Sep Shift 1) JEE Main Previous Year Paper (1) PN = 4 (2) MQ = 13 (3) MQ = 14 (4) PN = 3

Q56.Let the latus rectum of the parabola y2 = 4x be the common chord to the circles C1 and C2 each of them having radius 2√5. Then, the distance between the centres of the circles C1 and C2 is : (1) 12 (2) 8 (3) 8√5 (4) 4√5 = 1

Q56.If y = mx + 4 is a tangent to both the parabolas, y2 = 4x and x2 = 2by, then b is equal to (1) −32 (2) −64 (3) −128 (4) 128

Q57.Let e1 and e2 be the eccentricities of the ellipse x225 + y2b2 = 1 (b < 5) and the hyperbola x216 −y2b2 respectively satisfying e1e2 = 1. If α and β are the distances between the foci of the ellipse and the foci of the hyperbola respectively, then the ordered pair (α, β) is equal to: (1) (8, 10) (2) ( 203 , 12) (3) (8, 12) (4) ( 245 , 10) JEE Main 2020 (03 Sep Shift 2) JEE Main Previous Year Paper

Q57.If one end of a focal chord AB of the parabola y2 = 8x is at A( 12 , −2), then the equation of the tangent to it at B is: (1) 2x + y −24 = 0 (2) x −2y + 8 = 0 (3) x + 2y + 8 = 0 (4) 2x −y −24 = 0

Q57.The set of all possible values of θ in the interval (0, π) for which the points (1, 2) and (sin θ, cos θ) lie on the same side of the line x + y = 1 is? (1) (0, π2 ) (2) ( π4 , 3π4 ) (3) (0, 3π4 ) (4) (0, π4 )

Q57.Let x2 a2 + b2 = 1(a > b) be a given ellipse, length of whose latus rectum is 10. If its eccentricity is the maximum value of the function, ϕ(t) = 125 + t −t2 , then a2 + b2 is equal to : (1) 145 (2) 116 (3) 126 (4) 135

Q57.Which of the following statement is a tautology? (1) p ∨(~q) →p ∧q (2) ~(p ∧~q) →p ∨q (3) ~(p ∨~q) →p ∧q (4) ~(p ∨~q) →p ∨q JEE Main 2020 (08 Jan Shift 2) JEE Main Previous Year Paper

Q57.The locus of the mid-points of the perpendiculars drawn from points on the line x = 2y, to the line x = y, is. (1) 2x −3y = 0 (2) 5x −7y = 0 (3) 3x −2y = 0 (4) 7x −5y = 0

Q57.If the point P on the curve, 4x2 + 5y2 = 20 is farthest from the point Q(0, −4), then PQ2 is equal to (1) 36 (2) 48 (3) 21 (4) 29

Q57.If the distance between the foci of an ellipse is 6 and the distance between its directrix is 12, then the length of its latus rectum is (1) √3 (2) 3√2 (3) 3 (4) 2√3 √2

Q57.Let the line y = mx and the ellipse 2x2 + y2 = 1 intersect at a point P in the first quadrant. If the normal to , (0, β), then β is equal to 0) and this ellipse at P meets the co-ordinate axes at (− 3√21 (1) 2√2 (2) 2 3 √3 (3) 2 (4) √2 3 3 JEE Main 2020 (08 Jan Shift 1) JEE Main Previous Year Paper Q58. 3x2+2 x21 lim is equal to x→0 ( 7x2+2 ) (1) 1 (2) 1 e e2 (3) e2 (4) e

Q57.The contrapositive of the statement "If I reach the station in time, then I will catch the train" is (1) If I do not reach the station in time, then I will (2) If do not reach the station in time, then I will not catch the train. catch the train. (3) If I will catch the train, then I reach the station in (4) If I will not catch the train, then I do not reach time. the station in time.

Q57.If e1 and e2 are the eccentricities of the ellipse x218 + y24 = 1 9 −y24 = 1 (e1, e2) is a point on the ellipse 15x2 + 3y2 = k , then the value of k is equal to (1) 16 (2) 17 (3) 15 (4) 14

Q57.If the length of the chord of the circle, x2 + y2 = r2(r > 0) along the line, y −2x = 3 is r, then r2 is equal to: (1) 9 (2) 12 5 (3) 24 (4) 12 5 5 JEE Main 2020 (05 Sep Shift 2) JEE Main Previous Year Paper

Q57.Let x = 4 be a directrix to an ellipse whose centre is at the origin and its eccentricity is 12 . If P(1, β), β > 0 is a point on this ellipse, then the equation of the normal to it at P is JEE Main 2020 (04 Sep Shift 2) JEE Main Previous Year Paper (1) 4x–3y = 2 (2) 8x–2y = 5 (3) 7x–4y = 1 (4) 4x–2y = 1

Q57.If the normal at an end of latus rectum of an ellipse passes through an extremity of the minor axis, then the eccentricity e of the ellipse satisfies : (1) e4 + 2e2 −1 = 0 (2) e2 + e −1 = 0 (3) e4 + e2 −1 = 0 (4) e2 + 2e −1 = 0

Q57.Let L1 be a tangent to the parabola y2 = 4(x + 1) and L2 be a tangent to the parabola y2 = 8(x + 2) such that L1 and L2 intersect at right angles. Then L1 and L2 meet on the straight line: (1) x + 3 = 0 (2) 2x + 1 = 0 (3) x + 2 = 0 (4) x + 2y = 0

Q57.A hyperbola having the transverse axis of length, √2 has the same foci as that of the ellipse, 3x2 + 4y2 = 12 then this hyperbola does not pass through which of the following points? 2 , (1) ( √21 , 0) (2) (−√3 1) (3) (1, −1√2 ) (4) (√3 2 , √21 )

Q58.For two statements p and q , the logical statement (p →q) ∧(q →~p) is equivalent to (1) p (2) q (3) ~p (4) ~q JEE Main 2020 (07 Jan Shift 1) JEE Main Previous Year Paper Q59. ⎡ 1 1 1 ⎤ Let α be a root of the equation x2 + x + 1 = 0 and the matrix A = 1 1 α α2 , then the matrix A31 is √3 ⎣ 1 α2 α4 ⎦ equal to (1) A3 (2) I3 (3) A2 (4) A

Q58.If α is the positive root of the equation, p(x) = x2 −x −2 = 0, then lim √1−cosx+α−4p(x) is equal to x→α+ (1) 23 (2) √23 (3) 1 (4) 12 √2

Q58.Contrapositive of the statement : 'If a function f is differentiable at a , then it is also continuous at a ', is (1) If a function f is continuous at a , then it is not differentiable at a . (2) If a function f is not continuous at a , then it is not differentiable at a . (3) If a function f is not continuous at a . then it is differentiable at a . (4) If a function f is continuous at a , then it is differentiable at a .

Q58. (a+2x) 31 −(3x) 31 lim 1 1 (a ≠0) is equal to: x→a (3a+x) 3 −(4x) 3 (1) 2 2 31 (2) 2 34 ( 9 )( 3 ) ( 3 ) (3) 2 34 (4) 2 2 31 ( 9 ) ( 3 )( 9 )

Q58.Let P(3, 3) be a point on the hyperbola, x2 −y2 = 1. If the normal to it at P intersects the x-axis at (9, 0) a2 b2 and e is its eccentricity, then the ordered pair (a2, e2) is equal to: (1) ( 29 , 3) (2) ( 32 , 2) (3) ( 29 , 2) (4) (9, 3)

Q58.If the line y = m x + c is a common tangent to the hyperbola 100x2 −y264 = 1 and the circle x2 + y2 = 36, then which one of the following is true? (1) c2 = 369 (2) 5m = 4 (3) 4c2 = 369 (4) 8m + 5 = 0