Practice Questions

Q56.If the equation cos4 θ + sin4 θ + λ = 0 has real solutions for θ then λ lies in interval (1) (−54 , −1) (2) [−1, −12 ] (3) (−12 , −14 ] (4) [−32 , −54 ]

Q56.A line parallel to the straight line 2x −y = 0 is tangent to the hyperbola x24 −y22 = 1 at the point (x1, y1). Then x21 + 5y21 is equal to (1) 6 (2) 8 (3) 10 (4) 5 JEE Main 2020 (02 Sep Shift 1) JEE Main Previous Year Paper

Q56.If a hyperbola passes through the point P(10, 16), and it has vertices at (±6, 0), then the equation of the normal to it at P , is. (1) 3x + 4y = 94 (2) 2x + 5y = 100 (3) x + 2y = 42 (4) x + 3y = 58

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.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 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.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.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 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.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.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 )

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.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.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.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

Q58.Let X = {x ∈N : 1 ≤x ≤17} and Y = {ax + b : x ∈X and a, b ∈R, a > 0} . If mean and variance of elements of Y are 17 and 216 respectively then a + b is equal to (1) 7 (2) −7 (3) −27 (4) 9

Q58.Which of the following points lies on the locus of the foot of perpendicular drawn upon any tangent to the x2 y2 ellipse, 4 + 2 = 1 from any of its foci? (1) (−2, √3) (2) (−1, √2) (3) (−1, √3) (4) (1, 2)

Q58.Let [t] denote the greatest integer ≤t. If λ ε R −{0, 1}, lim 1−x+|x| = L, then L is equal to x→0 λ−x+[x] (1) 1 (2) 2 (3) 1 (4) 0 2

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.The length of the minor axis (along y-axis) of an ellipse in the standard form is 4 . If this ellipse touches the √3 line x + 6y = 8 then its eccentricity is: (1) 1 (2) 2 √113 √56 (3) 1 (4) 1 2 √53 3 √113

Q58.Let the tangents drawn from the origin to the circle, x2 + y2 −8x −4y + 16 = 0 touch it at the points A and B . Then (AB)2 is equal to (1) 52 (2) 56 5 5 (3) 64 (4) 32 5 5 y2

Q58.The mean and variance of 20 observations are found to be 10 and 4, respectively. On rechecking, it was found that an observation 9 was incorrect and the correct observation was 11, then the correct variance is (1) 3.99 (2) 4.01 (3) 4.02 (4) 3.98

Q59.Which one of the following is a tautology? (1) (p ∧(p →q)) →q (2) q →(p ∧(p →q)) (3) p ∧(p ∨q) (4) p ∨(p ∧q)

Q59.Given the following two statements: (S1) : (q ∨p) →(p ↔~q) is a tautology (S2) : ~q ∧(~p ↔q) is a fallacy. Then : (1) both (S1) and (S2) are not correct. (2) only (S1) is correct. (3) only (S2) is correct. (4) both (S1) and (S2) are correct.

Q59. x(e(√1+x2+x4−1)/x−1) lim x→0 √1+x2+x4−1 (1) is equal to √e (2) is equal to 1 (3) is equal to 0 (4) does not exist