Practice Questions

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 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 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 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.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.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.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.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 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

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.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.Consider the statement: "For an integer n, if n3 −1 is even, then n is odd". The contrapositive statement of this statement is: (1) For an integer n, if n is even, then n3 −1 is odd. (2) For an integer n, if n3 −1 is not even, then n is not odd. (3) For an integer n, if n is even, then n3 −1 is even.(4) For an integer n , if n is odd, then n3 −1 is even.

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.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.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.Negation of the statement: √5 is an integer or 5 is irrational is: (1) √5 is not an integer 5 is not irrational (2) √5 is not an integer and 5 is not irrational (3) √5 is irrational or 5 is an integer (4) √5 is an integer and 5 irrational JEE Main 2020 (09 Jan Shift 1) JEE Main Previous Year Paper

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.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

Q58.The area (in sq. units) of an equilateral triangle inscribed in the parabola y2 = 8x, with one of its vertices on the vertex of this parabola is (1) 64√3 (2) 256√3 (3) 192√3 (4) 128√3 JEE Main 2020 (02 Sep Shift 2) JEE Main Previous Year Paper

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

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 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

Q59.If 3x + 4y = 12√2 is a tangent o the ellipse x2 + 9 = 1 for some a ∈R, then the distance between the foci a2 of the ellipse is (1) 2√7 (2) 4 (3) 2√5 (4) 2√2

Q59.The proposition p →~(p ∧~q) is equivalent to : (1) q (2) (~p) ∨q (3) (~p) ∧q (4) (~p) ∨(~q)