Practice Questions

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.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 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.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.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.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 [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.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.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)

Q59.If p →(p ∧~q) is false, then the truth values of p and q are respectively (1) F, F (2) T, F (3) T, T (4) F, T JEE Main 2020 (09 Jan Shift 2) JEE Main Previous Year Paper

Q59.Let p, q, r be three statements such that the truth value of (p ∧q) →(~q ∨r) is F . Then the truth values of p, q, r are respectively : (1) T, T, F (2) T, T, T (3) T, F, T (4) F, T, F

Q59.If R = {(x, y) : x, y ∈Z, x2 + 3y2 ≤8} is a relation on the set of integers Z , then the domain of R−1 is (1) {−2, −1, 1, 2} (2) {0, 1} (3) {−2, −1, 0, 1, 2} (4) {−1, 0, 1}

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.For some θ ∈(0, π2 ), if the eccentricity of the hyperbola, x2 −y2 sec2 θ = 10 is √5 times the eccentricity of the ellipse, x2 sec2 θ + y2 = 5, then the length of the latus rectum of the ellipse, is (1) 2√6 (2) √30 (3) 2√5 (4) 4√5 3 3

Q59.The negation of the Boolean expression p ∨(~p ∧q) is equivalent to : (1) p ∧~q (2) ~p ∧~q (3) ~p ∨~q (4) ~p ∨q n n

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

Q59.Let the observation xi(1 ≤i ≤10) satisfy the equations ∑10i=1(xi −5) = 10 , ∑10i=1 (xi −5)2 = 40 . If μ and λ are the mean and the variance of the observations, x1 −3, x2 −3, . . . . , x10 −3, then the ordered pair (μ, λ) is equal to: (1) (3,3) (2) (6,3) (3) (6,6) (4) (3,6) Q60. ⎡1 1 2⎤ |adjB| If A = 1 3 4 , B = adjA and C = 3A, then is equal to ⎣1 −1 3⎦ |C| (1) 8 (2) 16 (3) 72 (4) 2

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

Q59.The angle of elevation of a cloud C from a point P, 200 m above a still take is 30o . If the angle of depression of the image of C in the lake from the point P is 60o , then PC (in m) is equal to (1) 100 (2) 200√3 (3) 400 (4) 400√3

Q59.The angle of elevation of the summit of a mountain from a point on the ground is 45° . After climbing up one km towards the summit at an inclination of 30° from the ground, the angle of elevation of the summit is found to be 60° . Then the height (in km) of the summit from the ground is : (1) √3−1 (2) √3+1 √3+1 √3−1 (3) 1 (4) 1 √3−1 √3+1 π

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.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.If A = (29 24 ) and I = (10 01 ), then 10 A−1 , is equal to. (1) A −4I (2) 6I −A (3) A −6I (4) 4I −A

Q59.The negation of the Boolean expression x ↔~y is equivalent to: (1) (~x ∧y) ∨(~x ∧~y) (2) (x ∧y) ∨(~x ∧~y) (3) (x ∧~y) ∨(~x ∧y) (4) (x ∧y) ∧(~x ∨~y)

Q60.The mean and the standard deviation (s.d.) of 10 observations are 20 and 2 respectively. Each of these 10 observations is multiplied by p and then reduced by q, where p ≠0 and q ≠0. If the new mean and new s.d. become half of their original values, then q is equal to (1) −5 (2) 10 (3) −20 (4) −10