Practice Questions

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.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.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.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 .

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. 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.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.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.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.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.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.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.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.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.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 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)

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.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

Q60.Let A be a 2 × 2 real matrix with entries from {0, 1} and |A| ≠0 . Consider the following two statements; (P) If A ≠l2 , then |A| = −1 (Q) If |A| = 1 , then tr(A) = 2 Where l2 denotes 2 × 2 identity matrix and tr(A) denotes the sum of the diagonal entries of A . Then (1) (P) is false and (Q) is true (2) Both (P) and (Q) are false (3) (P) is true and (Q) is false (4) Both (P) and (Q) are true

Q60.For the frequency distribution: Variate (x) : x1, x2, x3, … , x15 Frequency (f) : f1, f2, f3, … , f15 where 0 < x1 < x2 < x3 < … < x15 = 10 and ∑15i=1 fi > 0, the standard deviation cannot be (1) 4 (2) 1 (3) 6 (4) 2

Q60.Let 50∪ = ∪n = T , where each Xi contains 10 elements and each Yi contains 5 elements. If each element i=1Xi i=1Yi of the set T is an element of exactly 20 of sets Xi 's and exactly 6 of sets Yi 's then n is equal to : (1) 15 (2) 50 (3) 45 (4) 30

Q60.The statement (p →(q →p)) →(p →(p ∨q)) is : (1) equivalent to (p ∧q) ∨(~q) (2) a contradiction (3) equivalent to (p ∨q) ∧(~p) (4) a tautology

Q60.Let A, B, C and D be four non-empty sets. The contrapositive statement of “If A ⊆B and B ⊆D , then A ⊆C ” is (1) If A ⊈C , then A ⊆B and B ⊆D (2) If A ⊆C , then B ⊂A and D ⊂B (3) If A ⊈C , then A ⊈B and B ⊆D (4) If A ⊈C , then A ⊈B or B ⊈D