Practice Questions

Q67.The value of lim cos h−sin h) } h→0{ √3h(√3 (1) 43 (2) √32 (3) 23 (4) 43

Q67.Let F1(A, B, C) = (A ∧~B) ∨[~C ∧(A ∨B)] ∨~A and F2(A, B) = (A ∨B) ∨(B →~A) be two logical expressions. Then : (1) F1 is a tautology but F2 is not a tautology (2) F1 is not a tautology but F2 is a tautology (3) Both F1 and F2 are not tautologies (4) F1 and F2 both are tautologies

Q68.Let R = {(P, Q)|P and Q are at the same distance from the origin } be a relation, then the equivalence class of (1, −1) is the set JEE Main 2021 (26 Feb Shift 1) JEE Main Previous Year Paper (1) S = {(x, y) x2 + y2 = 1} (2) S = {(x, y) x2 + y2 = 2} (3) S = {(x, y) x2 + y2 = √2} (4) S = {(x, y) x2 + y2 = 4}

Q68.Let the equation of the pair of lines, y = px and y = qx, can be written as (y −px)(y −qx) = 0. Then the equation of the pair of the angle bisectors of the lines x2 −4xy −5y2 = 0 is: (1) x2 −3xy + y2 = 0 (2) x2 + 4xy −y2 = 0 (3) x2 + 3xy −y2 = 0 (4) x2 −3xy −y2 = 0

Q68.If P and Q are two statements, then which of the following compound statement is a tautology? (1) ((P ⇒Q) ∧~Q) ⇒Q (2) ((P ⇒Q) ∧~Q) ⇒~P (3) ((P ⇒Q) ∧~Q) ⇒P (4) ((P ⇒Q) ∧~Q) ⇒(P ∧Q)

Q68.If α, β are the distinct roots of x2 + bx + c = 0, then lim e2(x2+bx+c)−1−2(x2+bx+c) is equal to x→β (x−β)2 (1) 2(b2 + 4c) (2) b2 −4c (3) 2(b2 −4c) (4) b2 + 4c

Q68.A ray of light through (2, 1) is reflected at a point P on the y− axis and then passes through the point (5, 3). If this reflected ray is the directrix of an ellipse with eccentricity 1 and the distance of the nearer focus from this 3 directrix is 8 , then the equation of the other directrix can be: √53 JEE Main 2021 (27 Jul Shift 1) JEE Main Previous Year Paper (1) 11x + 7y + 8 = 0 or 11x + 7y −15 = 0 (2) 11x −7y −8 = 0 or 11x + 7y + 15 = 0 (3) 2x −7y + 29 = 0 or 2x −7y −7 = 0 (4) 2x −7y −39 = 0 or 2x −7y −7 = 0 x2f(2)−4f(x) is equal to:

Q68.Which of the following Boolean expression is a tautology ? (1) (p ∧q) ∨(p ∨q) (2) (p ∧q) ∨(p →q) (3) (p ∧q) ∧(p →q) (4) (p ∧q) →(p →q)

Q68.A spherical gas balloon of radius 16 meter subtends an angle 60° at the eye of the observer 𝐴 while the angle of elevation of its center from the eye of 𝐴 is 75°. Then the height (in meter) of the top most point of the balloon from the level of the observer's eye is : (1) 8 ( 2 + 2√3 + √2 ) (2) 8 ( √6 + √2 + 2 ) (3) 8 ( √2 + 2 + √3 ) (4) 8 ( √6 - √2 + 2 )

Q68.Let *, □∈{∧, ∨} be such that the Boolean expression (p*~q) ⇒(p □q) is a tautology. Then : (1) *= ∨, □= ∧ (2) *= ∨, □= ∨ (3) *= ∧, □= ∨ (4) *= ∧, □= ∧

Q68.The value of lim cos−1(x−[x]2)⋅sin−1(x−[x]2) , where [x] denotes the greatest integer ≤x is: x→0+ x−x3 (1) π (2) 0 (3) π (4) π 4 2

Q68.If the curves, x2 intersect each other at an angle of 90°, then which of the a + b = 1 and x2c + y2d = 1 following relations is TRUE? (1) a −c = b + d (2) a −b = c −d (3) a + b = c + d (4) ab = a+bc+d 1 1 n 1+ 2 +……+ n

Q68.Consider the following system of equations: x + 2y −3z = a 2x + 6y −11z = b x −2y + 7z = c where a, b and c are real constants. Then the system of equations : (1) has a unique solution when 5a = 2b + c (2) has no solution for all a, b and c (3) has infinite number of solutions when (4) has a unique solution for all a, b and c 5a = 2b + c

Q68.Consider the two statements : (S1) : (p →q) ∨(~q →p) is a tautology. (S2) : (p ∧~q) ∧(~p ∨q) is a fallacy. Then : (1) only (S1) is true. (2) both (S1) and (S2) are false. (3) only (S2) is true. (4) both (S1) and (S2) are true. Q69. ⎡ 1 0 0⎤ Let A = 0 1 1 . Then A2025 −A2020 is equal to ⎣ 1 0 0⎦ (1) A6 −A (2) A6 (3) A5 (4) A5 −A

Q68. sin2 x 1 + cos2 x cos 2x The maximum value of f(x) = 1 + sin2 x cos2 x cos 2x , x ∈R is sin2 x cos2 x sin 2x (1) √7 (2) 34 (3) √5 (4) 5

Q68.The number of integral values of m so that the abscissa of point of intersection of lines 3x + 4y = 9 and y = mx + 1 is also an integer, is: (1) 1 (2) 2 (3) 3 (4) 0

Q68.The value of x→0( (1) 0 (2) 4 (3) −4 (4) −1

Q68.Negation of the statement ( 𝑝∨𝑟) ⇒( 𝑞∨𝑟) is : (1) ~𝑝∧𝑞∧~𝑟 (2) ~𝑝∧𝑞∧𝑟 (3) 𝑝∧~𝑞∧~𝑟 (4) 𝑝∧𝑞∧𝑟

Q68.The value of lim [r]+[2r]+...+[nr] , where r is non-zero real number and [r] denotes the greatest integer less than n→∞ n2 or equal to r, is equal to : (1) r (2) r 2 (3) 2r (4) 0

Q68.Let in a right angled triangle, the smallest angle be θ. If a triangle formed by taking the reciprocal of its sides is also a right angled triangle, then sin θ is equal to: (1) √5+1 (2) √5−1 4 2 (3) √2−1 (4) √5−1 2 4

Q68.Two vertical poles are 150 m apart and the height of one is three times that of the other. If from the middle point of the line joining their feet, an observer finds the angles of elevation of their tops to be complementary, then the height of the shorter pole (in meters) is: (1) 25 (2) 30 (3) 20√3 (4) 25√3

Q68.Let A and B be 3 × 3 real matrices such that A is a symmetric matrix and B is a skew-symmetric matrix. Then the system of linear equations (A2 B2 −B2 A2)X = O, where X is a 3 × 1 column matrix of unknown variables and O is a 3 × 1 null matrix, has (1) exactly two solutions (2) infinitely many solutions (3) a unique solution (4) no solution is:

Q68.Which of the following is equivalent to the Boolean expression p ∧~q ? (1) ~p →~q (2) ~ ( q →p ) (3) ~ ( 𝑝→𝑞) (4) ~ ( 𝑝→~𝑞)

Q68.On the ellipse x2 8 + 4 = 1, let P be a point in the second quadrant such that the tangent at P to the ellipse is perpendicular to the line x + 2y = 0. Let S and S′ be the foci of the ellipse and e be its eccentricity. If A is the JEE Main 2021 (26 Aug Shift 1) JEE Main Previous Year Paper area of the triangle SPS′ , then the value of (5 −e2) ⋅A is (1) 12 (2) 6 (3) 14 (4) 24

Q68.If for the matrix, A = [ α1 −αβ ], (1) 3 (2) 1 (3) 2 (4) 4