Practice Questions

Q67.Let A = {(x, y) ∈R × R ∣2x2 + 2y2 −2x −2y = 1} B = {(x, y) ∈R × R ∣4x2 + 4y2 −16y + 7 = 0} and C = {(x, y) ∈R × R ∣x2 + y2 −4x −2y + 5 ≤r2}. Then the minimum value of |r| such that A ∪B ⊆C is equal to (1) 3+√10 (2) 2+√10 2 2 (3) 3+2√5 (4) 1 + √5 2

Q67.Let 𝜃 be the acute angle between the tangents to the ellipse 𝑥2 + 𝑦2 = 1 and the circle 𝑥2 + 𝑦2 = 3 at their 9 1 point of intersection in the first quadrant. Then tan𝜃 is equal to : (1) 5 (2) 4 2√3 √3 (3) 2 (4) 2 √3

Q67.The statement among the following that is a tautology is: JEE Main 2021 (24 Feb Shift 1) JEE Main Previous Year Paper (1) 𝐴∨𝐴∧𝐵 (2) 𝐴∧𝐴∨𝐵 (3) 𝐵→𝐴∧𝐴→𝐵 (4) 𝐴∧𝐴→𝐵→𝐵

Q67. lim sin2(π cos4 x) is equal to : x4 x→0 (1) 2π2 (2) π2 (3) 4π2 (4) 4π

Q67.The mean of 6 distinct observations is 6. 5 and their variance is 10. 25. If 4 out of 6 observations are 2, 4, 5 and 7, then the remaining two observations are: (1) 10, 11 (2) 3, 18 (3) 8, 13 (4) 1, 20

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

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

Q68.Let Z be the set of all integers, A = {(x, y) ∈Z × Z : (x −2)2 + y2 ≤4} B = {(x, y) ∈Z × Z : x2 + y2 ≤4} and C = {(x, y) ∈Z × Z : (x −2)2 + (y −2)2 ≤4} If the total number of relations from A ∩B to A ∩C is 2p , then the value of p is: (1) 25 (2) 9 (3) 16 (4) 49

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.Let A = [aij] be a real matrix of order 3 × 3, such that ai1 + ai2 + ai3 = 1, for i = 1, 2, 3 . Then, the sum of all the entries of the matrix A3 is equal to: (1) 2 (2) 1 (3) 3 (4) 9 JEE Main 2021 (22 Jul Shift 1) JEE Main Previous Year Paper

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

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