Practice Questions

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

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

Q68.If in a triangle ABC, AB = 5 units, ∠B = cos−1( 53 ) and radius of circumcircle of ΔABC is 5 units, then the area (in sq. units) of ΔABC is: (1) 10 + 6√2 (2) 8 + 2√2 (3) 6 + 8√3 (4) 4 + 2√3 a ∈R be written as P + Q where P is a symmetric matrix and Q is skew symmetric matrix.

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

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.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 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.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.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.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

Q69.Let A = [2a 30 ], If det (Q) = 9 , then the modulus of the sum of all possible values of determinant of P is equal to: (1) 36 (2) 24 (3) 45 (4) 18

Q69.A possible value of tan( 41 sin−1 √638 ) (1) 2√2 −1 (2) 1 2√2 (3) √7 −1 (4) 1 √7

Q69.The value of k ∈R, for which the following system of linear equations 3x −y + 4z = 3 x + 2y −3z = −2 JEE Main 2021 (20 Jul Shift 2) JEE Main Previous Year Paper 6x + 5y + kz = −3 has infinitely many solutions, is: (1) 3 (2) −5 (3) 5 (4) −3

Q69.The values of 𝑎 and 𝑏, for which the system of equations 2𝑥+ 3𝑦+ 6𝑧= 8 𝑥+ 2𝑦+ 𝑎𝑧= 5 3𝑥+ 5𝑦+ 9𝑧= 𝑏 JEE Main 2021 (25 Jul Shift 1) JEE Main Previous Year Paper has no solution, are : (1) 𝑎= 3, 𝑏≠13 (2) 𝑎≠3, 𝑏≠13 (3) 𝑎≠3, 𝑏= 3 (4) 𝑎= 3, 𝑏= 13

Q69.The system of linear equations 3𝑥- 2𝑦- 𝑘𝑧= 10 2𝑥- 4𝑦- 2𝑧= 6 𝑥+ 2𝑦- 𝑧= 5 𝑚 is inconsistent if : 4 4 (1) 𝑘= 3, 𝑚≠ (2) 𝑘= 3, 𝑚= 5 5 (3) 𝑘≠3, 𝑚∈𝑅 (4) 𝑘≠3, 𝑚≠4 5 1 2 Then the composition

Q69.Which of the following is the negation of the statement "for all M > 0, there exists x ∈S such that x ≥M ′′? (1) there exists M > 0, such that x < M for all (2) there exists M > 0, there exists x ∈S such that x ∈S x ≥M (3) there exists M > 0, there exists x ∈S such that (4) there exists M > 0 such that x ≥M for all x < M x ∈S

Q69.Let A be a symmetric matrix of order 2 with integer entries. If the sum of the diagonal elements of A2 is 1, then the possible number of such matrices is: (1) 12 (2) 4 (3) 1 (4) 6