Practice Questions

Q70. (a + 1)(a + 2) a + 2 1 The value of (a + 2)(a + 3) a + 3 1 is (a + 3)(a + 4) a + 4 1 (1) 0 (2) (a + 2)(a + 3)(a + 4) (3) −2 (4) (a + 1)(a + 2)(a + 3)

Q70.In a school, there are three types of games to be played. Some of the students play two types of games, but none play all the three games. Which Venn diagrams can justify the above statement? (1) P and Q (2) P and R (3) Q and R (4) None of these then a possible value of α is

Q70.The mean and standard deviation of 20 observations were calculated as 10 and 2. 5 respectively. It was found that by mistake one data value was taken as 25 instead of 35. If α and √β are the mean and standard deviation respectively for correct data, then (α, β) is: (1) (10. 5, 26) (2) (10. 5, 25) (3) (11, 25) (4) (11, 26)

Q70.Let sinsin BA = sin(C−B)sin(A−C) , where A, B, C are angles of a triangle ABC. If the lengths of the sides opposite these angles are a, b, c respectively, then (1) b2, c2, a2 are in A.P. (2) c2, a2, b2 are in A.P. (3) b2 −a2 = a2 + c2 (4) a2, b2, c2 are in A.P. satisfies A(A3 + 3I) = 2I, then the value of K is

Q70.Let the mean and variance of the frequency distribution x : x1 = 2 x2 = 6 x3 = 8 x4 = 9 JEE Main 2021 (27 Jul Shift 2) JEE Main Previous Year Paper f : 4 4 α β be 6 and 6. 8 respectively. If x3 is changed from 8 to 7, then the mean for the new data will be: (1) 4 (2) 5 (3) 17 (4) 16 3 3

Q70.If 𝛼+ 𝛽+ 𝛾= 2𝜋, then the system of equations 𝑥+ cos𝛾𝑦+ cos𝛽𝑧= 0 cos𝛾𝑥+ 𝑦+ cos𝛼𝑧= 0 cos𝛽𝑥+ cos𝛼𝑦+ 𝑧= 0 has : (1) infinitely many solutions (2) a unique solution (3) no solution (4) exactly two solutions

Q70.Let f(x) = sin−1 x and g(x) = x2−x−2 . If g(2) = lim g(x), then the domain of the function fog is 2x2−x−6 x→2 (1) (−∞, −1] ∪[2, ∞) (2) (−∞, −2] ∪[−32 , ∞) (3) (−∞, −2] ∪[−43 , ∞) (4) (−∞, −2] ∪[−1, ∞) Q71. 2 sin(−πx2 ), if x < −1 ⎧ Let f : R→R be defined as f(x) = ax2 + x + b , if −1 ≤x ≤1 ⎨ ⎩sin(πx), if x > 1 If f(x) is continuous on R, then a + b equals : (1) 1 (2) 3 (3) −3 (4) −1

Q70.The compound statement (P ∨Q) ∧(~P) ⇒Q equivalent to: (1) P ∨Q (2) P ∧~Q (3) ~(P ⇒Q) (4) ~(P ⇒Q) ⇔P ∧~Q

Q70.A pole stands vertically inside a triangular park ABC . Let the angle of elevation of the top of the pole from each corner of the park be π . If the radius of the circumcircle of ΔABC is 2 , then the height of the pole is 3 equal to : (1) 2√3 (2) 2√3 3 (3) √3 (4) 1 √3

Q70.The statement A →(B →A) is equivalent to : (1) A →(A ∧B) (2) A →(A ∨B) (3) A →(A ↔B) (4) A →(A →B)

Q70.Consider the statement "The match will be played only if the weather is good and ground is not wet". Select the correct negation from the following: (1) The match will not be played and weather is not (2) If the match will not be played, then either good and ground is wet. weather is not good or ground is wet. (3) The match will be played and weather is not (4) The match will not be played or weather is good good or ground is wet. and ground is not wet.

Q70.The following system of linear equations 2x + 3y + 2z = 9 3x + 2y + 2z = 9 x −y + 4z = 8 (1) has infinitely many solutions (2) has a unique solution (3) has a solution (α, β, γ) satisfying (4) does not have any solution α + β2 + γ 3 = 12

Q70.If the Boolean expression (p ∧q) ⊛(p ⊗q) is a tautology, then ⊛ and ⊗ are respectively given by (1) →, → (2) ∧, ∨ (3) ∨, → (4) ∧, →

Q70.The number of real roots of the equation tan−1 √x(x + 1) + sin−1 √x2 + x + 1 = π4 is: (1) 1 (2) 2 (3) 4 (4) 0

Q70.Which of the following is not correct for relation R on the set of real numbers? (1) (x, y) ∈R ⇔|x| −|y| ≤1 is reflexive but not (2) (x, y) ∈R ⇔|x −y| ≤1 is reflexive and symmetric. symmetric. (3) (x, y) ∈R ⇔0 < |x −y| ≤1 is symmetric and (4) (x, y) ∈R ⇔0 < |x| −|y| ≤1 is not transitive transitive. but symmetric.

Q70. cos−1(1−{x}2) sin−1(1−{x}) ⎧ , x ≠0 Let α ∈R be such that the function f(x) = {x}−{x}3 is continuous at x = 0, where ⎨ ⎩α, x = 0 {x} = x −[x], [x] is the greatest integer less than or equal to x. Then : (1) α = π (2) α = 0 √2 (3) no such α exists (4) α = π4

Q70.Let 𝑓: 𝑅→𝑅 be defined as 𝑓𝑥= 2 𝑥- 1 and 𝑔: 𝑅- 1 →𝑅. be defined as 𝑔𝑥= 𝑥- 𝑥- 1. function 𝑓𝑔𝑥 is: (1) neither one-one nor onto (2) one-one but not onto (3) onto but not one-one (4) both one-one and onto

Q70.The value of tan(2 tan−1( 53 ) + sin−1( 135 )) is equal to: (1) −181 (2) 220 69 21 (3) −291 (4) 151 76 63

Q71.Let f : R →R be defined as ⎧−55x, if x < −5 f(x) = −120x, if −5 ≤x ≤4 ⎨2x3 −3x2 ⎩2x3 −3x2 −36x −336, if x > 4 Let A = {x ∈R : f is increasing}. Then A is equal to: (1) (−5, ∞) (2) (−5, −4) ∪(4, ∞) (3) (−∞, −5) ∪(−4, ∞) (4) (−∞, −5) ∪(4, ∞)

Q71. a1 a2 a3 If ar = cos 2rπ9 + i sin 2rπ9 , r = 1, 2, 3, … , i = √−1, then the determinant a4 a5 a6 is equal to : a7 a8 a9 (1) a9 (2) a1a9 −a3a7 (3) a5 (4) a2a6 −a4a8

Q71.If 𝑓: 𝑅→𝑅 is a function defined by 𝑓𝑥= 𝑥- 1cos2𝑥- 1 𝜋, where · denotes the greatest integer function, then 𝑓 2 is: (1) discontinuous only at 𝑥= 1 (2) discontinuous at all integral values of 𝑥 except at 𝑥= 1 (3) continuous only at 𝑥= 1 (4) continuous for every real 𝑥

Q71.A man is observing, from the top of a tower, a boat speeding towards the tower from a certain point A , with uniform speed. At that point, angle of depression of the boat with the man's eye is 30° (Ignore man's height). After sailing for 20 seconds, towards the base of the tower (which is at the level of water), the boat has reached a point B, where the angle of depression is 45°. Then the time taken (in seconds) by the boat from B to reach the base of the tower is : JEE Main 2021 (25 Feb Shift 1) JEE Main Previous Year Paper (1) 10 (2) 10(√3 −1) + (3) 10√3 (4) 10(√3 1)

Q71.If x, y, z are in arithmetic progression with common difference d, x ≠3d, and the determinant of the matrix 3 4√2 x ⎡ ⎤ is zero, then the value of k2 is 4 5√2 y 5 k z ⎣ ⎦ (1) 72 (2) 12 (3) 36 (4) 6

Q71.The first of the two samples in a group has 100 items with mean 15 and standard deviation 3. If the whole group has 250 items with mean 15. 6 and standard deviation √13. 44, then the standard deviation of the second sample is: (1) 8 (2) 6 (3) 4 (4) 5 1 0 50 then P is: 1

Q71.Define a relation R over a class of n × n real matrices A and B as " ARB iff there exists a non-singular matrix P such that PAP −1 = B". Then which of the following is true ? (1) R is symmetric, transitive but not reflexive (2) R is reflexive, symmetric but not transitive (3) R is an equivalence relation (4) R is reflexive, transitive but not symmetric