Practice Questions

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.The mean and variance of 7 observations are 8 and 16 respetively. If two observations are 6 and 8, then the variance of the remaining 5 observations is : (1) 92 (2) 134 5 5 112 536 (3) (4) 5 25

Q69.The equation of one of the straight lines which passes through the point (1, 3) and makes an angles with the straight line, y + 1 = 3√2x is tan−1(√2) + + = 0 (1) 4√2x + 5y −(15 4√2) = 0 (2) 5√2x + 4y −(15 4√2) + = 0 (3) 4√2x + 5y −4√2 = 0 (4) 4√2x −5y −(5 4√2)

Q69.Let f : R →R be a function such that f(2) = 4 and f ′(2) = 1. Then, the value of lim x−2 x→2 (1) 4 (2) 8 (3) 16 (4) 12

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

Q70.Let 𝑔: 𝑁→𝑁 be defined as 𝑔( 3𝑛+ 1 ) = 3𝑛+ 2 𝑔( 3𝑛+ 2 ) = 3𝑛+ 3 𝑔( 3𝑛+ 3 ) = 3𝑛+ 1, for all 𝑛≥0 Then which of the following statements is true ? (1) There exists an onto function 𝑓: 𝑁→𝑁 such that (2) There exists a one-one function 𝑓: 𝑁→𝑁 such 𝑓𝑜𝑔= 𝑓 that 𝑓𝑜𝑔= 𝑓 (3) 𝑔𝑜𝑔𝑜𝑔= 𝑔 (4) There exists a function 𝑓: 𝑁→𝑁 such that 𝑔𝑜𝑓= 𝑓

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.Let [x] denote the greatest integer less than or equal to x. Then, the values of x ∈R satisfying the equation [ex]2 + [ex + 1] −3 = 0 lie in the interval: (1) [0, 1e ) (2) [loge 2, loge 3) (3) [1, e) (4) [0, loge 2)

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.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.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

Q70.For which of the following curves, the line x + √3y = 2√3 is the tangent at the point ( 3√32 , 12 )? (1) 2x2 −18y2 = 9 (2) y2 = 1 x 6√3 (3) x2 + 9y2 = 9 (4) x2 + y2 = 7

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.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 (cos( - 5) ) + sin-1 (sin(6) ) - tan-1 (tan(12) ) is equal to : (The inverse trigonometric functions take the principal values) (1) 3𝜋+ 1 (2) 3𝜋- 11 (3) 4𝜋- 11 (4) 4𝜋- 9

Q70.Choose the correct statement about two circles whose equations are given below: x2 + y2 −10x −10y + 41 = 0 x2 + y2 −22x −10y + 137 = 0 (1) circles have same centre (2) circles have no meeting point (3) circles have only one meeting point (4) circles have two meeting points

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