Practice Questions

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.A vertical pole fixed to the horizontal ground is divided in the ratio 3 : 7 by a mark on it with lower part shorter than the upper part. If the two parts subtend equal angles at a point on the ground 18 m away from the base of the pole, then the height of the pole (in meters) is : JEE Main 2021 (31 Aug Shift 1) JEE Main Previous Year Paper (1) 8√10 (2) 6√10 (3) 12√10 (4) 12√15

Q69.Let [λ] be the greatest integer less than or equal to λ. The set of all values of λ for which the system of linear equations x + y + z = 4, 3x + 2y + 5z = 3, 9x + 4y + (28 + [λ])z = [λ] has a solution is: (1) R (2) (−∞, −9) ∪[−8, ∞) (3) (−∞, −9) ∪(−9, ∞) (4) [−9, −8) Q70. ⎡[x + 1] [x + 2] [x + 3]⎤ Let A = [x] [x + 3] [x + 3] , where [x] denotes the greatest integer less than or equal to x. If ⎣ [x] [x + 2] [x + 4] ⎦ det (A)= 192 , then the set of values of x is in the interval: (1) [62, 63) (2) [65, 66) (3) [60, 61) (4) [68, 69) = x ∈( π2 , π), then dxdy at x = 5π6 is:

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

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

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.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.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.Two fair dice are thrown. The numbers on them are taken as λ and μ, and a system of linear equations x + y + z = 5 x + 2y + 3z = μ x + 3y + λz = 1 is constructed. If p is the probability that the system has a unique solution and q is the probability that the system has no solution, then: (1) p = 16 and q = 365 (2) p = 65 and q = 361 (3) p = 16 and q = 361 (4) p = 65 and q = 365

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.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.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.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.Let A = [ −ii −ii ], [ 648 ] (1) A unique solution (2) Infinitely many solutions (3) No solution (4) Exactly two solutions lim is equal to :

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)