Practice Questions

Q69. lim + n→∞(1 n2 ) is equal to (1) 1 (2) 0 e (3) 1 (4) 1 2

Q69.Let in a series of 2n observations, half of them are equal to a and remaining half are equal to −a. Also by adding a constant b in each of these observations, the mean and standard deviation of new set become 5 and 20 , respectively. Then the value of a2 + b2 is equal to : (1) 425 (2) 650 (3) 250 (4) 925

Q69.Let A be a 3 × 3 matrix with det (A) = 4. Let Ri denote the ith row of A . If a matrix B is obtained by performing the operation R2 →2R2 + 5R3 on 2 A , then det (B) is equal to : (1) 64 (2) 16 (3) 128 (4) 80

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.Consider the system of linear equations -𝑥+ 𝑦+ 2𝑧= 0 3𝑥- 𝑎𝑦+ 5𝑧= 1 2𝑥- 2𝑦- 𝑎𝑧= 7 Let 𝑆1 be the set of all 𝑎∈𝑅 for which the system is inconsistent and 𝑆2 be the set of all 𝑎∈𝑅 for which the system has infinitely many solutions. If nS1 and nS2 denote the number of elements in S1 and S2 respectively, then (1) nS1 = 2, nS2 = 0 (2) nS1 = 2, nS2 = 2 (3) nS1 = 0, nS2 = 2 (4) nS1 = 1, nS2 = 0

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.If the Boolean expression (p ⇒q) ⇔(q*(~p)) is a tautology, then the Boolean expression p*(~q) is equivalent to: (1) q ⇒p (2) ~q ⇒p (3) p ⇒~q (4) p ⇒q

Q69.The statement (p ∧(p →q) ∧(q →r)) →r is (1) a tautology (2) equivalent to q →~r (3) a fallacy (4) equivalent to p →~r JEE Main 2021 (27 Aug Shift 1) JEE Main Previous Year Paper

Q69.The value of the limit lim tan(π cos2 θ) is equal to : θ→0 sin(2π sin2 θ) (1) −12 (2) −14 (3) 0 (4) 14

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

Q69.Consider three observations a, b and c such that b = a + c . If the standard deviation of a + 2, c + 2 is d , then which of the following is true? (1) b2 = 3(a2 + c2) + 9d2 (2) b2 = a2 + c2 + 3d2 (3) b2 = 3(a2 + c2 + d2) (4) b2 = 3(a2 + c2) −9d2 has : i = √−1. Then, the system of linear equations = A8[ xy ]

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.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

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 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 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

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.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 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.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.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.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.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