Practice Questions

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.The values of λ and μ such that the system of equations x + y + z = 6, 3x + 5y + 5z = 26 and x + 2y + λz = μ has no solution, are: (1) λ = 3, μ = 5 (2) λ = 3, μ ≠10 (3) λ ≠2, μ = 10 (4) λ = 2, μ ≠10

Q69.Given that the inverse trigonometric functions take principal values only. Then, the number of real values of x which satisfy sin−1( 3x5 ) + sin−1( 4x5 ) = sin−1 x is equal to: (1) 2 (2) 1 (3) 3 (4) 0

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.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.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.If a tangent to the ellipse x2 + 4y2 = 4 meets the tangents at the extremities of its major axis at B and C, then the circle with BC as diameter passes through the point. (1) (√3, 0) (2) (√2, 0) (3) (1, 1) (4) (−1, 1)

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.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:

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.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 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.If the truth value of the Boolean expression ((p ∨q) ∧(q →r) ∧(~r)) →(p ∧q) is false, then the truth values of the statements p, q, r respectively can be: (1) FTF (2) TFF (3) TFT (4) FFT

Q69.Let A = {1, 2, 3, … , 10} and f : A →A be defined as + 1 if k is odd f(k) = {k k if k is even JEE Main 2021 (26 Feb Shift 2) JEE Main Previous Year Paper Then the number of possible functions g : A →A such that gof = f is: (1) 10C5 (2) 55 (3) 5! (4) 105

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

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.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

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.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.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.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.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.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