Practice Questions

Q68.Let A and B be 3 × 3 real matrices such that A is a symmetric matrix and B is a skew-symmetric matrix. Then the system of linear equations (A2 B2 −B2 A2)X = O, where X is a 3 × 1 column matrix of unknown variables and O is a 3 × 1 null matrix, has (1) exactly two solutions (2) infinitely many solutions (3) a unique solution (4) no solution is:

Q68. sin2 x 1 + cos2 x cos 2x The maximum value of f(x) = 1 + sin2 x cos2 x cos 2x , x ∈R is sin2 x cos2 x sin 2x (1) √7 (2) 34 (3) √5 (4) 5

Q68.The value of lim [r]+[2r]+...+[nr] , where r is non-zero real number and [r] denotes the greatest integer less than n→∞ n2 or equal to r, is equal to : (1) r (2) r 2 (3) 2r (4) 0

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.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.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.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.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.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.A possible value of tan( 41 sin−1 √638 ) (1) 2√2 −1 (2) 1 2√2 (3) √7 −1 (4) 1 √7

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

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.Which of the following is the negation of the statement "for all M > 0, there exists x ∈S such that x ≥M ′′? (1) there exists M > 0, such that x < M for all (2) there exists M > 0, there exists x ∈S such that x ∈S x ≥M (3) there exists M > 0, there exists x ∈S such that (4) there exists M > 0 such that x ≥M for all x < M x ∈S

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.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.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.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.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