Practice Questions

Q59.The angle of elevation of a cloud C from a point P, 200 m above a still take is 30o . If the angle of depression of the image of C in the lake from the point P is 60o , then PC (in m) is equal to (1) 100 (2) 200√3 (3) 400 (4) 400√3

Q59.The negation of the Boolean expression x ↔~y is equivalent to: (1) (~x ∧y) ∨(~x ∧~y) (2) (x ∧y) ∨(~x ∧~y) (3) (x ∧~y) ∨(~x ∧y) (4) (x ∧y) ∧(~x ∨~y)

Q59.If A = (29 24 ) and I = (10 01 ), then 10 A−1 , is equal to. (1) A −4I (2) 6I −A (3) A −6I (4) 4I −A

Q59.If 3x + 4y = 12√2 is a tangent o the ellipse x2 + 9 = 1 for some a ∈R, then the distance between the foci a2 of the ellipse is (1) 2√7 (2) 4 (3) 2√5 (4) 2√2

Q59.If R = {(x, y) : x, y ∈Z, x2 + 3y2 ≤8} is a relation on the set of integers Z , then the domain of R−1 is (1) {−2, −1, 1, 2} (2) {0, 1} (3) {−2, −1, 0, 1, 2} (4) {−1, 0, 1}

Q59.For some θ ∈(0, π2 ), if the eccentricity of the hyperbola, x2 −y2 sec2 θ = 10 is √5 times the eccentricity of the ellipse, x2 sec2 θ + y2 = 5, then the length of the latus rectum of the ellipse, is (1) 2√6 (2) √30 (3) 2√5 (4) 4√5 3 3

Q59.The angle of elevation of the summit of a mountain from a point on the ground is 45° . After climbing up one km towards the summit at an inclination of 30° from the ground, the angle of elevation of the summit is found to be 60° . Then the height (in km) of the summit from the ground is : (1) √3−1 (2) √3+1 √3+1 √3−1 (3) 1 (4) 1 √3−1 √3+1 π

Q59.Let the observation xi(1 ≤i ≤10) satisfy the equations ∑10i=1(xi −5) = 10 , ∑10i=1 (xi −5)2 = 40 . If μ and λ are the mean and the variance of the observations, x1 −3, x2 −3, . . . . , x10 −3, then the ordered pair (μ, λ) is equal to: (1) (3,3) (2) (6,3) (3) (6,6) (4) (3,6) Q60. ⎡1 1 2⎤ |adjB| If A = 1 3 4 , B = adjA and C = 3A, then is equal to ⎣1 −1 3⎦ |C| (1) 8 (2) 16 (3) 72 (4) 2

Q60.Let A, B, C and D be four non-empty sets. The contrapositive statement of “If A ⊆B and B ⊆D , then A ⊆C ” is (1) If A ⊈C , then A ⊆B and B ⊆D (2) If A ⊆C , then B ⊂A and D ⊂B (3) If A ⊈C , then A ⊈B and B ⊆D (4) If A ⊈C , then A ⊈B or B ⊈D

Q60.The mean and the standard deviation (s.d.) of 10 observations are 20 and 2 respectively. Each of these 10 observations is multiplied by p and then reduced by q, where p ≠0 and q ≠0. If the new mean and new s.d. become half of their original values, then q is equal to (1) −5 (2) 10 (3) −20 (4) −10

Q60.The system of linear equations λx + 2y + 2z = 5 2λx + 3y + 5z = 8 4x + λy + 6z = 10 has (1) no solution when λ = 8 (2) a unique solution when λ = −8 (3) no solution when λ = 2 (4) infinitely many solutions when λ = 2

Q60.The mean and variance of 8 observations are 10 and 13. 5, respectively. If 6 of these observations are 5, 7, 10, 12, 14, 15, then the absolute difference of the remaining two observations is : (1) 9 (2) 5 (3) 3 (4) 7

Q60.The mean and variance of 7 observations are 8 and 16, respectively. If five observations are 2, 4, 10, 12, 14 then the absolute difference of the remaining two observations is : (1) 1 (2) 4 (3) 2 (4) 3 JEE Main 2020 (05 Sep Shift 1) JEE Main Previous Year Paper

Q60.Let xi(1 ≤i ≤10) be ten observation of a random variable X . If ∑10i=1(xi −p) = 3 and ∑10i=1 (xi −p)2 = 9 where 0 ≠p ∈R, then the standard deviation of these observations is: (1) 4 (2) 5 √35 (3) 9 (4) 7 10 10

Q60.The following system of linear equations 7x + 6y −2z = 0 3x + 4y + 2z = 0 x −2y −6z = 0, has (1) infinitely many solutions, (x, y, z) satisfying (2) no solution y = 2z (3) infinitely many solutions, (x, y, z) satisfying (4) only the trivial solution x = 2z

Q60.Let θ = and A = . If B = A + A4 , then det (B) : 5 [−sinθcosθ cosθsinθ ] (1) is one (2) lies in (2, 3) (3) is zero (4) lies in (1, 2)

Q60. lim (tan( π4 + x))1/x is equal to x→0 (1) e (2) 2 (3) 1 (4) e2

Q60.The statement (p →(q →p)) →(p →(p ∨q)) is : (1) equivalent to (p ∧q) ∨(~q) (2) a contradiction (3) equivalent to (p ∨q) ∧(~p) (4) a tautology

Q60.For the frequency distribution: Variate (x) : x1, x2, x3, … , x15 Frequency (f) : f1, f2, f3, … , f15 where 0 < x1 < x2 < x3 < … < x15 = 10 and ∑15i=1 fi > 0, the standard deviation cannot be (1) 4 (2) 1 (3) 6 (4) 2

Q60.If Σ −a) = n and Σ −a)2 = na, (n, a > 1), then the standard deviation of n observations i=1(xi i=1(xi x1, x2, … , xn is JEE Main 2020 (06 Sep Shift 1) JEE Main Previous Year Paper (1) a −1 (2) n√(a −1) (3) √n(a −1) (4) √(a −1)

Q61.For a suitably chosen real constant a, let a function, f : R −{−a} →R be defined by f(x) = a+xa−x . Further supposed that for any real number x ≠−a,and f(x) ≠−a, (fof)(x) = x. Then f(−12 ) is equal to : (1) 3 1 (2) −13 (3) −3 (4) 3

Q61.A survey shows that 73% of the persons working in an office like coffee, whereas 65% like tea. If x denotes the percentage of them, who like both coffee and tea, then x cannot be: (1) 63 (2) 36 (3) 54 (4) 38

Q61.Let R1 and R2 be two relations defined as follows : R1 = {(a, b) ∈R2 : a2 + b2 ∈Q} and R2 = {(a, b) ∈R2 : a2 + b2 ∉Q} , where Q is the set of all rational numbers, then (1) R1 is transitive but R2 is not transitive. (2) R2 is transitive but R1 is not transitive. (3) Neither R1 nor R2 is transitive. (4) R1 and R2 are both transitive. Q62. ⎡ 2 −1 1 ⎤ Let A be a 3 × 3 matrix such that adj A = −1 0 2 and B =adj (adjA). If |A| = λ and ⎣ 1 −2 −1 ⎦ (B−1) ⊤= μ, then the ordered pair (|λ|, μ) is equal to (1) (3, 811 ) (2) (9, 91 ) (3) (3, 81) (4) (9, 811 )

Q61. x −2 2x −3 3x −4 If Δ = 2x −3 3x −4 4x −5 = Ax3 + Bx2 + Cx + D , then B + C is equal to : 3x −5 5x −8 10x −17 (1) −1 (2) 1 (3) −3 (4) 9 Q62. 2π −(sin−1 45 + sin−1 135 + sin−1 1665 ) is equal to : (1) π (2) 5π 2 4 (3) 3π (4) 7π 2 4

Q61. cos2 x 1 + sin2 x sin 2x Let m and M be respectively the minimum and maximum value values of 1 + cos2 x sin2 x sin 2x cos2 x sin2 x 1 + sin 2x Then the ordered pair (m, M) is equal to: (1) (3, 3) (2) (−3, −1) (3) (4, 1) (4) (1, 3)