Practice Questions

Q75.If the angle of elevation of a cloud from a point P which is 25m above a lake be 30o and the angle of depression of reflection of the could in the lake from P be 60o , then the height of the cloud (in meters) from JEE Main 2019 (12 Jan Shift 2) JEE Main Previous Year Paper the surface of the lake is : (1) 50 (2) 60 (3) 45 (4) 42 and B = {x ∈Z : −3 < 2x −1 < 9},

Q75.The mean of five observations is 5 and their variance is 9.20. If three of the given five observations are 1, 3 and 8, then a ratio of other two observations is (1) 10 : 3 (2) 4 : 9 (3) 6 : 7 (4) 5 : 8

Q75.Two vertical poles of height, 20 𝑚 and 80 𝑚 stand apart on a horizontal plane. The height (in meters) of the point of intersection of the lines joining the top of each pole to the foot of the other, from this horizontal plane is: (1) 16 (2) 12 (3) 18 (4) 15

Q75.If 𝐴 is a symmetric matrix and 𝐵 is skew- symmetric matrix such that 𝐴+ 𝐵= 2 3 , then 𝐴𝐵 is equal to: 5 -1 (1) -4 2 (2) 4 -2 1 4 1 -4 (3) 4 -2 (4) -4 -2 -1 -4 -1 4

Q75.Let A and B be two invertible matrices of order 3 × 3. If det (ABAT) (BA−1 BT) is equal to (1) 1 (2) 1 4 (3) 1 (4) 16 16

Q76.Two poles standing on a horizontal ground are of heights 5 m and 10 m respectively. The line joining their tops makes an angle of 15° with the ground. Then the distance (in m) between the poles, is + (1) 10(√3 −1) (2) 52 (2 √3) + + (3) 5(2 √3) (4) 5(√3 1) Q77. ⎛ 0 2y 1 ⎞ The total number of matrices A = 2x y −1 , (x, y ∈R, x ≠y) for which ATA = 3I3 is: ⎝ 2x −y 1 ⎠ (1) 6 (2) 3 (3) 4 (4) 2

Q76.If the function f : R −{1, −1} →A defined by f(x) = x2 , is surjective, then A is equal to 1−x2 (1) [0, ∞) (2) R −{−1} (3) R −[−1, 0) (4) R −(−1, 0)

Q76.Consider a triangular plot ABC with sides AB = 7 m, BC = 5 m and CA = 6 m. A vertical lamp-post at the mid-point D of AC subtends an angle 30° at B. The height (in m ) of the lamp-post is: (1) 2√21 (2) 23 √21 (3) 3 2 √21 (4) 7√3 JEE Main 2019 (10 Jan Shift 1) JEE Main Previous Year Paper

Q76.A data consists of n observations: x1, x2, … , xn. If ∑ni=1 (xi + 1)2 = 9n and ∑ni=1 (xi −1)2 = 5n, then the standard deviation of this data is JEE Main 2019 (09 Jan Shift 2) JEE Main Previous Year Paper (1) 5 (2) √7 (3) √5 (4) 2

Q76.If the system of linear equations x + y + z = 5 , x + 2y + 2z = 6 , x + 3y + λz = µ, (λ, µ ∈R) , has infinitely many solutions, then the value of λ + µ is: (1) 7 (2) 10 (3) 12 (4) 9

Q76.The angle of the top of a vertical tower standing on a horizontal plane is observed to be 45° from a point A on the plane. Let B be the point 30 m vertically above the point A. If the angle of elevation of the top of the tower from B be 30° , then the distance (in m) of the foot of the tower from the point A is: + + (1) 15(3 √3) (2) 15(1 √3) (3) 15(5 −√3) (4) 15(3 −√3)

Q76.Let Z be the set of integers. If A = {x ∈Z : 2(x+2)(x2−5x+6) = 1} then the number of subsets of the set A × B, is : (1) 212 (2) 210 (3) 218 (4) 215 Q77. ⎡ 1 sin θ 1 ⎤ 3π 5π If A = −sin θ 1 sin θ , then for all θ ∈( 4 , 4 ), det(A) lies in the interval : ⎣ −1 −sin θ 1 ⎦ (1) (1, 52 ] (2) [ 52 , 4) (3) ( 23 , 3] (4) (0, 32 ]

Q76.The outcome of each of 30 items was observed; 10 items gave an outcome 1 2 −d each, 10 items gave outcome 1 each and the remaining 10 items gave outcome 2 2 1 + d each. If the variance of this outcome data is 34 then |d| equals: (1) 2 (2) 2 3 (3) √5 (4) √2 2 Q77. ⎛0 2q r ⎞ Let A = p q −r . If AAT = I3, then |p| is: ⎝p −q r ⎠ (1) 1 (2) 1 √5 √3 (3) 1 (4) 1 √2 √6

Q76.If 𝐴= cos𝜃-sin𝜃 , then the matrix 𝐴-50 when 𝜃= 𝜋 is equal to: sin𝜃 cos𝜃 12, (1) √3 1 (2) 1 √3 2 2 2 2 -1 √3 -√3 1 2 2 2 2 (3) √3 -1 (4) 1 -√3 2 2 2 2 1 √3 √3 1 2 2 2 2

Q76.The greatest value of 𝑐∈𝑅 for which the system of linear equations 𝑥- 𝑐𝑦- 𝑐𝑧= 0, 𝑐𝑥- 𝑦+ 𝑐𝑧= 0, 𝑐𝑥+ 𝑐𝑦- 𝑧= 0 has a non-trivial solution, is (1) -1 (2) 2 (3) 1 (4) 0 2

Q76.The angles 𝐴, 𝐵 & 𝐶 of a ∆𝐴𝐵𝐶 are in 𝐴. 𝑃. and 𝑎: 𝑏= 1: √3 . If 𝑐= 4 𝑐𝑚, then the area (in 𝑠𝑞. 𝑐𝑚) of this triangle is: 2 (1) 2√3 (2) √3 4 (3) (4) 4√3 √3

Q76.The value of sin-1⁡12 - sin-1⁡3 is equal to: 13 5 33 𝜋 9 (1) 𝜋- cos-1⁡ (2) - cos-1⁡ 65 2 65 𝜋 56 (3) 𝜋- sin-163 (4) - sin-1⁡ 65 2 65

Q76.All x satisfying the inequality (cot−1 x)2 −7 (cot−1 x) + 10 > 0 , lie in the interval : (1) (−∞, cot 5) ∪(cot 4, cot 2) (2) (cot 2, ∞) (3) (−∞, cot 5) ∪(cot 2, ∞) (4) (cot 5, cot 4)

Q77.The system of linear equations 𝑥+ 𝑦+ 𝑧= 2 2𝑥+ 3𝑦+ 2𝑧= 5 2𝑥+ 3𝑦+ 𝑎2 - 1𝑧= 𝑎+ 1 (1) is inconsistent when 𝑎= √3 (2) has a unique solution for 𝑎= √3 (3) has infinitely many solutions for 𝑎= 4 (4) is inconsistent when 𝑎= 4

Q77.If 𝛼= cos-13 , 𝛽= tan-11 , where 0 < 𝛼, 𝛽< 𝜋 then 𝛼- 𝛽 is equal to 5 3 2, (1) tan-1 9 (2) cos-1 9 (3) sin-1⁡ 9 (4) tan-1 9 14 5√10 5√10 5√10 2𝑥 is equal to 𝑥< 1, then 𝑓

Q77.Let a function f : (0, ∞) →(0, ∞) be defined by f(x) = 1 −1x . Then f is : (1) not injective but it is surjective (2) injective only (3) neither injective nor surjective (4) None of the above

Q77.For 𝑥∈𝑅, Let [𝑥] denotes the greatest integer ≤𝑥, then the sum of the series -1 + -1 - 1 + -1 - 2 + . . . . . + -1 - 99 is 3 3 100 3 100 3 100 (1) -131 (2) -153 (3) -135 (4) -133

Q77. x sinθ cosθ x sin2θ cos2θ If Δ1 = −sinθ −x 1 and Δ2 = −sin2θ −x 1 , x ≠0; then for all θ ∈(0, π2 ) : cosθ 1 x cos2θ 1 x (1) Δ1 + Δ2 = −2(x3 + x −1) (2) Δ1 −Δ2 = x(cos2θ −cos4θ) (3) Δ1 + Δ2 = −2x3 (4) Δ1 −Δ2 = −2x3

Q77.Let A, B and C be sets such that ϕ ≠A ∩B ⊆C. Then which of the following statements is not true? (1) B ∩C ≠ϕ (2) (C ∪A) ∩(C ∪B) = C (3) If (A −B) ⊆C, then A ⊆C (4) If (A −C) ⊆B, then A ⊆B Q78. 1 + cos2θ sin2θ 4 cos6θ A value of θ ∈(0, π3 ), for which cos2θ 1 + sin2θ 4 cos6θ = 0, is cos2θ sin2θ 1 + 4 cos6θ (1) π (2) 7π 9 24 (3) 7π (4) π 36 18

Q77.Let f(x) = 15–|x –10|; x ∈R. Then the set of all values of x, at which the function g(x) = f(f(x)) is not differentiable, is: (1) {5, 10, 15} (2) {10} (3) {10, 15} (4) {5, 10, 15, 20} √2cosx−1 π cotx−1 , x ≠π 4 is continuous, then k is equal to