Practice Questions

Q75.Let 𝐴= cos𝛼-sin𝛼 𝑎∈𝑅 such that 𝐴32 = 0 -1 . Then, a value of 𝛼 is: sin𝛼 cos𝛼, 1 0 (1) 0 (2) 𝜋 (3) 𝜋 (4) 𝜋 16 64 32 JEE Main 2019 (08 Apr Shift 1) JEE Main Previous Year Paper

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. y + 1 α β Let α and β be the roots of the equation x2 + x + 1 = 0. Then for y ≠0 in R, α y + β 1 is equal β 1 y + α to (1) y3 (2) y(y2–1) (3) y3–1 (4) y(y2–3)

Q75.The mean and the median of the following ten numbers in increasing order 10, 22, 26, 29, 34, x, 42, 67, 70, y are 42 and 35 respectively, then xy is equal to: (1) 9 (2) 7 4 3 (3) 7 (4) 8 2 3

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.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 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.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 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 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.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)

Q76.Let a1, a2, a3 … , a10 be in G. P. with ai > 0 for i = 1, 2, … , 10 and S be the set of pairs (r, k), r, k ∈N (the set of natural numbers) for which JEE Main 2019 (10 Jan Shift 2) JEE Main Previous Year Paper loge ar1 ak2 loge ar2ak3 loge ar3ak4 loge ar4 ak5 loge ar5ak6 loge ar6ak7 = 0 loge ar7ak8 loge ar8ak9 loge ar9ak10 Then the number of elements in S, is: (1) Infinitely many (2) 4 (3) 10 (4) 2

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 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.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.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.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.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.If the sum of the deviations of 50 observations from 30 is 50, then the mean of these observations is : (1) 30 (2) 51 (3) 50 (4) 31 Q77. ⎡ 1 0 0⎤ 5 q21+q31 Let P = 3 1 0 and Q = [qij] be two 3 × 3 matrices such that Q −P = I3 . Then q32 is equal to : ⎣ 9 3 1⎦ (1) 10 (2) 9 (3) 15 (4) 135

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.The value of cot(∑19n=1 cot−1(1 + ∑np=1 2p)) is: (1) 21 (2) 19 19 21 (3) 2223 (4) 2223

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.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. et e−tcos t e−t sin t If A = ⎡et −e−t cos t −e−t sin t −e−t sin t + e−t cos t ⎤, then A is: et 2e−t sin t −2e−t cos t ⎣ ⎦ (1) Invertible only if t = π (2) Not invertible for any t ∈R (3) Invertible only if t = π2 (4) Invertible for all t ∈R