Practice Questions

Q75.The logical statement [~(~p ∨q) ∨(p ∧r)] ∧(~q ∧r) is equivalent to (1) (~p ∧~q) ∧r (2) (p ∧r) ∧~q (3) (p ∧~q) ∨r (4) ~p ∨r

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 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.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

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.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.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 lengths of the sides of a triangle are in A.P and the greatest angle is double the smallest, then a ratio of lengths of the sides of this triangle is: (1) 3: 4: 5 (2) 5: 6: 7 (3) 5: 9: 13 (4) 4: 5: 6 Q77. 1 1 1 Let the numbers 2, 𝑏, 𝑐 be in an A.P. and 𝐴= 2 𝑏 𝑐 . If det ( 𝐴) ∈[2,16], then 𝑐 lies in the interval: 4 𝑏2 𝑐2 (1) 2,3 (2) 4,6 3 (3) 3,2 + 2 4 (4) 2 + 2 34, 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.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.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.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 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.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.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.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

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. 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

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. 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.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.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.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