Practice Questions

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 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.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.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.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 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.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.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)

Q77.The sum of the real roots of the equation 𝑥 -6 -1 2 -3𝑥 𝑥- 3 = 0, is equal to: -3 2𝑥 𝑥+ 2 (1) 0 (2) -4 (3) 6 (4) 1 JEE Main 2019 (10 Apr Shift 2) JEE Main Previous Year Paper

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.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 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

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.In a class of 140 students numbered 1 to 140 , all even numbered students opted Mathematics course, those whose number is divisible by 3 opted Physics course and those whose number is divisible by 5 opted Chemistry course. Then the number of students who did not opt for any of the three courses is: (1) 42 (2) 1 (3) 38 (4) 102

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.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.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

Q78.If the system of linear equations 2x + 2y + 3z = a 3x −y + 5z = b x −3y + 2z = c where, a, b, care non- zero real numbers, has more than onc solution, then (1) b −c + a = 0 (2) b −c −a = 0 (3) a + b + c = 0 (4) b + c −a = 0

Q78.If the system of equations x + y + z = 5, x + 2y + 3z = 9, x + 3y + αz = β has inifinitely many solutions, then β −α equals (1) 8 (2) 21 (3) 5 (4) 18 Q79. ⎡−2 4 + d (sin θ) −2 ⎤ Let d ∈R, and A = 1 (sin θ) + 2 d , θ ∈[0, 2π]. If the minimum value of det(A) is ⎣ 5 (2 sin θ) −d (−sin θ) + 2 + 2d ⎦ 8, then a value of d is: + + (1) 2(√2 2) (2) 2(√2 1) (3) −5 (4) −7 . Let S be the set of points in the interval (−4, 4) at which f is not

Q78.An ordered pair (α, β) for which the system of linear equations (1 + α)x + βy + z = 2 αx + (1 + β)y + z = 3 αx + βy + 2z = 2 has a unique solution, is : (1) (−3, 1) (2) (1, −3) (3) (2, 4) (4) (−4, 2) JEE Main 2019 (12 Jan Shift 1) JEE Main Previous Year Paper

Q78.For 𝑥 𝜖 (0, 3 ), let 𝑓𝑥= 𝑔𝑥= tan𝑥 and ℎ𝑥= 1 - 𝑥2 . If ϕ𝑥= ( ho𝑓) og ) ( 𝑥) , then ϕ 𝜋 is equal to: 2 √𝑥, 1 + 𝑥2 3 𝜋 5𝜋 (1) tan⁡ (2) tan⁡ 12 12 7𝜋 11𝜋 (3) tan⁡ (4) tan⁡ 12 12

Q78.If the system of linear equations 𝑥- 2𝑦+ 𝑘𝑧= 1 2𝑥+ 𝑦+ 𝑧= 2 3𝑥- 𝑦- 𝑘𝑧= 3 has a solution 𝑥, 𝑦, 𝑧, 𝑧≠0, then 𝑥, 𝑦 lies on the straight line whose equation is: (1) 4𝑥- 3𝑦- 4 = 0 (2) 3𝑥- 4𝑦- 4 = 0 (3) 3𝑥- 4𝑦- 1 = 0 (4) 4𝑥- 3𝑦- 1 = 0