Practice Questions

Q83.The integral ∫π/4π/6 sin 2x(tan5dxx+cot5 x) equals: (1) 20 1 tan−1 ( 9√31 ) (2) 101 ( π4 −tan−1 ( 9√31 )) (3) π (4) 1 40 5 ( π4 −tan−1 ( 3√31 ))

Q83.Let 𝑓: 𝑅→𝑅 be a continuous and differentiable function such that 𝑓2 = 6 and 𝑓'2 = 48.1 If 𝑓( 𝑥) ∫6 4𝑡3𝑑𝑡= 𝑥- 2𝑔𝑥, then 𝑥→2𝑔𝑥lim is equal to (1) 24 (2) 18 (3) 12 (4) 36 π Q84. 2 cot𝑥 If ∫ 𝑚(π + 𝑛), then 𝑚𝑛 is equal to cot𝑥+ cosec𝑥𝑑𝑥= 0 (1) 1 (2) 1 2 1 (3) -1 (4) - 2

Q83.If ∫ √1−x2x4 dx = A(x)(√1 −x2) m constant of integration, then (A(x))m equals : (1) −1 (2) −1 27x9 3x3 (3) 1 (4) 1 27x6 9x4 x dx (where [x] denotes the greatest integer less than or equal to x) is x 1

Q83.The value of the integral ∫10 xcot−1(1 −x2 + x4)dx is (1) π 4 −12 loge2 (2) π4 −loge2 (3) π 2 −loge2 (4) π2 −12 loge2

Q83.The value of ∫2π [sin 2x(1 + cos 3x)]dx , where [t] denotes the greatest integer function is 0 (1) π (2) 2π (3) −π (4) −2π (n+1)1/3 (n+2)1/3 (2n)1/3

Q84.If ∫ 𝑑𝑥 2 = 𝑥𝑓𝑥1 + 𝑥6 3 + 𝐶, where 𝐶 is a constant of integration, then the function 𝑓𝑥 is equal to 𝑥31 + 𝑥6 3 (1) 3 (2) - 1 𝑥2 2𝑥3 1 1 (3) - (4) - 6𝑥3 2𝑥2 𝑥 𝑥

Q85.Let 𝑓𝑥= ∫ 𝑔𝑡𝑑𝑡, where 𝑔 is a non-zero even function. If 𝑓𝑥+ 5 = 𝑔𝑥, then ∫ 𝑓( 𝑡) 𝑑𝑡 equals 0 0 𝑥+ 5 5 (1) (2) ∫ 𝑔( 𝑡) 𝑑𝑡 ∫ 𝑔( 𝑡) 𝑑𝑡 5 𝑥+ 5 5 𝑥+ 5 (3) (4) 5 ∫ 𝑔( 𝑡) 𝑑𝑡 2 ∫ 𝑔( 𝑡) 𝑑𝑡 𝑥+ 5 5

Q86.Let √3^i + ^j,^i + √3^j and β^i + (1 −β)^j respectively be the position vectors of the points A, B and C with respect to the origin O. If the distance of C from the bisector of the acute angle between OA and OB is 3 , √2 then the sum of all possible values of β is: (1) 4 (2) 3 (3) 2 (4) 1

Q86.Let f(x) be a differentiable function such that f ′(x) = 7 −34 f(x)x , (x > 0) and f(1) ≠4. Then lim x→0+ (1) does not exist. (2) exists and equals 4 . (3) exists and equals 4 . (4) exists and equals 0 . 7 → → → → →

Q87.Let is parallel to α and α = 3ˆi + ˆj and β = 2ˆi −ˆj + 3ˆk. If β = α, β1 −β2, β2 is perpendicular to where β1 −−→ → then β1 × β2 is equal to: (1) 1 2 (−3ˆi + 9ˆj + 5ˆk) (2) 3ˆi −9ˆj −5ˆk (3) −3ˆi + 9ˆj + 5ˆk (4) 1 + 2 (3ˆi −9ˆj 5ˆk)

Q87.If the volume of parallelepiped formed by the vectors ^𝑖+ 𝜆^𝑗+ ^𝑘, ^𝑗+ 𝜆^𝑘 and 𝜆^𝑖+ ^𝑘 is minimum, then 𝜆 is equal to: 1 (1) - (2) -√3 √3 1 (3) √3 (4) √3

Q88.The vertices B and C of a ΔABC lie on the line, x+2 3 = y−10 = 4z such that BC = 5 units. Then the area (in sq. units) of this triangle, given the point A(1, −1, 2), is (1) 6 (2) 2√34 (3) √34 (4) 5√17

Q88.A tetrahedron has vertices P(1, 2, 1), Q(2, 1, 3), R(−1, 1, 2) and O(0, 0, 0). The angle between the faces OPQ and PQR is JEE Main 2019 (12 Jan Shift 1) JEE Main Previous Year Paper (1) cos−1( 317 ) (2) cos−1( 3117 ) (3) cos−1( 3519 ) (4) cos−1( 359 )

Q88.The length of the perpendicular drawn from the point (2, 1, 4) to the plane containing the lines is + + + + →r= (ˆi ˆj) λ(ˆi + 2ˆj −ˆk) and →r= (ˆi ˆj) μ(−ˆi + ˆj −2ˆk) (1) 1 (2) 3 3 (3) √3 (4) 1 √3

Q88.A plane passing though the points (0, −1, 0) and (0, 0, 1) and making an angle π4 with the plane y–z + 5 = 0, also passes through the point −1, 1, (1) (√2, 4) (2) (√2, 4) −1, 1, (3) (−√2, −4) (4) (−√2, −4)

Q88.The plane containing the line x−3 2 = y+2−1 = z−13 and also containing its projection on the plane 2x + 3y −z = 5 , contains which one of the following points? (1) (2,2,0) (2) (-2,2,2) (3) (0,-2,2) (4) (2,0,-2)

Q89.Let S = {1, 2, … . . , 20}. A subset B of S is said to be "nice", if the sum of the elements of B is 203 . Than the probability that a randomly chosen subset of S is "nice" is : JEE Main 2019 (11 Jan Shift 2) JEE Main Previous Year Paper (1) 7 (2) 5 220 220 (3) 4 (4) None of the above 220

Q89.The equation of the line passing through -4, 3, 1, parallel to the plane 𝑥+ 2𝑦- 𝑧- 5 = 0 and intersecting the 𝑥 + 1 𝑦- 3 𝑧- 2 line = = is -3 2 -1 𝑥+ 4 𝑦- 3 𝑧- 1 𝑥+ 4 𝑦- 3 𝑧- 1 (1) = = (2) = = 3 -1 1 1 1 3 (3) 𝑥+ 4 = 𝑦- 3 = 𝑧- 1 (4) 𝑥- 4 = 𝑦+ 3 = 𝑧+ 1 -1 1 1 2 1 4

Q90.Assume that each born child is equally likely to be a boy or girl. If two families have two children each, then the conditional probability that all children are girls given that at least two are girls is: (1) 1 (2) 1 12 10 (3) 1 (4) 1 11 17 JEE Main 2019 (10 Apr Shift 1) JEE Main Previous Year Paper

Q90.In a random experiment, a fair die is rolled until two fours are obtained in succession. The probability that the experiment will end in the fifth throw of the die is equal to : (1) 150 (2) 175 65 65 (3) 225 (4) 200 65 65 JEE Main 2019 (12 Jan Shift 1) JEE Main Previous Year Paper

Q2. A man in a car at location Q on a straight highway is moving with speed v . He decides to reach a point P in a field at a distance d from highway (point M ) as shown in the figure.Speed of the car in the field is half to that on the highway. What should be the distance RM, so that the time taken to reach P is minimum? (1) d (2) d √3 2 (3) d (4) d √2

Q4. A particle is moving in a circular path of radius a under the action of an attractive potential U = −k . Its total 2r2 energy is: (1) −32 a2k (2) – 4a2k (3) k (4) Zero 2a2

Q6. A proton of mass m collides elastically with a particle of unknown mass at rest. After the collision, the proton and the unknown particle are seen moving at an angle of 90∘ with respect to each other. The mass of unknown particle is: (1) m (2) m √3 2 (3) 2 m (4) m

Q7. A body of mass m is moving in a circular orbit of radius R about a planet of mass M . At some instant, it splits into two equal masses. The first mass moves in a circular orbit of radius R . And the other mass, in a circular 2 orbit of radius 3R 2 . The difference between the final and the initial total energies is (1) + Gm6R (2) −Gm2R (3) −Gm6R (4) Gm2R

Q8. Seven identical circular planar disks, each of mass M and radius R are welded symmetrically as shown. The moment of inertia of the arrangement about the axis normal to the plane and passing through the point P is: (1) 181 MR2 (2) 19 MR2 2 2 (3) 55 MR2 (4) 73 MR2 2 2