Practice Questions

Q87.Let the plane P : 8x + α1y + α2z + 12 = 0 be parallel to the line L : x+22 = y−33 = z+45 . If the intercept of P on the y-axis is 1 , then the distance between P and L is (1) √27 (2) √146 (3) √72 (4) √14 JEE Main 2023 (31 Jan Shift 2) JEE Main Previous Year Paper

Q87.Let the plane containing the line of intersection of the planes P1 : x + (λ + 4)y + z = 1 and P2 : 2x + y + z = 2 pass through the points (0, 1, 0) and (1, 0, 1) . Then the distance of the point (2λ, λ, −λ) from the plane P2 is (1) 5√6 (2) 4√6 (3) 2√6 (4) 3√6

Q88.If the area bounded by the curve 2y2 = 3x, lines x + y = 3, y = 0 and outside the circle (x −3)2 + y2 = 2 is A, then 4(π + 4A) is equal to __________.

Q88.Let →𝑎 and →𝑏 be two vector such that →𝑎= √14, →𝑏= √6 and →𝑎× →𝑏= √48. Then →𝑎· →𝑏 is equal to _____ . 𝑥- 1 𝑦+ 1 𝑧- 3

Q88.If the shortest distance between the line joining the points (1, 2, 3) and (2, 3, 4), and the line x−1 2 = y+1−1 = z−20 is α, then 28α2 is equal to _____ .

Q88.If the equation of the plane containing the line x + 2y + 3z −4 = 0 = 2x + y −z + 5 and perpendicular to + + + ax + by + cz = 4 then (a −b + c) is equal to the plane→r= (ˆi −ˆj) λ(ˆi + ˆj + ˆk) μ(ˆi −2ˆj 3ˆk) is (1) 18 (2) 22 (3) 20 (4) 24

Q88.If the foot of the perpendicular drawn from (1, 9, 7) to the line passing through the point (3, 2, 1) and parallel the planes x + 2y + z = 0 and 3y −z = 3 is (α, β, γ), then α + β + γ is equal to (1) −1 (2) 3 (3) 1 (4) 5 y−√6

Q88.Let P be the plane, passing through the point (1, −1, −5) and perpendicular to the line joining the points (4, 1, −3) and (2, 4, 3). Then the distance of P from the point (3, −2, 2) is (1) 6 (2) 4 (3) 5 (4) 7

Q88.Let 𝛼 be the area of the larger region bounded by the curve 𝑦2 = 8𝑥 and the lines 𝑦= 𝑥 and 𝑥= 2, which lies in the first quadrant. Then the value of 3𝛼 is equal to

Q88.If the area of the region 𝑆= ( 𝑥, 𝑦) : 2𝑦- 𝑦2 ≤𝑥2 ≤2𝑦, 𝑥≥𝑦 is equal to 𝑛+ 2 - 𝜋 then the natural number 𝑛+ 1 𝑛- 1, 𝑛 is equal to _______ JEE Main 2023 (06 Apr Shift 1) JEE Main Previous Year Paper

Q88.The distance of the point (−1, 2, 3) from the plane →r⋅(ˆi −2ˆj + 3ˆk) is + + + + distance between the lines→r= (ˆi −ˆj) λ(2ˆi ˆk) and →r= (2ˆi −ˆj) μ(ˆi −ˆj ˆk) (1) 4√6 (2) 2√5 (3) 2√6 (4) 3√5

Q88.Let 𝑓: ℝ→ℝ be a differentiable function such that 𝑓'𝑥+ 𝑓𝑥= ∫0 𝑓𝑡𝑑𝑡. If 𝑓0 = 𝑒-2, then 2𝑓0 - 𝑓2 is equal to _____ .

Q88.If a plane passes through the points (−1, k, 0), (2, k, −1), (1, 1, 2) and is parallel to the line x−11 = 2y+12 = z+1−1 , then the value of (k−1)(k−2)k2+1 is (1) 17 (2) 5 5 17 (3) 6 (4) 13 13 6

Q88.Let P be the plane passing through the line x−1 1 = y−2−3 = z+57 and the point (2, 4, −3). If the image of the point (−1, 3, 4) in the plane P is (α, β, γ), then α + β + γ is equal to (1) 10 (2) 12 (3) 9 (4) 11

Q88.Let the co-ordinates of one vertex of ΔABC be A(0, 2, α) and the other two vertices lie on the line x+α 5 = y−12 = z+43 . For α ∈Z , if the area of ΔABC is 21 sq. units and the line segment BC has length 2√21 units, then α2 is equal to _______.

Q88.Let the line passing through the points P(2, −1, 2) and Q(5, 3, 4) meet the plane x −y + z = 4 at the point R. Then the distance of the point R from the plane x + 2y + 3z + 2 = 0 measured parallel to the line x−7 2 = y+32 = z−21 is (1) √61 (2) √189 (3) √31 (4) 3

Q88.If the area of the region 𝑥, 𝑦: 𝑥2 - 2 ≤𝑦≤𝑥 is A, then 6𝐴+ 16√2 is equal to ______________ JEE Main 2023 (10 Apr Shift 2) JEE Main Previous Year Paper 1

Q88.Let the plane P : 4x −y + z = 10 be rotated by an angle π2 about its line of intersection with the plane x + y −z = 4 . If α is the distance of the point (2, 3, −4) from the new position of the plane P , then 35α is equal to (1) 85 (2) 105 (3) 126 (4) 90

Q88.Let αx + βy + γz = 1 be the equation of a plane passing through the point (3, −2, 5) and perpendicular to the line joining the points (1, 2, 3) and (−2, 3, 5). Then the value of α β y is equal to _____ .

Q88.The plane 2x −y + z = 4 intersects the line segment joining the points A(a, −2, 4) and B(2, b, −3) at the point C in the ratio 2 : 1 and the distance of the point C from the origin is √5 . If ab < 0 and P is the point (a −b, b, 2 b −a) then CP2 is equal to: (1) 17 (2) 16 3 3 (3) 73 (4) 97 3 3

Q88.The number of elements in the set 𝑛∈ℤ: 𝑛2 - 10𝑛+ 19 < 6 is _______ .

Q88.The value of 8 𝜋∫0 sin𝑥2023 + cos𝑥2023𝑑𝑥 is ______. 3

Q88.A plane P contains the line of intersection of the plane →r⋅(ˆi + ˆj + ˆk) = 6 and →r⋅(2ˆi + 3ˆj + 4ˆk) passes through the point (0, 2, −2),then the square of distance of the point (12, 12, 18) from the plane P is (1) 620 (2) 155 (3) 310 (4) 1240

Q88.For 𝑥∈( - 1, 1], the number of solutions of the equation sin-1𝑥= 2tan-1𝑥 is equal to 𝜋

Q88.Consider the lines L1 and L2 given by L1 : x−12 = y−31 = z−22 L2 : x−21 = y−22 = z−33 A line L3 having direction ratios 1, −1, −2, intersects L1 and L2 at the points P and Q respectively. Then the length of line segment PQ is (1) 2√6 (2) 3√2 (3) 4√3 (4) 4