Practice Questions

Q87.Let the tangent to the curve 𝑥2 + 2𝑥- 4𝑦+ 9 = 0 at the point 𝑃1, 3 on it meet the 𝑦- axis at 𝐴. Let the line passing through 𝑃 and parallel to the line 𝑥- 3𝑦= 6 meet the parabola 𝑦2 = 4𝑥 at 𝐵. If 𝐵 lies on the line 2𝑥- 3𝑦= 8, then 𝐴𝐵2 is equal to _______ .

Q87.A vector →vin the first octant is inclined to the x axis at 60° , to the y-axis at 45° and to the z-axis at an acute −1, (a, b, c), is normal to →v, then 1) and angle. If a plane passing through the points (√2, (1) √2a + b + c = 1 (2) a + b + √2c = 1 (3) a + √2b + c = 1 (4) √2a −b + c = 1

Q87. lim 48 𝑥 𝑡3 is equal to 𝑥→0 𝑥4 ∫0 𝑡6 + 1𝑑𝑡

Q87.Let 𝐴= { - 4, - 3, - 2, 0, 1, 3, 4} and 𝑅= { ( 𝑎, 𝑏) ∈𝐴× 𝐴 : 𝑏= | 𝑎| or 𝑏2 = 𝑎+ 1 be a relation on 𝐴. Then the minimum number of elements, that must be added to the relation 𝑅 so that it becomes reflexive and symmetric, is

Q87.Shortest distance between the lines x−1 2 = y+8−7 = z−45 and x−12 = y−21 = z−6−3 is JEE Main 2023 (29 Jan Shift 2) JEE Main Previous Year Paper (1) 2√3 (2) 4√3 (3) 3√3 (4) 5√3

Q87.The shortest distance between the lines x−4 4 = y+25 = z+33 and x−13 = y−34 = z−42 is (1) 6√3 (2) 2√6 (3) 6√2 (4) 3√6

Q87.Let the equation of the plane P containing the line x + 10 = 8−y2 = z be ax + by + 3z = 2(a + b) and the distance of the plane P from the point (1, 27, 7) be c . Then a2 + b2 + c2 is equal to

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

Q87.Let 𝐴 be the area bounded by the curve 𝑦= 𝑥𝑥- 3, the 𝑥-axis and the ordinates 𝑥= - 1 and 𝑥= 2. Then 12 𝐴 is equal to _____ . 2

Q87.The foot of perpendicular of the point (2, 0, 5) on the line x+12 = y−15 = z+1−1 is (α, β, γ). Then. Which of the following is NOT correct? (1) αβ γ = 154 (2) αβ = −8 (3) β γ = −5 (4) αγ = 85

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 equation of plane passing through the line of intersection of the planes x + 2y + az = 2 and x −y + z = 3 be 5x −11y + bz = 6a −1. For c ∈Z, if the distance of this plane from the point (a, −c, c) is 2 , then a+bc is equal to √a (1) 2 (2) 4 (3) −4 (4) −2 = 10 parallel to the line of the shortest

Q87.If the mean of the frequency distribution Class : 0 - 10 10 - 20 20 - 30 30 - 40 40 - 50 Frequency : 2 3 𝑥 5 4 is 28, then its variance is ________ .

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

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.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.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 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 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 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.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