Practice Questions

Q87.Let 𝐶 be the largest circle centred at 2, 0 and inscribed in the ellipse 𝑥2 + 𝑦2 = 1. If 1, 𝛼 lies on 𝐶, then 10𝛼2 is 36 16 equal to ______ 𝜋 2 cos𝑥2023

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

Q87.Let the quadratic curve passing through the point -1, 0 and touching the line 𝑦= 𝑥 at 1, 1 be 𝑦= 𝑓𝑥. Then the 𝑥-intercept of the normal to the curve at the point 𝛼, 𝛼+ 1 in the first quadrant is

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

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

Q89.Let the line L : x = 1−y−2 = z−3λ , λ ∈R meet the plane P : x + 2 y + 3 z = 4 at the point (α, β, γ). If the angle between the line L and the plane P is , then α + 2β + 6γ is equal to 14 cos−1(√5 )

Q89.If the equation of the plane passing through the point ( 1, 1, 2 ) and perpendicular to the line 𝑥- 3𝑦+ 2𝑧- 1 = 0 = 4𝑥- 𝑦+ 𝑧 is 𝐴𝑥+ 𝐵𝑦+ 𝐶𝑧= 1, then 140 ( 𝐶- 𝐵+ 𝐴) is equal to

Q89.Let →𝑣= 𝛼 ^𝑖+ 2 ^𝑗- 3 ^𝑘, →𝑤= 2𝛼 ^𝑖+ ^𝑗- ^𝑘, and →𝑢 be a vector such that →𝑢= 𝛼> 0. If the minimum value of the 2 𝑚 where 𝑚 and 𝑛 are coprime natural numbers, then scalar triple product →𝑢 →𝑣 →𝑤 is -𝛼√3401, and →𝑢. ^𝑖 = 𝑛 𝑚+ 𝑛 is equal to _____ . JEE Main 2023 (01 Feb Shift 1) JEE Main Previous Year Paper Q90.𝐴2, 6, 2, 𝐵-4, 0, 𝜆, 𝐶2, 3, - 1 and 𝐷4, 5, 0, 𝜆≤5 are the vertices of a quadrilateral 𝐴𝐵𝐶𝐷. If its area is 18 square units, then 5 - 6𝜆 is equal to _____ . JEE Main 2023 (01 Feb Shift 1) JEE Main Previous Year Paper

Q89.If the line x = y = z intersects the line x sin A + y sin B + z sin C −18 = 0 = x sin 2A + y sin 2B + z sin 2C −9, where A, B, C are the angles of a triangle ABC , then 80(sin A2 sin B2 sin C2 ) is equal to _________.

Q89.Let 𝑓𝑛= ∫0 2 ∑𝑘=𝑛 1 sin𝑘- 1𝑥∑𝑘=𝑛 1 (2𝑘- 1)sin𝑘- 1𝑥cos𝑥𝑑𝑥, 𝑛∈ℕ. Then 𝑓21 - 𝑓20 is equal to

Q89.The point of intersection C of the plane 8x + y + 2z = 0 and the line joining the points A(−3, −6, 1) and B(2, 4, −3) divides the line segment AB internally in the ratio k : 1. If a, b, c (|a|, |b|, |c| are coprime) are the direction ratios of the perpendicular from the point C on the line 1−x 1 = y+42 = z+23 , then |a + b + c| is equal to _____ .

Q89.Let the image of the point ( 53 , 53 , 83 ) in the plane x − 2y + z–2 = 0 be P. If the distance of the point Q(6, −2, α), α > 0, from P is 13, then α is equal to _______. JEE Main 2023 (13 Apr Shift 1) JEE Main Previous Year Paper