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.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 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.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 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.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.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 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 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

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 shortest distance between the lines x+√6 2 = 3 = z−√64 and x−λ3 = y−2√64 = z+2√65 is 6 , then sum of squares of all possible values(s) of λ is

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

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.For 𝑚, 𝑛> 0, let 𝛼𝑚, 𝑛= ∫0 𝑡𝑚1 + 3𝑡𝑛𝑑𝑡. If ,11𝛼10, 6 + 18𝛼11, 5 = 𝑝146, then 𝑝 is equal to JEE Main 2023 (11 Apr Shift 1) JEE Main Previous Year Paper

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.If the lines x−1 2 = 2−y−3 = z−3α and x−45 = y−12 = βz intersect, then the magnitude of the minimum value of 8αβ is _____. JEE Main 2023 (06 Apr Shift 2) JEE Main Previous Year Paper

Q90.The shortest distance between the lines = = and = = is equal to ______ 3 2 2 3 2 0 JEE Main 2023 (24 Jan Shift 1) JEE Main Previous Year Paper

Q90.Let the image of the point 𝑃( 1, 2, 3 ) in the plane 2𝑥– 𝑦+ 𝑧= 9 be 𝑄. If the coordinates of the point 𝑅 are ( 6, 10, 7 ) , then the square of the area of the triangle 𝑃𝑄𝑅 is _______ . JEE Main 2023 (06 Apr Shift 1) JEE Main Previous Year Paper

Q90.A fair n (n > 1) faces die is rolled repeatedly until a number less than n appears. If the mean of the number of tosses required is n , then n is equal to 9 JEE Main 2023 (12 Apr Shift 1) JEE Main Previous Year Paper

Q90.Let 𝜃 be the angle between the planes 𝑃1 = →𝑟· ^𝑖+ ^𝑗+ 2 ^𝑘= 9 and 𝑃2 = →𝑟· 2 ^𝑖- ^𝑗+ ^𝑘= 15. Let L be the line that meets 𝑃2 at the point 4, - 2, 5 and makes an angle 𝜃 with the normal of 𝑃2. If 𝛼 is the angle between 𝐿 and 𝑃2 then tan2𝜃cot2𝛼 is equal to _____ . JEE Main 2023 (31 Jan Shift 1) JEE Main Previous Year Paper

Q90.Let the probability of getting head for a biased coin be 1 . It is tossed repeatedly until a head appears. Let N 4 be the number of tosses required. If the probability that the equation 64x2 + 5Nx + 1 = 0 has no real root is p , where p and q are co-prime, then q −p is equal to.......... q JEE Main 2023 (11 Apr Shift 2) JEE Main Previous Year Paper

Q90.A bag contains six balls of different colours. Two balls are drawn in succession with replacement. The probability that both the balls are of the same colour is p. Next four balls are drawn in succession with replacement and the probability that exactly three balls are of the same colours is q . If p : q = m : n, where m and n are co-prime, then m + n is equal to JEE Main 2023 (30 Jan Shift 2) JEE Main Previous Year Paper

Q90.Let A be the event that the absolute difference between two randomly chosen real numbers in the sample space [0, 60] is less than or equal to a. If P(A) = 1136 , then a is equal to _____ . JEE Main 2023 (31 Jan Shift 2) JEE Main Previous Year Paper

Q2. The velocity of the bullet becomes one third after it penetrates 4 cm in a wooden block. Assuming that bullet is facing a constant resistance during its motion in the block. The bullet stops completely after travelling at (4 + x) cm inside the block. The value of x is (1) 2. 0 (2) 1. 0 (3) 0. 5 (4) 1. 5