Practice Questions

Q86.Let a common tangent to the curves 𝑦2 = 4𝑥 and 𝑥- 42 + 𝑦2 = 16 touch the curves at the points 𝑃 and 𝑄. Then 𝑃𝑄2 is equal to ________.

Q86.Let 𝐻𝑛: 𝑥2 𝑦2 1, 𝑛∈ℕ. Let 𝑘 be the smallest even value of 𝑛 such that the eccentricity of 𝐻𝑘 is a 1 + 𝑛- 3 + 𝑛= rational number. If 𝑙 is the length of the latus rectum of 𝐻𝑘, then 21𝑙 is equal to

Q86.If 1 1 / 1 𝑛 𝑥21 + 𝑥14 + 𝑥72𝑥14 + 3𝑥7 + 6 7𝑑𝑥= where 𝑙, 𝑚, 𝑛∈𝑁, 𝑚 and 𝑛 are co-prime then 𝑙+ 𝑚+ 𝑛 ∫0 𝑙11𝑚/ is equal to _____ .

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

Q87.For a, b ∈Z and |a −b| ≤10 , let the angle between the plane P : a x + y −z = b and the line L : x −1 = a −y = z + 1 be cos−1( 13 ) If the distance of the point (6, −6, 4) from the plane P is 3√6 , then a4 + b2 is equal to (1) 32 (2) 85 (3) 25 (4) 48

Q87.Let the line L pass through the point (0, 1, 2), intersect the line x−12 = y−23 = z−34 and be parallel to the plane 2x + y −3z = 4. Then the distance of the point P(1, –9, 2) from the line L is (1) √74 (2) √69 (3) √54 (4) 9 = −5. If P

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.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 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.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.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 f(x) = ∫ 2 . If f(0) = 0 and f(1) = αβ1 tan−1( αβ ), √3 (3+4x2)√4−3x2 equal to _______.

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 ________ .

Q87.The number of ordered triplets of the truth values of 𝑝, 𝑞 and 𝑟 such that the truth value of the statement 𝑝∨𝑞∧𝑝∨𝑟⇒𝑞∨𝑟 is True, is equal to Q88. 0 1 2 Let 𝐴= 𝑎0 3 , where 𝑎, 𝑐∈𝑅. If 𝐴3 = 𝐴 and the positive value of 𝑎 belongs to the interval ( 𝑛- 1, 𝑛], 1 𝑐 0 where 𝑛∈ℕ, then 𝑛 is equal to ____. 2

Q87.Let the plane P pass through the intersection of the planes 2x + 3y −z = 2 and x + 2y + 3z = 6, and be perpendicular to the plane 2x + y −z + 1 = 0. If d is the distance of P from the point (−7, 1, 1), then d2 is equal to : (1) 250 (2) 15 83 53 (3) 25 (4) 250 83 82

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.The distance of the point P(4, 6, −2) from the line passing through the point (−3, 2, 3) and parallel to a line with direction ratios 3, 3, −1 is equal to: (1) 3 (2) √6 (3) 2√3 (4) √14

Q87.Let the lines L1 : x+53 = y+41 = z−α−2 and L2 : 3x + 2y + z −2 = 0 = x −3y + 2z −13 be coplanar. If the point P(a, b, c) on L1 is nearest to the point Q(−4, −3, 2), then |a| + |b| + |c| is equal to (1) 12 (2) 14 (3) 8 (4) 10

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

Q87.Let P be the plane passing through the points (5, 3, 0), (13, 3, −2) and (1, 6, 2). For α ∈N, if the distance of the points A(3, 4, α) and B(2, α, a) from the plane P are 2 and 3 respectively, then the positive value of a is (1) 6 (2) 3 (3) 5 (4) 4

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 for 𝑥∈𝑅, 𝑓𝑥= 𝑥+ 𝑥 and 𝑔𝑥= 𝑥, 𝑥< 0 . Then area bounded by the curve 𝑦= 𝑓𝑜𝑔𝑥 and the lines 2 𝑥2, 𝑥≥0 𝑦= 0, 2𝑦- 𝑥= 15 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