Practice Questions

Q85.The mean of the coefficients of 𝑥, 𝑥2, … … , 𝑥7 in the binomial expression of ( 2 + 𝑥) 9 is _________

Q86.The sum of all values of α, for which the points whose position vectors are ˆi −2ˆj + 3ˆk, 2ˆi −3ˆj + 4ˆk, (α + 1)ˆi + 2ˆk and 9ˆi + (α −8)ˆj + 6ˆk are coplanar, is equal to (1) −2 (2) 2 (3) 6 (4) 4

Q86.Let →a, b and →cbe three non-zero non-coplanar vectors. Let the position vectors of four points A, B, C and D −−−→ → → → → → → be →a− b +→c, λ→a−3 b + 4→c, −→a+ 2 b −3→cand 2→a−4 b + 6→crespectively. If AB , AC and AD are coplanar, then λ is : JEE Main 2023 (29 Jan Shift 1) JEE Main Previous Year Paper

Q86.Let a tangent to the curve 9𝑥2 + 16𝑦2 = 144 intersect the coordinate axes at the points 𝐴 and 𝐵. Then, the minimum length of the line segment 𝐴𝐵 is ______

Q86.Let →a = 6ˆi + 9ˆj + 12ˆk, b = αˆi + 11ˆj −2ˆk and →cbe vectors such that →a×→c=→a× b If →a⋅→c= −12, and →c⋅(ˆi −2ˆj + ˆk) = 5 then →c⋅(ˆi + ˆj + ˆk) is equal to _______

Q86.The shortest distance between the lines x + 1 = 2 y = −12z and x = y + 2 = 6z −6 is (1) 2 (2) 3 (3) 5 (4) 3 2 2

Q86.Let →aand→b be two vectors. Let →a = 1, →b = 4 and →a⋅→b = 2 . If →c= (2→a →b) (1) −24 (2) −48 (3) −84 (4) −60

Q86.The mean and standard deviation of the marks of 10 students were found to be 50 and 12 respectively. Later, it was observed that two marks 20 and 25 were wrongly read as 45 and 50 respectively. Then the correct variance is JEE Main 2023 (13 Apr Shift 2) JEE Main Previous Year Paper

Q86.Let A = {1, 2, 3, 4} and R be a relation on the set A × A defined by R = {((a, b), (c, d)) : 2a + 3b = 4c + 5d} . Then the number of elements in R is _________. α, β > 0 , then α2 + β2 is dx , |x| <

Q86.The area of the quadrilateral ABCD with vertices A(2, 1, 1), B(1, 2, 5), C(−2, −3, 5) and D(1, −6, −7) is equal to (1) 48 (2) 8√38 (3) 54 (4) 9√38

Q86.Let 𝑓1𝑥= 3𝑥+ 2 𝑥∈𝑅- - 3 For 𝑛≥2, define 𝑓𝑛𝑥= 𝑓1𝑜𝑓𝑛- 1𝑥. If 𝑓5𝑥= 𝑎𝑥+ 𝑏 gcd𝑎, 𝑏= 1, then 𝑎+ 𝑏 is 2𝑥+ 3, 2. 𝑏𝑥+ 𝑎, equal to ________

Q86.Let the plane x + 3y −2z + 6 = 0 meet the co-ordinate axes at the points A, B, C . If the orthocenter of the triangle ABC is (α, β, 76 ), then 98(α + β)2 is equal to __________.

Q86.Let →a = 4ˆi + 3ˆj and→b = 3ˆi −4ˆj + 5ˆk and→cis a vector such that →c⋅(→a → b) + 25 = 0,→c⋅(ˆi ˆk) → and projection of →con →a is 1 , then the projection of →con b equals: (1) 5 (2) 1 √2 5 (3) 1 (4) 3 √2 √2

Q86.Let →a,→b,→cbe three vectors such that →a = √31, 4 →b = →c = 2 and 2(→a →b) → 2 2π →a×→c , then is equal to _____ . between b and →cis → 3 b ) ( →a⋅

Q86.Let →a = 3ˆi +ˆj −ˆk and →c= 2ˆi −3ˆj + 3ˆk. If b is a vector such that →a = b ×→c and b = 50, then → 2 72 − b +→c 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 the variance of the frequency distribution 𝑥𝑖 2 3 4 5 6 7 8 Frequency 𝑓i 3 6 16 𝛼 9 5 6 is 3, then 𝛼 is equal to

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

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 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 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