Practice Questions

Q83.If the second, third and fourth terms in the expansion of (x + y)n are 135,30 and 103 , respectively, then 6 (n3 + x2 + y) is equal to _______

Q83.Let A, B and C be three points on the parabola y2 = 6x and let the line segment AB meet the line L through C parallel to the x-axis at the point D. Let M and N respectively be the feet of the perpendiculars from A and AM⋅BN 2 B on L. Then ( CD ) is equal to _________

Q83.If the sum of squares of all real values of α, for which the lines 2x - y + 3 = 0, 6x + 3y + 1 = 0 and αx + 2y - 2 = 0 do not form a triangle is p, then the greatest integer less than or equal to p is ________.

Q83.Let the centre of a circle, passing through the points (0, 0), (1, 0) and touching the circle x2 + y2 = 9, be (h, k) . Then for all possible values of the coordinates of the centre (h, k), 4 (h2 + k2) is equal to_________

Q83.If the coefficient of 𝑥30 in the expansion of 1 + 1 + 𝑥271 −𝑥38; 𝑥≠0 is 𝛼, then 𝛼 equals _________. 𝑥 JEE Main 2024 (01 Feb Shift 1) JEE Main Previous Year Paper

Q83.Let 𝑆𝑛 be the sum to n-terms of an arithmetic progression 3, 7, 11, … … , if 40 < 𝑛( 𝑛+ 1 ) ∑𝑘= 1 𝑆𝑘< 42, then 𝑛 equals ____________. 𝑛C𝑘 𝑛C𝑘+ 1 𝑛 𝑛C𝑘 2

Q83.Number of integral terms in the expansion of 1 1 824 is equal to ______. 2 ) + 11( )} {7(

Q83.Let 𝐴−2, −1, 𝐵1, 0, 𝐶𝛼, 𝛽 and 𝐷𝛾, 𝛿 be the vertices of a parallelogram 𝐴𝐵𝐶𝐷. If the point 𝐶 lies on 2𝑥−𝑦= 5 and the point 𝐷 lies on 3𝑥−2𝑦= 6, then the value of 𝛼+ 𝛽+ 𝛾+ 𝛿 is equal to ______.

Q83.In the expansion of 1 + 𝑥1 −𝑥21 + + , 𝑥≠0, the sum of the coefficient of 𝑥3 and 𝑥-13 is equal to 𝑥+ 𝑥2 𝑥3 ______

Q83.If 11C1 2 + 3 + … . . + 10 = mn with gcd (n, m) = 1, then n + m is equal to

Q83.Let a ray of light passing through the point (3, 10) reflects on the line 2x + y = 6 and the reflected ray passes through the point (7, 2). If the equation of the incident ray is ax + by + 1 = 0, then a2 + b2 + 3ab is equal to_________ , on the positive x-axis. Let C be the circle with its centre at

Q84.Let A be a 2 × 2 symmetric matrix such that A [ 11] [ 37] where I is an identity matrix of order 2 × 2 , then α + β equals _______

Q84.In a triangle ABC, BC = 7, AC = 8, AB = α ∈N and cos A = 32 . If 49 cos(3C) + 42 = mn , where gcd(m, n) = 1, then m + n is equal to________ Q85. 2x + 7y + λz = 3 If the system of equations 3x + 2y + 5z = 4 has infinitely many solutions, then (λ −μ) is equal x + μy + 32z = −1 to________

Q84.Let the foci and length of the latus rectum of an ellipse 𝑥2 + 𝑦2 = 1, 𝑎> 𝑏 be ±5, 0 and √50, respectively. 𝑎2 𝑏2 𝑥2 𝑦2 Then, the square of the eccentricity of the hyperbola − = 1 equals 𝑏2 𝑎2𝑏2

Q84.Let P(α, β) be a point on the parabola y2 = 4x. If P also lies on the chord of the parabola x2 = 8y whose mid point is (1, 54 ), then (α −28)(β −8) is equal to _______.

Q84.Let 𝛼= and 𝛽= 𝑛- 1 ∑𝑘= 0 𝑘+ 1 ∑𝑘= 0 𝑘+ 2 . If 5𝛼= 6𝛽, then 𝑛 equals

Q84.Consider a circle 𝑥- 𝛼2 + 𝑦- 𝛽2 = 50, where 𝛼, 𝛽> 0. If the circle touches the line 𝑦+ 𝑥= 0 at the point P, whose distance from the origin is 4√2 , then ( 𝛼+ 𝛽) 2 is equal to _______.

Q84.Let S be the focus of the hyperbola x23 −y25 = 1 A(√6, √5) and passing through the point S . If O is the origin and SAB is a diameter of C , then the square of the area of the triangle OSB is equal to___________

Q84.Equations of two diameters of a circle are 2x −3y = 5 and 3x −4y = 7. The line joining the points (−227 , −4) and (−17 , 3) intersects the circle at only one point P(α, β). Then 17β −α is equal to = 1 lie on the curve y2 = 3x2 ,

Q84.If limx→1 (5x+1)1/3−(x+5)1/3 = m√5 , where gcd(m, n) = 1, then 8 m + 12n is equal to______ (2x+3)1/2−(x+4)1/2 n(2n)2/3

Q84.Let a line perpendicular to the line 2x −y = 10 touch the parabola y2 = 4(x −9) at the point P . The distance of the point P from the centre of the circle x2 + y2 −14x −8y + 56 = 0 is __________ = α + β√17, where

Q84.Let 𝐴= 𝐼2 −2𝑀𝑀𝑇, where 𝑀 is real matrix of order 2 × 1 such that the relation 𝑀𝑇𝑀= 𝐼1 holds. If 𝜆 is a real number such that the relation 𝐴𝑋= 𝜆𝑋 holds for some non-zero real matrix 𝑋 of order 2 × 1, then the sum of squares of all possible values of 𝜆 is equal to:

Q85.Let 𝐴= 1, 2, 3, . ...100 . Let 𝑅 be a relation on 𝐴 defined by 𝑥, 𝑦∈𝑅 if and only if 2𝑥= 3𝑦. Let 𝑅1 be a symmetric relation on 𝐴 such that 𝑅⊂𝑅1 and the number of elements in 𝑅1 is 𝑛. Then the minimum value of 𝑛 is _______.

Q85.In a survey of 220 students of a higher secondary school, it was found that at least 125 and at most 130 students studied Mathematics; at least 85 and at most 95 studied Physics; at least 75 and at most 90 studied Chemistry; 30 studied both Physics and Chemistry; 50 studied both Chemistry and Mathematics; 40 studied both Mathematics and Physics and 10 studied none of these subjects. Let m and n respectively be the least and the most number of students who studied all the three subjects. Then m + n is equal to ______ Q86. ⎡ 1⎤ ⎡1⎤ Let A be a 3 × 3 matrix of non-negative real elements such that A 1 = 3 1 . Then the maximum value of ⎣ 1⎦ ⎣1⎦ det(A) is ______ π a, b ∈N, then a + b is equal to_________

Q85.Consider the function f : R →R defined by f(x) = 2x . If the composition of √1+9x2 f, (f ∘f ∘f ∘⋯∘f) (x) = 210x , then the value of √3α + 1 is equal to ______ √1+9αx2 10 times 