Practice Questions

Q83.Consider a triangle ABC having the vertices A(1, 2), B(α, β) and C(γ, δ) and angles ∠ABC = π6 and ∠BAC = 2π3 . If the points B and C lie on the line y = x + 4, then α2 + γ 2 is equal to ________ = and the determinant of A be 1 . If A−1 = αA + βI ,

Q83.Let α = ∑nr=0 (4r2 + 2r + 1)nCr and β = (∑nr=0 r+1nCr ) _______

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

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.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.Remainder when 643232 is divided by 9 is equal to _____.

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

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:

Q84.Let the line 𝐿: √2𝑥+ 𝑦= 𝛼 pass through the point of the intersection 𝑃(in the first quadrant)of the circle 𝑥2 + 𝑦2 = 3 and the parabola 𝑥2 = 2𝑦. Let the line 𝐿 touch two circles 𝐶1 and 𝐶2 of equal radius 2√3. If the centres 𝑄1 and 𝑄2 of the circles 𝐶1 and 𝐶2 lie on the 𝑦- axis, then the square of the area of the triangle 𝑃𝑄1𝑄2 is equal 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 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.Consider the circle C : x2 + y2 = 4 and the parabola P : y2 = 8x. If the set of all values of α, for which three chords of the circle C on three distinct lines passing through the point (α, 0) are bisected by the parabola P is the interval (p, q), then (2q −p)2 is equal to ________ JEE Main 2024 (09 Apr Shift 2) JEE Main Previous Year Paper

Q84.If lim 𝑎𝑥2𝑒𝑥−𝑏log𝑒1 + 𝑥+ 𝑐𝑥𝑒−𝑥 = 1, then 16𝑎2 + 𝑏2 + 𝑐2 is equal to ______. 𝑥→0 𝑥2sin𝑥 JEE Main 2024 (31 Jan Shift 2) JEE Main Previous Year Paper

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.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.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 conic C pass through the point (4, −2) and P(x, y), x ≥3, be any point on C . Let the slope of the line touching the conic C only at a single point P be half the slope of the line joining the points P and (3, −5). If the focal distance of the point (7, 1) on C is d, then 12d 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.If the orthocentre of the triangle formed by the lines 2x + 3y −1 = 0, x + 2y −1 = 0 and ax + by −1 = 0, is the centroid of another triangle, whose circumcentre and orthocentre respectively are (3, 4) and (−6, −8), then the value of |a −b| is_______ is

Q84.Suppose AB is a focal chord of the parabola y2 = 12x of length l and slope m < √3 . If the distance of the chord AB from the origin is d , then l d2 is equal to _______ for

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 ,