Practice Questions

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

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.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.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 the latus rectum of the hyperbola x2 = 1 subtend an angle of π3 at the centre of the hyperbola. If b2 9 −y2b2 is equal to l (1 + √n), where l and m are co-prime numbers, then l2 + m2 + n2 is equal to __________. m

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.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 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.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.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.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 n − 2n + n − 8n + … + n − 2n⋅n2 be πk , limn→∞( √n4+1 (n2+1)√n4+1 √n4+16 (n2+4)√n4+16 √n4+n4 (n2+n2)√n4+n4 ) using only the principal values of the inverse trigonometric functions. Then k2 is equal to ________

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.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.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.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.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.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

Q85.Consider two circles 𝐶1: 𝑥2 + 𝑦2 = 25 and 𝐶2: ( 𝑥- 𝛼) 2 + 𝑦2 = 16, where 𝛼∈( 5, 9 ) . Let the angle between . If the the two radii (one to each circle) drawn from one of the intersection points of C1and C2 be sin-1√638 length of common chord of 𝐶1 and 𝐶2 is 𝛽, then the value of ( 𝛼𝛽) 2 equals _________. JEE Main 2024 (30 Jan Shift 2) JEE Main Previous Year Paper

Q85.A group of 40 students appeared in an examination of 3 subjects - Mathematics, Physics & Chemistry. It was found that all students passed in at least one of the subjects, 20 students passed in Mathematics, 25 students passed in Physics, 16 students passed in Chemistry, at most 11 students passed in both Mathematics and Physics, at most 15 students passed in both Physics and Chemistry, at most 15 students passed in both Mathematics and Chemistry. The maximum number of students passed in all the three subjects is _____.

Q85.The mean and standard deviation of 15 observations were found to be 12 and 3 respectively. On rechecking it was found that an observation was read as 10 in place of 12 . If 𝜇 and 𝜎2 denote the mean and variance of the correct observations respectively, then 15𝜇+ 𝜇2 + 𝜎2 is equal to _________. JEE Main 2024 (27 Jan Shift 2) JEE Main Previous Year Paper

Q85.Let the slope of the line 45x + 5y + 3 = 0 be 27r1 + 9r22 for some r1, r2 ∈R. Then 8t2 is equal to ______. lim 3 3r2x x→3(∫x 2 −r2x2−r1x3−3x dt)

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 