Practice Questions

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

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_________

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

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 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.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 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.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, 4 and 𝑅= ( 1, 2 ) , ( 2, 3 ) , ( 1, 4 ) be a relation on 𝐴. Let 𝑆 be the equivalence relation on 𝐴 such that 𝑅⊂𝑆 and the number of elements in 𝑆 is 𝑛. Then, the minimum value of 𝑛 is _______ 4𝑥

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 

Q85.If 𝑦= √𝑥+ 1𝑥2 −√𝑥 1 then 96𝑦'𝜋 is equal to: 𝑥√𝑥+ 𝑥+ √𝑥+ 153cos2𝑥−5cos3𝑥, 6 𝑥

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.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.Consider the matrices : A = [ 23 −5m ], B = [ 20m ] and X = [ xy ] . Let the set of all m, for which the system of equations AX = B has a negative solution (i.e., x < 0 and y < 0 ), be the interval (a, b). Then 8 ∫ba |A|dm 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_________

Q86.Let A = [ 21 −11 ] . If the sum of the diagonal elements of A13 is 3n , then n is equal to_________

Q86.Let A be a 2 × 2 real matrix and I be the identity matrix of order 2 . If the roots of the equation |A - xI | = 0 be -1 and 3, then the sum of the diagonal elements of the matrix A2 is _____. π

Q86.Let [t] denote the greatest integer less than or equal to t. Let f : [0, ∞) →R be a function defined by f(x) = [ x2 + 3] −[√x]. Let S be the set of all points in the interval [0, 8] at which f is not continuous. Then ∑a∈S a is equal to _______

Q86.The number of elements in the set 𝑆= 𝑥, 𝑦, 𝑧: 𝑥, 𝑦, 𝑧∈𝑍, 𝑥+ 2𝑦+ 3𝑧= 42, 𝑥, 𝑦, 𝑧≥0 equals ________