Practice Questions

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

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 ________

Q85.If α = limx→0+ e√tan x−e√x and β = limx→0(1 + sin x) 1 ( √tan x−√x ) 2 cot x are the roots of the quadratic equation ax2 + bx −√e = 0, then 12 loge(a + b) is equal to__________

Q85.Let f(x) = x3 + x2f ′(1) + xf ′′(2) + f ′′′(3), x ∈R. Then f ′(10) is equal to + x −y, ∀x, y ∈(0, ∞). Then

Q85.Let 𝑥 denote the fractional part of 𝑥 and 𝑓𝑥= cos−11 −𝑥2sin−11 −𝑥 , 𝑥≠0. If 𝐿 and 𝑅 respectively denotes the 𝑥−𝑥3 32 left hand limit and the right hand limit of 𝑓𝑥 at 𝑥= 0, then 𝜋2𝐿2 + 𝑅2 is equal to __________.

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.Let a > 0 be a root of the equation 2x2 + x −2 = 0. If limx→1a 16(1−cos(2+x−2x2))(1−ax)2 α, β ∈Z , then α + β is equal to_______

Q85.Let L1, L2 be the lines passing through the point P(0, 1) and touching the parabola 9x2 + 12x + 18y −14 = 0. Let Q and R be the points on the lines L1 and L2 such that the △PQR is an isosceles triangle with base QR. If the slopes of the lines QR are m1 and m2 , then 16 (m21 + m22) is equal to _______

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

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 

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.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.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.If the points of intersection of two distinct conics x2 + y2 = 4b and x216 + y2b2 then 3√3 times the area of the rectangle formed by the intersection points is _______.

Q85.Let A = {2, 3, 6, 7} and B = {4, 5, 6, 8}. Let R be a relation defined on A × B by (a1, b1)R (a2, b2) if and only if a1 + a2 = b1 + b2 . Then the number of elements in R is _________

Q85.Let f be a differentiable function in the interval (0, ∞) such that f(1) = 1 and limt→x t2f(x)−x2f(t)t−x = 1 each x > 0 . Then 2f(2) + 3f(3) is equal to _______

Q85.The value of limx→0 2 ( 1−cos x√cos 2x3√cosx2 3x……10√cos 10x )

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_________

Q86.Let f : R →R be a thrice differentiable function such that f(0) = 0, f(1) = 1, f(2) = −1, f(3) = 2 and f(4) = −2. Then, the minimum number of zeros of (3f ′f ′′ + ff ′′′)(x) is _______

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.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