Practice Questions

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

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 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 __________.

Q86.If the mean and variance of the data 65, 68, 58, 44, 48, 45, 60, α, β, 60 where α > β are 56 and 66. 2 respectively, then α2 + β2 is equal to JEE Main 2024 (29 Jan Shift 1) JEE Main Previous Year Paper

Q86.Let αβγ = 45; α, β, γ ∈R. If x(α, 1, 2) + y(1, β, 2) + z(2, 3, γ) = (0, 0, 0) for some x, y, z ∈R, xyz ≠0, then 6α + 4β + γ is equal to _______

Q86.Let 𝑓: 0, ∞→𝑅 and 𝐹𝑥= ∫ 𝑡𝑓𝑡𝑑𝑡. If 𝐹𝑥2 = 𝑥4 + 𝑥5, then 12 𝑓𝑟2 is equal to: ∑𝑟= 1 0

Q86.Let 𝐴 be a 3 × 3 matrix and det𝐴= 2. If 𝑛= det𝑎𝑑𝑗𝑎𝑑𝑗.⏟ ... 𝑎𝑑𝑗𝐴 , then the remainder when 𝑛 is divided by 9 2024 −times is equal to __________. 𝜋 Q87. 120 𝑥2sin𝑥cos𝑥 ∫ is equal to ______. 𝜋3 0 sin4𝑥+ cos4𝑥𝑑𝑥

Q86.Let the inverse trigonometric functions take principal values. The number of real solutions of the equation 2 sin−1 x + 3 cos−1 x = 2π5 , is _______

Q86.If the variance 𝜎2 of the data xi 0 1 5 6 10 12 17 is 𝑘 then the value of 𝑘 is ______ {where . fi 3 2 3 2 6 3 3 denotes the greatest integer funciton}

Q86.From a lot of 10 items, which include 3 defective items, a sample of 5 items is drawn at random. Let the random variable X denote the number of defective items in the sample. If the variance of X is σ2 , then 96σ2 is equal to ______

Q86.Let a, b, c ∈N and a < b < c. Let the mean, the mean deviation about the mean and the variance of the 5 observations 9, 25, a, b, c be 18,4 and 1365 , respectively. Then 2a + b −c is equal to__________

Q86.Let for a differentiable function f : (0, ∞) →R, f(x) −f(y) ≥loge( xy ) ∑20n=1 f ′( n21 ) is equal to

Q86.Let 𝑓: ℝ→ℝ be a function defined by 𝑓𝑥= 𝑓1 −𝑎 and 𝑀= ∫ 𝑥sin4𝑥1 −𝑥𝑑𝑥, 4𝑥+ 2 𝑓𝑎 𝑓1 −𝑎 𝑁= 𝛼𝑀= 𝛽𝑁, 𝛼, 𝛽∈ℕ, then the least value of 𝛼2 + 𝛽2 is equal to ______ ∫ sin4𝑥1 −𝑥𝑑𝑥; 𝑎≠12. If 𝑓𝑎 𝑥

Q86.Let A = {1, 2, 3, … . 7} and let P(A) denote the power set of A . If the number of functions f : A →P(A) such that a ∈f(a), ∀a ∈A is mn, m and n ∈N and m is least, then m + n is equal to ______. 1 , |x| ≥2 |x|