Practice Questions

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

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 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.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 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.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.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 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.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 _______

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 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 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 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|

Q87.Let the maximum and minimum values of −x2 −12 2 + (x −7)2, x ∈R be M and m , (√8x −4) respectively. Then M2 −m2 is equal to _________ π

Q87.Let 𝑆= −1, ∞ and 𝑓: 𝑆→ℝ be defined as 𝑓𝑥= ∫ 𝑒𝑡−1112𝑡−15𝑡−27𝑡−3122𝑡−1061𝑑𝑡. Let 𝑝= Sum −1 of square of the values of 𝑥, where 𝑓𝑥 attains local maxima on 𝑆. and 𝑞= Sum of the values of 𝑥, where 𝑓𝑥 attains local minima on 𝑆. Then, the value of 𝑝2 + 2𝑞 is ________ 𝜋 1 Q88. 2 11 5 If the integral 525 ∫ sin2𝑥 cos 2 𝑥1 + cos 2𝑥 2𝑑𝑥 is equal to 𝑛√2 −64, then 𝑛 is equal to ________ 0 → → → → →

Q87.Three points 𝑂0, 0, 𝑃𝑎, 𝑎2, 𝑄−𝑏, 𝑏2, 𝑎> 0, 𝑏> 0, are on the parabola 𝑦= 𝑥2. Let 𝑆1 be the area of the region bounded by the line 𝑃𝑄 and the parabola, and 𝑆2 be the area of the triangle 𝑂𝑃𝑄. If the minimum value 𝑆1 𝑚 of is 𝑛, gcd𝑚, 𝑛= 1, then 𝑚+ 𝑛 is equal to: 𝑆2 JEE Main 2024 (01 Feb Shift 2) JEE Main Previous Year Paper

Q87.If ∫ 0 4 1+sinsin2x xcos x dx = 1a loge ( 3a ) + b√3π , where

Q87.Let [t] denote the largest integer less than or equal to t. If + = a + b√2 −√3 −√5 + c√6 −√7, where a, b, c ∈Z, then a + b + c is equal ∫30 ([x2] [ x22 ])dx to_______

Q87.Let f(x) = √limr→x{ 2r2[(f(r))2−f(x)f(r)] f(r) the value of ae , such that f(a) = 0, is equal to ______. π

Q87.Let fx = x 1 - t where g is a continuous odd function. If 2 fx + x2cosx dx = π 2 - α, then α is equal ∫0 gtloge 1 + tdt, ∫-π 1 + ex α 2 to _____.

Q87.If ∫cosec5 xdx = α cot x cosec x (cossc2 x + 32 ) + β logϵ tan x2 + C where α, β ∈R and C is the constant of integration, then the value of 8(α + β) equals _______

Q87.If a function f satisfies f( m + n) = f( m) + f(n) for all m, n ∈N and f(1) = 1, then the largest natural number λ such that ∑2022k=1 f(λ + k) ≤(2022)2 is equal to _________ JEE Main 2024 (09 Apr Shift 1) JEE Main Previous Year Paper Q88. ⎧ ( 78 ) tantan 7x8x , 0 < π x < 2 a −8, x = π2 Let f : (0, π) →R be a function given by f(x) = ⎨ b | tan π x < π ⎩ (1 + | cot x|) x|, 2 < where a, b ∈Z. If f is continuous at x = π2 , then a2 + b2 is equal to

Q88.For a differentiable function f : R →R, suppose f ′(x) = 3f(x) + α, where α ∈R, f(0) = 1 and limx→−∞f(x) = 7. Then 9f (−loge 3) is equal to_________

Q88.The area of the region enclosed by the parabola ( 𝑦- 2 ) 2 = 𝑥- 1, the line 𝑥- 2 𝑦+ 4 = 0 and the positive coordinate axes is __________.