Practice Questions

Q88.Let f(x) = min{[x −1], [x −2], … , [x −10]} where [t] denotes the greatest integer ≤t. Then ∫100 f(x)dx + ∫100 (f(x))2dx + ∫100 |f(x)|dx is equal _______. to x > 0 and f(1) = √3 . If y = f(x)

Q88.If the tangent to the curve 𝑦= 𝑥3 - 𝑥2 + 𝑥 at the point 𝑎, 𝑏 is also tangent to the curve 𝑦= 5𝑥2 + 2x - 25 at the point 2, - 1, then 2𝑎+ 9𝑏 is equal to ______. 2 2 2 2 2

Q88.The value of 𝑏> 3 for which 12 𝑏 1 49 is equal to _____. ∫3 𝑥2 - 1𝑥2 - 4𝑑𝑥= log𝑒 40,

Q88.If the sum of all the roots of the equation e2x −11ex −45e−x + 812 = 0 is loge P , then P is equal to _____.

Q88.Let the tangents at the points P and Q on the ellipse x2 S is 2 + 4 = 1 meet at the point R(√2, 2√2 −2). If the focus of the ellipse on its negative major axis, then SP 2 + SQ2 is equal to π dx is equal to

Q88.For real numbers a, b(a > b > 0), let x2 y2 = 30π Area {(x, y) : x2 + y2 ≤a2 and a2 + b2 ≥1} and x2 y2 = 18π Area {(x, y) : x2 + y2 ≥b2 and a2 + b2 ≤1} Then the value of (a −b)2 is equal to _____.

Q88.Let A = {1, a1, a2 … … a18, 77} be a set of integers with 1 < a1 < a2 < … . . < a18 < 77. Let the set A + A = {x + y : x, y ∈A} contain exactly 39 elements. Then, the value of a1 + a2 + … . . +a18 is equal to ______.

Q88.Let 𝑓𝑥= 4𝑥2 - 8𝑥+ 5, if 8𝑥2 - 6𝑥+ 1 ≥0 , where 𝛼 denotes the greatest integer less than or equal to 𝛼. 4𝑥2 - 8𝑥+ 5, if 8𝑥2 - 6𝑥+ 1 < 0 Then the number of points in 𝑅 where 𝑓 is not differentiable is _____ . 1 𝑛+ 1𝑘- 1

Q88.Let f be a twice differentiable function on R. If f ′(0) = 4 and f(x) + ∫x0 (x −t)f ′(t)dt = (e2x + e−2x) cos 2x + a2 x, then (2a + 1)5a2 is equal to _______. n ∈N . Then the sum of all the elements of the set

Q88.Let M and N be the number of points on the curve y5 −9xy + 2x = 0 , where the tangents to the curve are parallel to x-axis and y-axis, respectively. Then the value of M + N equals _______.

Q88.Let y = y(x) be the solution of the differential equation dx 2 2 cos4 x−cos 2x with y( π4 ) = π232 . If y( π3 ) = π218 e−tan−1(α) , then the value of 3α2 is equal to ______.

Q88.If n(2n + 1) ∫10 (1 −xn)2ndx = 1177 ∫10 (1 −xn)2n+1dx, then JEE Main 2022 (26 Jul Shift 1) JEE Main Previous Year Paper

Q88.Suppose 𝑦= 𝑦𝑥 be the solution curve to the differential equation 𝑑𝑦 𝑦= 2 - 𝑒-𝑥 such that lim is finite. 𝑑𝑥- 𝑥→∞𝑦𝑥 If 𝑎 and 𝑏 are respectively the 𝑥- and 𝑦- intercept of the tangent to the curve at 𝑥= 0, then the value of 𝑎- 4𝑏 is equal to _______.

Q88.Let S = {(−10 ab ); 100 elements in n=1Tn∩ is _____.

Q88.Let f : R →R satisfy f(x + y) = 2xf(y) + 4y(f(x), ∀x, y ∈R. If f(2) = 3 , then 14 ⋅ff ′(4)′(2) (2−x2) dx √2

Q88.The number of matrices of order 3 × 3, whose entries are either 0 or 1 and the sum of all the entries is a prime number, is _______.

Q88.Let 𝑓: 0, 1 →𝑅 be a twice differentiable function in 0, 1 such that 𝑓0 = 3 and 𝑓1 = 5. If the line 𝑦= 2𝑥+ 3 intersects the graph of 𝑓 at only two distinct points in 0, 1, then the least number of points 𝑥∈0, 1, at which 𝑓''𝑥= 0, is √3 15𝑥3

Q88.The value of the integral dx is equal to ______. π4 48 ∫π0 ( 3πx22 −x3) 1+cos2sin x x

Q88.If the system of linear equations 2x −3y = γ + 5 αx + 5y = β + 1 , where α, β, γ ∈R has infinitely many solutions, then the value of |9α + 3β + 5γ| is equal to

Q89.Let y = y(x), x > 1 , be the solution of the differential equation (x −1) dxdy + 2xy = x−11 , with y(2) = 1+e42e4 . If y(3) = eα+1βeα . then the value of α + β is equal to ______. → → , then the value of is b 3(→c.→a)

Q89.Let l be a line which is normal to the curve y = 2x2 + x + 2 at a point P on the curve. If the point Q(6, 4) lies on the line l and O is origin, then the area of the triangle OPQ is equal to _____. → →

Q89.The value of the integral ∫ 0 2 60 sin(6x)sin x JEE Main 2022 (28 Jul Shift 2) JEE Main Previous Year Paper

Q89.Let 𝜃 be the angle between the vectors →𝑎 and →𝑏, where →𝑎= 4, →𝑏= 3 and 𝜃∈𝜋 𝜋 Then 4, 3. 2 2 →𝑎- →𝑏× →𝑎+ →𝑏 + 4→𝑎· →𝑏 is equal to ______

Q89.The integral 24 is equal to ______. π ∫ 0 (2+x2)√4+x4

Q89.Let y = y(x) be the solution of the differential equation −1 < x < 1 (1 −x2)dy = (xy + (x3 + 2)√1 −x2)dx, 1 and y(0) = 0. If ∫ 2 √1 −x2y(x)dx = k then k−1 is equal to −12