Practice Questions

Q87.Let c, k ∈R. If f(x) = (c + 1)x2 + (1 −c2)x + 2k and f(x + y) = f(x) + f(y) −xy, for all x, y ∈R, then the value of |2(f(1) + f(2) + f(3) + … … + f(20))| is equal to ______. √2y π dy + = xetan−1(√2 cot 2x), 0 < x <

Q87.Let the area enclosed by the x-axis, and the tangent and normal drawn to the curve 4x3 −3xy2 + 6x2 −5xy −8y2 + 9x + 14 = 0 at the point (−2, 3) be A . Then 8A is equal to _______.

Q87.Let the mean and the variance of 20 observations x1, x2, … x20 be 15 and 9, respectively. For α ∈R, if the mean of (x1 + α)2, (x2 + α)2, … , (x20 + α)2 is 178, then the square of the maximum value of α is equal to JEE Main 2022 (29 Jul Shift 1) JEE Main Previous Year Paper ______.

Q87.If y(x) = (xx)x, x > 0 then d2x + 20 at x = 1 is equal to dy2 2 2 + y 3 ≤1, x + y ≥0, y y) : x 3 is A , then 256Aπ is ≥0}

Q87.Let 𝐴= 1 -1 and 𝐵= 𝛽1 , 𝛼, 𝛽∈𝑅. Let 𝛼1 be the value of 𝛼 which satisfies 𝐴+ 𝐵2 = 𝐴2 + 2 2 and 2 𝛼 1 0 2 2 𝛼2 be the value of 𝛼 which satisfies 𝐴+ 𝐵2 = 𝐵2. Then 𝛼1 - 𝛼2 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 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.Let S = {(−10 ab ); 100 elements in n=1Tn∩ is _____.

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.If the area of the region {(x,

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.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.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.Let S be the region bounded by the curves y = x3 and y2 = x. The curve y = 2|x| divides S into two regions of areas R1 and R2 . If max|R1, R2| = R2 , then R1R2 is equal to ______

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

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 A1 = {(x, y) : |x| ≤y2, |x| + 2y ≤8} and A2 = {(x, y) : |x| + |y| ≤k}. If 27 (Area A1 ) = 5 (Area A2 ), then k is equal to

Q89.Let →𝑎 and →𝑏 be two vectors such that →𝑎+ →𝑏 = →𝑎 + 2 →𝑏 , →𝑎· →𝑏= 3 and →𝑎× →𝑏 = 75. Then →𝑎 is equal to ______.

Q89.If 𝑙𝑖𝑚 lim ( 𝑛𝑘+ 1 ) + ( 𝑛𝑘+ 2 ) + … + 𝑛𝑘+ 𝑛= 33 . 1 1 · 1𝑘+ 2𝑘+ 3𝑘+ … + 𝑛𝑘, then the 𝑛𝑘+ 𝑛𝑘+ 𝑛→∞ 𝑛→∞ integral value of 𝑘 is equal to _____ . 𝑥- 2 𝑦- 1 𝑧 𝑥- 3 𝑦- 5 𝑧- 1

Q89.Let d be the distance between the foot of perpendiculars of the points P(1, 2 −1) and Q(2, −1, 3) on the plane −x + y + z = 1 . Then d2 is equal to ______. JEE Main 2022 (29 Jun Shift 1) JEE Main Previous Year Paper = 4 be a plane. Let P2 be another plane which passes through the points

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.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.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 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.Let the solution curve y = y(x) of the differential equation (4 + x2)dy −2x(x2 + 3y + 4)dx = 0 pass through the origin. Then y(2) is equal to _____.