Practice Questions

Q73.Let λ* be the largest value of λ for which the function fλ(x) = 4λx3 −36λx2 + 36x + 48 is increasing for all x ∈R. Then fλ*(1) + fλ,*(−1) is equal to: (1) 36 (2) 48 (3) 64 (4) 72 π

Q73.The number of bijective function f(1, 3, 5, 7, ⋯, 99) →(2, 4, 6, 8, ⋯, 100) if f(3) > f(5) > f(7) ⋯> f(99) is (1) 50C1 (2) 50C2 (3) 50! (4) 50C3 × 3! 2

Q73.Let f : R →R be a function defined by f(x) = (x −3)n1(x −5)n2, n1, n2 ∈N . The, which of the following is NOT true? (1) For n1 = 3, n2 = 4 , there exists α ∈(3, 5) (2) For n1 = 4, n2 = 3, there exists α ∈(3, 5) where f attains local maxima. where f attains local maxima. (3) For n1 = 3, n2 = 5 , there exists α ∈(3, 5) (4) For n1 = 4, n2 = 6, there exists α ∈(3, 5) where f attains local maxima. where f attains local maxima.

Q73.The integral ∫ 0 2 3+2 sin1x+cos x dx is equal to: (1) tan−1(2) (2) tan−1(2) −π4 (3) 1 2 tan−1(2) −π8 (4) 21 α > 0, then f(e3) + f(e−3) is equal to

Q73.The sum of the absolute minimum and the absolute maximum values of the function f(x) = 3x −x2 + 2 −x in the interval [−1, 2] is (1) √17+3 (2) √17+5 2 2 (3) 5 (4) 9−√17 2

Q74.The area of the region S = {(x, y) : y2 ≤8x, y ≥√2x, x ≥1} is (1) 5√2 (2) 19√2 6 6 (3) 13√2 (4) 11√2 6 6 pass + e x = x + + e x y ]x dxdy y ]y

Q74. lim 2n1 1 + 1 + 1 + … . + 1 is equal to n→∞ ( √1−12n √1−22n √1−32n √1−2n−12n ) (1) 1 (2) 1 2 (3) 2 (4) −2

Q74.The area of the region given by 𝐴= 𝑥, 𝑦: 𝑥2 ≤𝑦≤min𝑥+ 2, 4 - 3𝑥 is (1) 31 (2) 17 8 6 19 27 (3) (4) 6 8 JEE Main 2022 (25 Jul Shift 1) JEE Main Previous Year Paper

Q74.If the line 𝑦= 4 + 𝑘𝑥, 𝑘> 0, is the tangent to the parabola 𝑦= 𝑥- 𝑥2 at the point 𝑃 and 𝑉 is the vertex of the parabola, then the slope of the line through 𝑃 and 𝑉 is (1) 3 (2) 26 2 9 5 23 (3) (4) 2 6

Q74.If 𝑡 denotes the greatest integer ≤t, then the value of ∫0 2𝑥- 3𝑥2 - 5𝑥+ 2 + 1𝑑𝑥 is JEE Main 2022 (29 Jul Shift 2) JEE Main Previous Year Paper (1) √37 + √13 - 4 (2) √37 - √13 - 4 6 6 (3) -√37 - √13 + 4 (4) -√37 + √13 + 4 6 6

Q74. I = ∫ π 3 ( 8 sin x−sinx 2x )dx. Then 4 (1) π 2 < I < 3π4 (2) π5 < I < 5π12 (3) 5π 12 < I < √23 π (4) 3π4 < I < π

Q74.Let f be a differentiable function in (0, π2 ). If ∫1cos x t2f(t)dt = sin3 x + cos x, then √31 f ′( √31 ) (1) 6 −9√2 (2) 6 + 9 √2 (3) 6 − 9 (4) 3 + √2 √2 dx, where [⋅] denotes the greatest integer function, is equal to

Q74.If ∫1x √1−x1+x + π3 (1) loge( √3+1√3−1 ) + π3 (2) loge( √3+1√3−1 ) (3) loge( √3−1√3+1 ) −π3 (4) 13 loge( √3−1√3+1 ) −π6

Q74.The area enclosed by the curves y = loge(x + e2), x = loge( 2y ) and (1) 2 + e −loge 2 (2) 1 + e −loge 2 (3) e −loge 2 (4) 1 + loge 2 dy +

Q74.If the tangent at the point (x1, y1) on the curve y = x3 + 3x2 + 5 passes through the origin, then (x1, y1) does NOT lie on the curve (1) x2 + 81y2 = 2 (2) y29 −x2 = 8 (3) y = 4x2 + 5 (4) x3 −y2 = 2

Q74.If a = n→∞∑n (1) 2√2f( a2 ) = f ′( a2 ) (2) f( a2 )f ′( a2 ) = √2 (3) √2f( a2 ) = f ′( a2 ) (4) f( a2 ) = √2f ′( a2 )

Q74.If f(α) = ∫α1 log101+t t dt, (1) 9 (2) 92 (3) 9 (4) 9 loge(10) 2 loge(10) is equal to

Q74.Let S be the set of all the natural numbers, for which the line xa + yb = 2 is a tangent to the curve ( xa ) n + ( yb ) n = 2 at the point (a, b), ab ≠0. Then (1) S = ϕ (2) n(S) = 1 (3) S = {2k : k ∈N} (4) S = N

Q74.The value of the integral ∫ −π2 2 (1+ex)(sin6dxx+cos6 x) is equal to (1) 2π (2) 0 (3) π (4) π 2

Q75.If ∫20 (√2x −√2x −x2)dx + I , then I equal to + ∫21 (2 −y22 )dy ∫10 (1 −√1 −y2 −y22 )dy −y2 + + √1 −y2)dy (2) ∫10 ( y22 −√1 1)dy (1) ∫10 (1 + √1 −y2 + 1)dy (3) ∫10 (1 −√1 −y2)dy (4) ∫10 ( y22

Q75.The sum of absolute maximum and absolute minimum values of the function f(x) = 2x2 + 3x −2 + sin x cos x in the interval [0, 1] is 1 sin(1) cos2( (1) 2 ) (2) 3 + 12 (1 + 2 cos(1)) sin(1) 3 + 2 (3) 5 + 12 (sin(1) + sin(2)) (4) 2 + sin( 21 ) cos( 12 )

Q75.The area of the bounded region enclosed by the curve y = 3 −x −12 −|x + 1| and the x-axis is (1) 9 (2) 45 4 16 (3) 278 (4) 1663 x x −4xe y2 = 0 such that x(1) = 0.

Q75.A wire of length 22m is to be cut into two pieces. One of the pieces is to be made into a square and the other into an equilateral triangle. Then, the length of the side of the equilateral triangle, so that the combined area of the square and the equilateral triangle is minimum, is (1) 22 (2) 66 9+4√3 9+4√3 (3) 22 (4) 66 4+9√3 4+9√3 t, is equal toQ76. ∫50 cos(π(x −[ x2 ]))dx, where [t] denotes greatest integer less than or equal to (1) 0 (2) 2 (3) −3 (4) 4 JEE Main 2022 (29 Jun Shift 1) JEE Main Previous Year Paper

Q75.Let the solution curve y = y(x) of the differential equation, [ √x2−y2x [ √x2−y2x through the points (1, 0) and (2α, α), α > 0 . Then α is equal to (1) 2 1 exp( π6 + √e −1) (2) 12 exp( π3 + √e −1) (3) exp( π6 + √e + 1) (4) 2 exp( π3 + √e −1)

Q75.The value of ∫0 1 + cos2𝑥ecos𝑥+ e-cos𝑥d𝑥 is equal to (1) 𝜋2 (2) 𝜋 4 4 (3) 𝜋 (4) 𝜋2 6 2