Practice Questions

Q73.Let f(x) = { −2xx3 −x2+ log2(b2+ 10x −4),−7, x ≤1 Then the set of all values of b, for which f(x) has maximum value at x = 1 , is: (1) (−6, −2) (2) (2, 6) (3) [−6, −2) ∪(2, 6] (4) [−√6, −2) ∪(2, √6] , x ∈(0, 1), then: lim k=1 n2+k22n and f(x) = √1−cos1+cos xx

Q73.For any real number 𝑥, let 𝑥 denote the largest integer less than or equal to 𝑥. Let 𝑓 be a real-valued function defined on the interval -10, 10 by 𝑥- 𝑥, if 𝑥 is odd 𝑓𝑥= 1 + 𝑥- 𝑥, if 𝑥 is even π2 10 Then, the value of 10 ∫-10 𝑓𝑥 cosπ𝑥𝑑𝑥 is (1) 4 (2) 2 (3) 1 (4) 0

Q73.Considering only the principal values of the inverse trigonometric functions, the domain of the function 𝑥2 - 4𝑥+ 2 𝑓𝑥= cos-1 is 𝑥2 + 3 1 1 (1) - ∞, (2) - ∞ 4 4, (3) -1 ∞ (4) - ∞, 1 3, 3

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 𝑃 and 𝑄 be any points on the curves 𝑥- 12 + 𝑦+ 12 = 1 and 𝑦= 𝑥2, respectively. The distance between 𝑃 and 𝑄 is minimum for some value of the abscissa of 𝑃 in the interval 1 1 3 (1) 0, (2) 4 2, 4 1 1 3 (3) 4, 2 (4) 4, 1

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 : R →R be continuous function satisfying f(x) + f(x + k) = n, for all x ∈R where k > 0 and n is a positive integer. If I1 = ∫4nk0 f(x)dx and I2 = ∫3k−k f(x)dx, then (1) I1 + 2I2 = 4nk (2) I1 + 2I2 = 2nk (3) I1 + nI2 = 4n2 K (4) I1 + nI2 = 6n2k

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.Let f be a real valued continuous function on [0, 1] and f(x) = x + ∫10 (x −t)f(t)dt. Then which of the following points (x, y) lies on the curve y = f(x)? (1) (2, 4) (2) (1, 2) (3) (4, 17) (4) (6, 8) JEE Main 2022 (29 Jun Shift 2) JEE Main Previous Year Paper =

Q74.The value of the integral ∫2−2 (ex|x|+1)x3+x (1) 5e2 (2) 3e−2 (3) 4 (4) 6 dy ax−by+a

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

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.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.Let 𝑔: 0, ∞→𝑅 be a differentiable function such that ∫ + d𝑥= + 𝐶, for all 𝑥> 0 e𝑥+ 1 e𝑥+ 12 e𝑥+ 1 , where 𝐶 is an arbitrary constant. Then 𝜋 𝜋 (1) 𝑔 is decreasing in 0, (2) 𝑔- 𝑔' is increasing in 0, 4 2 (3) 𝑔' is increasing in 0, 𝜋 (4) 𝑔+ 𝑔' is increasing in 0, 𝜋 4 2 𝜋 ecos𝑥sin𝑥

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 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.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.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.The minimum value of the twice differentiable function 𝑓𝑥= 𝑥𝑒𝑥- 𝑡𝑓'𝑡𝑑𝑡- 𝑥2 - 𝑥+ 1𝑒𝑥, 𝑥∈𝑅, is ∫0 2 (1) - (2) -2√𝑒 √𝑒 2 (3) -√𝑒 (4) √𝑒

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. max{t3 −3t}; x ≤2 t≤x ⎧ x2 + 2x −6; 2 < x < 3 Let f : R →R be a function defined by : f(x) = ⎨ [x −3] + 9; 3 ≤x ≤5 2x + 1; x > 5 ⎩ Where [t] is the greatest integer less than or equal to t. Let m be the number of points where f is not differentiable and I = ∫2−2 f(x)dx. Then the ordered pair (m, I) is equal to (1) (3, 274 ) (2) (3, 234 ) (3) (4, 274 ) (4) (4, 234 )

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.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.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 +