Practice Questions

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.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 𝐼𝑥= ∫sec2𝑥- 2022 if 𝐼𝜋 = 21011, then sin2022𝑥𝑑𝑥, 4 𝜋 𝜋 𝜋 𝜋 (1) 31010𝐼 - 𝐼 = 0 (2) 31010𝐼 - 𝐼 = 0 3 6 6 3 (3) 31011𝐼𝜋 - 𝐼𝜋 = 0 (4) 31011𝐼𝜋 - 𝐼𝜋 = 0 3 6 6 3 1

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.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 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 differentiable function such that f( π4 ) = √2, f( π2 ) = 0 and f ′( π2 ) = 1 and let π lim g(x) = ∫ x4 (f ′(t) sec t + tan t sec tf(t))dt for x ∈[ π4 , π2 ). Then π x→( 2 )−g(x) is equal to (1) 2 (2) 3 (3) 4 (4) −3

Q73.Let 𝑓: 𝑅→𝑅 and 𝑔: 𝑅→𝑅 be two functions defined by 𝑓𝑥= 1 - 2e2𝑥 loge𝑥2 + 1 - e-𝑥+ 1 and 𝑔𝑥= e𝑥 · Then, for 𝛼- 12 5 which of the following range of 𝛼, the inequality 𝑓𝑔 > 𝑓𝑔𝛼- holds? 3 3 (1) -2, - 1 (2) 2, 3 (3) 1, 2 (4) -1, 1 𝑥cos𝑥- sin𝑥 𝑔𝑥e𝑥+ 1 - 𝑥e𝑥 𝑥𝑔𝑥

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.For the function f(x) = 4 loge(x −1) −2x2 + 4x + 5, x > 1 , which one of the following is NOT correct? JEE Main 2022 (24 Jun Shift 1) JEE Main Previous Year Paper (1) f(x) is increasing in (1, 2) and decreasing in (2) f(x) = −1 has exactly two solutions (2, ∞) (3) f ′(e) −f ′′(2) < 0 (4) f(x) = 0 has a root in the interval (e, e + 1)

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 domain of the function 2 sin−1( is π cos−1( ) , , ∞) ∞) (1) (−∞, −1√2 ] ∪[ √21 ∪{0} (2) (−∞, −1√2 ] ∪[ √21 ∪( 12 , ∞) ∪{0} (4) R −{−12 , 12 } (3) (−∞, −1√2 )

Q73.Let a function f : R →R be defined as: 0 (5 −|t −3|)dt, x > 4 f(x) = {∫xx2 + bx, x ≤4 where b ∈R. If f is continuous at x = 4, then which of the following statements is NOT true? (1) f is not differentiable at x = 4 (2) f ′(3) + f ′(5) = 354 (3) f is increasing in (−∞, 81 ) ∪(8, ∞) (4) f has a local minima at x = 81 π

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 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 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. 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 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. 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 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 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.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. 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.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