Practice Questions

Q71.The number of points, where the function f : R →R, f(x) = |x −1| cos|x −2| sin|x −1| + (x −3) x2 −5x + 4 , is NOT differentiable, is (1) 1 (2) 2 (3) 3 (4) 4

Q72.Let f, g : N −{1} →N be functions defined by f(a) = α, where α is the maximum of the powers of those primes p such that pα divides a, and g(a) = a + 1, for all a ∈N −{1}. Then, the function f + g is (1) one-one but not onto (2) onto but not one-one (3) both one-one and onto (4) neither one-one nor onto

Q72.The curve 𝑦𝑥= 𝑎𝑥3 + 𝑏𝑥2 + 𝑐𝑥+ 5 touches the 𝑥-axis at the point 𝑃-2, 0 and cuts the 𝑦-axis at the point $\mathrm{Q}$, where 𝑦' is equal to 3. Then the local maximum value of 𝑦𝑥 is (1) 27 (2) 29 4 4 37 9 (3) (4) 4 2

Q72.The domain of f(x) = cos−1(log(x2−3x+2)x2−5x+6 (1) x ∈[ −12 , 1) ∪(2, ∞) −{3} (2) x ∈[ −12 , 1] ∪(2, ∞) −{3} (3) x ∈( −12 , 1) ∪[2, ∞) −{3} (4) x ∈[ −12 , 1) ∪[2, ∞) −{3}

Q72.Let f, g : R →R be functions defined by , x < 0 f(x) = and {[x]|1 −x| , x ≥0 JEE Main 2022 (28 Jun Shift 2) JEE Main Previous Year Paper ex −x, x < 0 g(x) = { (x −1)2 −1, x ≥0 where [x] denote the greatest integer less than or equal to x. Then, the function fog is discontinuous at exactly (1) one point (2) two points (3) three points (4) four points

Q72.If for p ≠q ≠0 , then function f(x) = 7√p(729+x)−3 is continuous at x = 0 , then 3√729+qx−9 (1) 7pqf(0) −1 = 0 (2) 63qf(0) −p2 = 0 (3) 21qf(0) −p2 = 0 (4) 7pq f(0) −9 = 0

Q72.Let f(x) = min{1, 1 + x sin x}, 0 ≤x ≤2π. If m is the number of points, where f is not differentiable and n is the number of points, where f is not continuous, then the ordered pair (m, n) is equal to (1) (2, 0) (2) (1, 0) (3) (1, 1) (4) (2, 1) JEE Main 2022 (26 Jun Shift 2) JEE Main Previous Year Paper

Q72.The value of cot(∑50n=1 tan−1( 1+n+n21 )) (1) 25 (2) 50 26 51 (3) 26 (4) 52 25 51 JEE Main 2022 (27 Jun Shift 2) JEE Main Previous Year Paper

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

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 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

Q73.Water is being filled at the rate of 1cm3sec-1 in a right circular conical vessel (vertex downwards) of height 35cm and diameter 14cm. When the height of the water level is 10cm, the rate (in cm2 sec-1) at which the JEE Main 2022 (25 Jun Shift 2) JEE Main Previous Year Paper wet conical surface area of the vessel increases is (1) 5 (2) √21 5 (3) √26 (4) √26 5 10

Q73.Let In(x) = ∫x0 (t2+5)n1 (1) 50I6 −9I5 = xI 5′ (2) 50I6 −11I5 = xI 5′ (3) 50I6 −9I5 = I 5′ (4) 50I6 −11I5 = I 5′ x = loge 2 , above the line y = 1 is

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.Consider a cuboid of sides 2x, 4x and 5x and a closed hemisphere of radius r. If the sum of their surface areas is constant k, then the ratio x : r, for which the sum of their volumes is maximum, is (1) 2 : 5 (2) 19 : 45 (3) 3 : 8 (4) 19 : 15 dx = g(x) + c, g(1) = 0 , then g( 12 ) 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 the maximum value of 𝑎, for which the function 𝑓𝑎𝑥= tan-12𝑥- 3𝑎𝑥+ 7 is non-decreasing in -𝜋 𝜋 is ¯𝑎, 6, 6, 𝜋 then 𝑓¯𝑎 8 is equal to (1) 8 - 9𝜋 (2) 8 - 4𝜋 49 + 𝜋2 94 + 𝜋2 1 + 𝜋2 𝜋 (4) 8 - (3) 8 4 9 + 𝜋2 JEE Main 2022 (26 Jul Shift 2) JEE Main Previous Year Paper Q75. 1 - 1 √3cos𝑥- sin𝑥 The integral ∫ 2 is equal to 1 + √3sin2𝑥𝑑𝑥 𝜋 𝜋 tan𝑥 + tan𝑥 + 2 12 2 (1) 1 (2) 𝑥 𝜋 + 𝐶 2log𝑒 6 + 𝜋 + 𝐶 log𝑒 𝑥 + 2 6 2 3 𝜋 𝜋 tan𝑥 + tan𝑥 - 2 2 12 (3) 1 6 (4) 1 𝑥 𝜋 + 𝐶 2log𝑒 + 𝜋 + 𝐶 2log𝑒 tan𝑥 - 2 3 2 6 Q76. 20𝜋sin𝑥+ cos𝑥2𝑑𝑥 is equal to: ∫0 (1) 10𝜋+ 4 (2) 10𝜋+ 2 (3) 20𝜋- 2 (4) 20𝜋+ 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.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.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 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

Q75.Let [t] denote the greatest integer less than or equal to t. Then the value of the integral ∫101−3 ([sin(πx)] + e[cos(2πx)])dx is equal to (1) 52(1−e) (2) 52 e e (3) 52(2+e) (4) 104 e e

Q75.The area of the smaller region enclosed by the curves y2 = 8x + 4 and x2 + y2 + 4√3x −4 = 0 is equal to (1) 1 + + 3 (2 −12√3 8π) (2) 13 (2 −12√3 6π) (3) 1 −12√3 + −12√3 + 3 (4 8π) (4) 13 (4 6π)

Q75.Let f(x) = 2 cos−1 x + 4 cot−1 x −3x2 −2x + 10, x ∈[−1, 1]. If [a, b] is the range of the function, then 4a −b is equal to (1) 11 (2) 11 −π (3) 11 + π (4) 15 −π

Q75.The area of the region {(x, y) : |x −1| ≤y ≤√5 −x2} (1) 5 2 sin−1( 53 ) −12 (2) 5π4 −32 (3) 3π 4 + 23 (4) 5π4 −12 + = 1 pass through the point