Practice Questions

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 value of tan-1cos15𝜋 is equal to sin𝜋 4 𝜋 𝜋 (1) - (2) - 4 8 (3) -5𝜋 (4) -4𝜋 12 9

Q72.The lengths of the sides of a triangle are 10 + x2 , 10 + x2 and 20 −2x2 . If for x = k, the area of the triangle is maximum, then 3k2 is equal to (1) 5 (2) 12 (3) 10 (4) 20 d3f dx = f(x)ex + C , where C is a constant, then at x = 1 is equal to Q73. ∫ (x2+1)ex dx3 (x+1)2 (1) 3 (2) 3 4 8 (3) −32 (4) 78 dx is equal to

Q72.Let 𝛼, 𝛽 and 𝛾 be three positive real numbers. Let 𝑓𝑥= 𝛼x5 + 𝛽x3 + 𝛾x, x ∈R and 𝑔: 𝑅→𝑅 be such that 𝑔𝑓𝑥= 𝑥 for all 𝑥∈𝑅. If 𝑎1, 𝑎2, 𝑎3, … , 𝑎𝑛 be in arithmetic progression with mean zero, then the value of 1 𝑛 𝑓𝑔 𝑛∑𝑖= 1 𝑓𝑎𝑖 is equal to (1) 0 (2) 3 (3) 9 (4) 27

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.If f(x) = {x|x+−4|,a, xx >≤00 { x(x+−4)21, + b, xx <≥00 (gof)(2) + (fog)(−2) is equal to: (1) −10 (2) 10 (3) 8 (4) −8 x > 1

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) = 3(x2−2)3+4, x ∈R. Then which of the following statements are true? P : x = 0 is a point of local minima of f Q : x = √2 is a point of inflection of f R : f ′ is increasing for x > √2 (1) Only P and Q (2) Only P and R (3) Only Q and R (4) All P, Q and R π

Q72.If the system of linear equations 2x + y −z = 7 x −3y + 2z = 1 x + 4y + δz = k, where δ, k ∈R has infinitely many solutions, then δ + k is equal to (1) −3 (2) 3 (3) 6 (4) 9 1 ) 4x2−1

Q72.The number of distinct real roots of the equation x7 −7x −2 = 0 is (1) 5 (2) 7 (3) 1 (4) 3

Q72. log𝑒1 + 5𝑥- log𝑒1 + 𝛼𝑥 if 𝑥≠0 Let the function 𝑓𝑥= 𝑥 be continuous at 𝑥= 0. Then 𝛼 is equal to 10 if 𝑥= 0 (1) 10 (2) -10 (3) 5 (4) -5

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.The value of d 𝑥 at 𝑥= 𝜋 is log𝑒2 dxlogcos𝑥cosec 4 (1) -2√2 (2) 2√2 (3) -4 (4) 4

Q72.Let 𝑓: 𝑅→𝑅 be defined as 𝑓𝑥= 𝑥3 + 𝑥- 5. If 𝑔𝑥 is a function such that 𝑓𝑔𝑥= 𝑥, ∀𝑥∈𝑅, then 𝑔'63 is equal to ______ (1) 49 (2) 1 49 43 3 (3) (4) 49 49

Q72.The number of real solutions of x7 + 5x3 + 3x + 1 = 0 is equal to _____. (1) 0 (2) 1 (3) 3 (4) 5

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 sum of the absolute maximum and absolute minimum values of the function f(x) = tan−1(sin x −cos x) in the interval [0, π] is (1) 0 (2) tan−1( √21 ) −π4 12 (3) cos−1( √31 ) −π4 (4) −π dt, n = 1, 2, 3, … . Then

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

Q73.Let f(x) = 2 + |x| −|x −1| + |x + 1|, x ∈R. Consider (S1) : f ′(−32 ) + f ′(−12 ) + f ′( 12 ) + f ′( 32 ) = 2 (S2) : ∫2−2 f(x)dx = 12 Then, (1) both (S1) and (S2) are correct (2) both (S1) and (S2) are wrong (3) only (S1) is correct (4) only (S2) is correct Q74. ∫20 ( 2x2 −3x + [x −12 ])dx, where [t] is the greatest integer function, is equal to (1) 7 (2) 19 6 12 (3) 31 (4) 3 12 2

Q73.If m and n respectively are the number of local maximum and local minimum points of the function dt, then the ordered pair (m, n) is equal to f(x) = ∫x20 t2−5t+42+et (1) (2, 3) (2) (3, 2) (3) (2, 2) (4) (3, 4) is equal to

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.Let [t] denote the greatest integer less than or equal to t. Then, the value of the integral ∫10 [−8x2 + 6x −1]dx is equal to (1) −1 (2) −54 (3) √17−13 (4) √17−16 8 8

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

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.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