Practice Questions

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

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.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 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

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 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

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.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.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.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) = { −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.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 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.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.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.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.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.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.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 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.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.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 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