Practice Questions

Q68.Let the system of linear equations x + y + az = 2 3x + y + z = 4 x + 2z = 1 have a unique solution ( x∗, y∗, z∗). If ( (a, x∗), (y∗, α) and ( x∗, −y∗) are collinear points, then the sum of absolute values of all possible values of α is: (1) 4 (2) 3 (3) 2 (4) 1

Q69.For 𝛼∈𝑁, consider a relation 𝑅 on 𝑁 given by 𝑅= {𝑥, 𝑦: 3𝑥+ 𝛼𝑦 is a multiple of 7}. The relation 𝑅 is an equivalence relation if and only if (1) 𝛼= 14 (2) 𝛼 is a multiple of 4 (3) 4is the remainder when 𝛼 is divided by 10 (4) 4 is the remainder when 𝛼 is divided by 7 Q70. 0 1 0 Let the matrix 𝐴= 1 0 0 and the matrix 𝐵0 = 𝐴49 + 2𝐴98. If 𝐵𝑛= Adj𝐵𝑛- 1 for all 𝑛≥1, then det 𝐵4 is 0 0 1 equal to (1) 328 (2) 330 (3) 332 (4) 336

Q69. is equal to lim x→π4 √2−√2 sin 2x (1) 14 (2) 7 (3) 14√2 (4) 7√2

Q69.The number of 𝜃∈0, 4𝜋 for which the system of linear equations 3sin3𝜃𝑥- 𝑦+ 𝑧= 2 3cos2𝜃𝑥+ 4𝑦+ 3𝑧= 3 6𝑥+ 7𝑦+ 7𝑧= 9 has no solution is (1) 6 (2) 7 (3) 8 (4) 9

Q70.Let f : R →R be a continuous function such that f(3x) −f(x) = x. If f(8) = 7 , then f(14) is equal to: (1) 4 (2) 10 (3) 11 (4) 16

Q71.The probability that a randomly chosen one-one function from the set {a, b, c, d} to the set {1, 2, 3, 4, 5} satisfied f(a) + 2 f(b) −f(c) = f(d) is (1) 1 (2) 1 24 40 (3) 1 (4) 1 30 20

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.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.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 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.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 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.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

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(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.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

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 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.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 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.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 = , where a, b, c are constants. represent a circle passing through the point (2, 5). Then the dx bx+cy+a shortest distance of the point (11, 6) from this circle is (1) 10 (2) 8 (3) 7 (4) 5 dy 2x−y(2y−1)