Practice Questions

Q9. Let [x] denote the greatest integer function, and let m and n respectively be the numbers of the points, where the function f(x) = [x] + |x −2|, −2 < x < 3, is not continuous and not differentiable. Then m + n is equal to : (1) 6 (2) 8 (3) 9 (4) 7

Q9. Let f(x) be a real differentiable function such that f(0) = 1 and f(x + y) = f(x)f ′(y) + f ′(x)f(y) for all x, y ∈R. Then ∑100n=1 loge f(n) is equal to : (1) 2525 (2) 5220 (3) 2384 (4) 2406

Q9. Let f : [0, 3] → A be defined by f(x) = 2x3 −15x2 + 36x + 7 and g : [0, ∞) →B be defined by g(x) = x2025 . If both the functions are onto and S = {x ∈Z : x ∈ A or x ∈ B}, then n(S) is equal to : x2025+1 2025 (28 Jan Shift 2) JEE Main Previous Year Paper (1) 29 (2) 30 (3) 31 (4) 36

Q9. The length of the chord of the ellipse x2 4 + 2 = 1, whose mid-point is (1, 12 ), is : (1) 5 3 √15 (2) 13 √15 (3) 2 3 √15 (4) √15

Q9. Let P be the foot of the perpendicular from the point Q(10, −3, −1) on the line x−37 = y−2−1 = z+1−2 . Then the area of the right angled triangle PQR, where R is the point (3, −2, 1), is (1) 9√15 (2) √30 (3) 8√15 (4) 3√30

Q9. If the image of the point (4, 4, 3) in the line x−12 = y−21 = z−13 is (α, β, γ), then α + β + γ is equal to (1) 9 (2) 12 (3) 7 (4) 8

Q9. The integral 80 ∫ 0 4 ( 9+16sin θ+cossin 2θθ )dθ is equal to : (1) 3 loge 4 (2) 4 loge 3 (3) 6 loge 4 (4) 2 loge 3 2025 (29 Jan Shift 1) JEE Main Previous Year Paper y2 1 . Let the + = 1, A < B have same eccentricity = 1, a > b and E2 : x2 B2

Q10. x + y + z = 6 The system of equations x + 2y + 5z = 9, has no solution if x + 5y + λz = μ, (1) λ = 15, μ ≠17 (2) λ ≠17, μ ≠18 (3) λ = 17, μ ≠18 (4) λ = 17, μ = 18

Q10.For a statistical data x1, x2, … , x10 of 10 values, a student obtained the mean as 5.5 and ∑10i=1 x2i = 371. He later found that he had noted two values in the data incorrectly as 4 and 5 , instead of the correct values 6 and 8 , respectively. The variance of the corrected data is (1) 9 (2) 5 (3) 7 (4) 4

Q10.Let →a and →b be two unit vectors such that the angle between them is . If λ→a + 2→b and 3→a −λ→b are 3 perpendicular to each other, then the number of values of λ in [−1, 3] is : (1) 2 (2) 1 (3) 0 (4) 3 e 1 x x loge α

Q10.Let the ellipse E1 : x2a2 + y2b2 A2 √3 product of their lengths of latus rectums be 32 , and the distance between the foci of E1 be 4. If E1 and E2 √3 meet at A, B, C and D , then the area of the quadrilateral ABCD equals : (1) 12√6 (2) 6√6 5 (3) 18√6 (4) 24√6 5 5

Q10. cos (sin−1 35 + sin−1 135 + sin−1 3365 ) is equal to: (1) 1 (2) 0 (3) 32 (4) 33 65 65

Q10.Bag B1 contains 6 white and 4 blue balls, Bag B2 contains 4 white and 6 blue balls, and Bag B3 contains 5 white and 5 blue balls. One of the bags is selected at random and a ball is drawn from it. If the ball is white, then the probability, that the ball is drawn from Bag B2 , is : (1) 4 (2) 1 15 3 (3) 2 (4) 2 5 3

Q10.Let A = [aij] be a square matrix of order 2 with entries either 0 or 1 . Let E be the event that A is an invertible matrix. Then the probability P(E) is : 2025 (24 Jan Shift 2) JEE Main Previous Year Paper (1) 3 (2) 5 16 8 (3) 3 (4) 1 8 8

Q10.From all the English alphabets, five letters are chosen and are arranged in alphabetical order. The total number of ways, in which the middle letter is ' M ', is : 2025 (22 Jan Shift 1) JEE Main Previous Year Paper (1) 5148 (2) 6084 (3) 4356 (4) 14950

Q10.Let the function f(x) = (x2 + 1) x2 −ax + 2 + cos |x| be not differentiable at the two points x = α = 2 and x = β . Then the distance of the point (α, β) from the line 12x + 5y + 10 = 0 is equal to : (1) 5 (2) 4 (3) 3 (4) 2

Q11.The area of the region {(x, y) : x2 + 4x + 2 ≤y ≤|x + 2|} is equal to (1) 7 (2) 5 (3) 24/5 (4) 20/3

Q11.Let f(x) = loge x and g(x) = x4−2x3+3x2−2x+22x2−2x+1 . Then the domain of (1) [0, ∞) (2) [1, ∞) (3) (0, ∞) (4) R

Q11.Let the area enclosed between the curves |y| = 1 −x2 and x2 + y2 = 1 be α. If 9α = βπ + γ; β, γ are integers, then the value of |β −γ| equals. (1) 27 (2) 33 (3) 15 (4) 18

Q11. Let f : R →R be a twice differentiable function such that f(2) = 1. If F(x) = xf(x) for all x ∈R, ∫20 x F′(x)dx = 6 and ∫20 x2 F′′(x)dx = 40, then F′(2) + ∫20 F(x)dx is equal to : (1) 11 (2) 13 (3) 15 (4) 9 507S2025 is :

Q11.Let A(x, y, z) be a point in xy-plane, which is equidistant from three points (0, 3, 2), (2, 0, 3) and ( 0, 0, 1 ). Let B = (1, 4, −1) and C = (2, 0, −2). Then among the statements (S1) : △ABC is an isosceles right angled triangle, and (S2) : the area of △ABC is 9√22 , (1) both are true (2) only (S2) is true (3) only (S1) is true (4) both are false

Q11.Let the range of the function f(x) = 6 + 16 cos x ⋅cos ( π3 −x) ⋅cos ( π3 + x) ⋅sin 3x ⋅cos 6x, x ∈R be [α, β] . Then the distance of the point (α, β) from the line 3x + 4y + 12 = 0 is : (1) 11 (2) 8 (3) 10 (4) 9 sin y > 0 and x(1) = π2 . Then

Q11.If limx→∞(( 1−e ) ( e − 1+x )) = α, then the value of 1+loge α equals : (1) e−1 (2) e2 (3) e−2 (4) e

Q11.Using the principal values of the inverse trigonometric functions, the sum of the maximum and the minimum is : values of 16 ((sec−1 x)2 + (cosec−1 x)2) (1) 24π2 (2) 22π2 (3) 31π2 (4) 18π2

Q11.Let A = [aij] = [ log5log51288 log4log4255 ] . If Aij is the cofactor of aij, Cij = ∑2k=1 aikAjk, 1 ≤i, j ≤2, and C = [Cij], then 8|C| is equal to : (1) 288 (2) 222 (3) 242 (4) 262