Practice Questions

Q8. If the line 3x −2y + 12 = 0 intersects the parabola 4y = 3x2 at the points A and B, then at the vertex of the parabola, the line segment AB subtends an angle equal to ⎪ ⎪ 2025 (23 Jan Shift 1) JEE Main Previous Year Paper (1) tan−1 ( 45 ) (2) tan−1 ( 97 ) (3) tan−1 ( 119 ) (4) π2 −tan−1 ( 32 )

Q8. Let the lines 3x −4y −α = 0, 8x −11y −33 = 0, and 2x −3y + λ = 0 be concurrent. If the image of the point (1, 2) in the line 2x −3y + λ = 0 is ( 5713 , −4013 ), then |αλ| is equal to (1) 84 (2) 113 (3) 91 (4) 101

Q8. If 7 = 5 + 17 (5 + α) + 721 (5 + 2α) + 731 (5 + 3α)+ ∞, then the value of α is : (1) 6 (2) 6 7 (3) 1 (4) 1 7

Q8. Two number k1 and k2 are randomly chosen from the set of natural numbers. Then, the probability that the value of ik1 + ik2, (i = √−1) is non-zero, equals ⎪ ⎪ 2025 (28 Jan Shift 1) JEE Main Previous Year Paper (1) 1 (2) 3 2 4 (3) 1 (4) 2 4 3

Q8. If the set of all a ∈R, for which the equation 2x2 + (a −5)x + 15 = 3a has no real root, is the interval (α, β), and X = {x ∈Z : α < x < β}, then ∑x∈X x2 is equal to : (1) 2109 (2) 2129 (3) 2119 (4) 2139

Q8. Let the point A divide the line segment joining the points P(−1, −1, 2) and Q(5, 5, 10) internally in the ratio −−−−→ → → → r : 1(r > 0). If O is the origin and (OQ ⋅OA) −15 |OP × OA|2 = 10, then the value of r is : (1) √7 (2) 14 (3) 3 (4) 7 2025 (23 Jan Shift 2) JEE Main Previous Year Paper y2

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. Let P(4, 4√3) be a point on the parabola y2 = 4ax and PQ be a focal chord of the parabola. If M and N are the foot of perpendiculars drawn from P and Q respectively on the directrix of the parabola, then the area of the quadrilateral PQMN is equal to : 2025 (22 Jan Shift 2) JEE Main Previous Year Paper (1) 17√3 (2) 263√3 8 (3) 34√3 (4) 343√3 3 8 π

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

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.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

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 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 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.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 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

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

Q12.For positive integers n, if 4an = (n2 + 5n + 6) and Sn = ∑nk=1 ( ak1 ), then the value of (1) 540 (2) 675 (3) 1350 (4) 135

Q12.Let |z1 −8 −2i| ≤1 and |z2 −2 + 6i| ≤2, z1, z2 ∈C . Then the minimum value of |z1 −z2| is : (1) 13 (2) 10 (3) 3 (4) 7