Practice Questions

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. 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. 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. 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. 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. 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(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. 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. 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. 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

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

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

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.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.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

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

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.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

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

Q12.Let A = {1, 2, 3, 4} and B = {1, 4, 9, 16}. Then the number of many-one functions f : A →B such that 1 ∈f( A) is equal to : (1) 151 (2) 139 (3) 163 (4) 127

Q12.The area (in sq. units) of the region {(x, y) : 0 ≤y ≤2|x| + 1, 0 ≤y ≤x2 + 1, |x| ≤3} is (1) 80 (2) 64 3 3 (3) 32 (4) 17 3 3