UFRGS Mathematics Past Paper 2024 PDF

## MATEMÁTICA NESTA PROVA, SERÃO UTILIZADOS OS SEGUINTES SÍMBOLOS E CONCEITOS COM OS RESPECTIVOS SIGNIFICADOS: - [z]: módulo do número complexo z. - log (x): logaritmo de x na base 10. - Cn,p: combinação de "n" elementos tomados "p" a "p". ### 46. O valor de [(1 + 1/1) * (1 + 1/2) * (1 + 1/3)......

## MATEMÁTICA NESTA PROVA, SERÃO UTILIZADOS OS SEGUINTES SÍMBOLOS E CONCEITOS COM OS RESPECTIVOS SIGNIFICADOS: - [z]: módulo do número complexo z. - log (x): logaritmo de x na base 10. - Cn,p: combinação de "n" elementos tomados "p" a "p". ### 46. O valor de [(1 + 1/1) * (1 + 1/2) * (1 + 1/3)...(1 + 1/99)]^2 é (A) múltiplo de 4. (B) múltiplo de 5. (C) múltiplo de 6. (D) múltiplo de 7. (E) múltiplo de 8. ### 47. Considere as seguintes afirmações sobre números e suas operações. I. 1/1+1/2+1/4+... > 2/1+2/3+2/5+... II. √(√7 + 10) < √(√7 + √10). III. 6.5^10 < 5.6^10. Quais estão corretas? (A) Apenas I. (B) Apenas II. (C) Apenas I e III. (D) Apenas II e III. (E) I, II e III. ### 48. Se a e b são as raízes da equação x^2 + 2x -15 = 0, então o valor de (ab)^a+b é (A) -225. (B) 1/225 (C) -30. (D) 1/225 (E) 225. ### 49. Considere as seguintes afirmações sobre números complexos. I. O módulo de z = 3 + 4i é |z| = 5. II. Se u = 1 + i e v = 1 – i, então |uv| = |u||v|. III. Para que w = (x - 3) + (x + 4) i seja um número real, é necessário e suficiente que x = 3. Quais estão corretas? (A) Apenas I. (B) Apenas III. (C) Apenas I e II. (D) Apenas II e III. (E) I, II e III. ### 50. The image shows a sequence of squares, where the sides of the first three squares are log(2), log(√2) and log(√2), respectively. The sum of the areas of this infinite sequence of squares is (A) 1/2 * [log (2)]^2. (B) 1/3 * [log (2)]^2. (C) 1/3 * [log (2)]^2. (D) log (2 + √2 + √2). (E) log (2√2 - √2). ### 51. The image shows a triangle ABC, where AB = BC = CA = 4√3/3. The height AM of the triangle ABC is equal to the diameter of the circle. The length of AN is (A) 3√3. (B) √3. (C) √3/3. (D) 2. (E) 1. ### 52. The image shows a square ABCD with side length 4. There are four larger circles inside the square, tangent to the sides of the square and tangent to each other. A smaller circle is tangent to the larger circles. The area of the smaller circle is (A) π(3 - 2√2). (B) 2π(3 - √2). (C) 2π(3 - 2√2). (D) 4π(3 - √2). (E) 4π(3 - 2√2). ### 53. The image shows a square ABCD with side length 1. M and N are the midpoints of AB and BC respectively. P is the point of intersection of the segments DM and AN. Knowing that the angle APD is right, the area of the shaded region is (A) 1/3. (B) 2/3. (C) 1/1-√5. (D) 2/√5. (E) 2-√5/3-√5. ### 54. From each vertex of a cube with side length a, a pyramid is cut. The image shows the vertices of one of the pyramids, where I, J and K are the midpoints of edges and A is a vertex of the cube. After all the pyramids are removed, the volume of the remaining solid is (A) a^3/2. (B) a^3/3. (C) a^3/6. (D) 2a^3/3. (E) 5a^3/6. ### 55. The image shows a quadrilateral ABCD and the line r that passes through points A and B. The lengths AB and AD are equal to 1 and the length of side DC is equal to 2. The volume of the solid generated by rotating quadrilateral ABCD around line r is (A) π/3. (B) π/2. (C) 5π/3. (D) π/2. (E) π. ### 56. Consider the real functions f, g and h defined by f(x) = 2x, g(x) = -1/x and h(x) = √4 - x^2. The area of the region enclosed by the graphs of functions f(x), g(x) and h(x) is (A) π/4. (B) π/2. (C) π. (D) 2π. (E) 4π. ### 57. The image shows an equilateral triangle ABC with side length 6. M is the midpoint of side AB and P is a point on segment AM. M is also the midpoint of PQ. The rectangle PQRS is formed with vertices R and S on sides BC and AC, respectively. Defining x as the length of segment AP, consider A(x) as the function that expresses the area of rectangle PQRS in terms of x. For x ∈ [0,3], A(x) is (A) A(x) = x√3. (6 – 2x). (B) A(x) = 2x√3. (6 – 2x). (C) A(x) = x√3. (3 - 2x). (D) A(x) = x√3. (3 - x) (E) A(x) = 2x√3. (6 + 2x). ### 58. The infographic shows the number of new cancer cases in Brazil for men and women. Based on the data presented in the infographic, consider the following statements: I. In women, the number of lung cancer cases is less than 20% of the number of breast cancer cases. II. In men, the number of bladder cancer cases is less than 10% of the number of prostate cancer cases. III. In men, the number of lung cancer cases exceeds the number of lung cancer cases in women by more than 30%. Which statements are correct? (A) Only I. (B) Only II. (C) Only III. (D) Only I and II. (E) I, II and III. ### 59. A soccer team has 20 players, and 3 of them are goalkeepers. The number of possible teams, with 5 players, where only the goalkeeper plays a fixed position is (A) C17,4. (B) C20,4. (C) C20,5. (D) C3,1 + C17,4. (E) C3,1 * C17,4. ### 60. Consider a fair coin with a head and a tail side. When tossing this coin 5 times, the probability of getting at least 3 heads is (A) 1/2. (B) 1/6. (C) 1/4. (D) 1/1-√5. (E) 1/2.