Prova de Lógica e Matemática Básica Pg-ita 2022 1 584

4. Two statements are given about three numbers x, y and z:\nI. \\( x + \\frac{1}{y} = 3 \\)\nII. \\( \\frac{y}{x} + \\frac{1}{z} = 5 \\)\n\nIf it is asked to determine the value of x, one can say that\n(a) statement I alone is sufficient, but statement II alone is not sufficient to answer the question asked;\n(b) statement II alone is sufficient, but statement I alone is not sufficient to answer the question asked;\n(c) both statements I and II together are sufficient to answer the question asked, but neither statement alone is sufficient;\n(d) each statement alone is sufficient to answer the question asked;\n(e) statements I and II together are not sufficient to answer the question asked, and additional data specific to the problem are needed.\n\n5. What is the correct formula to calculate \\( sec(a + b) \\)?\n(a) \\( \\frac{1}{tan(a) + tan(b)} \\)\n(b) \\( sec(a) + sec(b) \\)\n(c) \\( sec(a) + sec(b) \\)\n(d) \\( 1 - sec(b) + sec(a) \\)\n(e) \\( sec(a) cos(b) + sec(b) csc(a) \\)\n\n6. About the roots of the polynomial equation \\( x^6 - 64 = 0 \\), one can say that\n(a) it has 6 non rational roots\n(b) it has 4 non integer roots\n(c) it has 2 integer roots\n(d) it has 6 natural roots\n(e) it is not possible to discuss the roots of this polynomial.\n\n7. Among the options below, which one produces a rational number when divided by \\( ln(x + 1) \\) for all \\( x > 1 \\)?\n(a) \\( ln(x^4 - 4x^3 + 6x^2 - 4x + 1) \\)\n(b) \\( ln(x^2 - 1) \\)\n(c) \\( ln(x^2 - 2x + 1) \\)\n(d) \\( ln(\\sqrt{2} - 1) \\)\n(e) \\( ln(x^3 - 1) - ln(x^2 - x + 1) \\) 2. Figure 1 shows a half right circular cylinder of radius r, which is cut by a plane tilted by 45° in relation to the cylinder basis. A plane perpendicular to the cylinder basis is drawn and its intersection to cylinder basis is parallel to the intersection of this basis and the tilted plane. The intersection of this plane and the cut cylinder is the rectangle shown in the cited Figure coloured in gray. If the rectangle has the maximal area A in these conditions, one could say that\n(a) A = \\frac{r^2}{2}\n(b) A = r^2\n(c) A = r^2 cos(1)sen(1), with trigonometric functions arguments expressed in radians\n(d) A = 2r^2\n(e) None of the above\n\n3. In order to complete Table 1 of four numbers A, B, C and D in relation to their sum, mark the wrong statement:\n(a) Their sum is 300\n(b) A corresponds to 20% of the sum\n(c) B = 150\n(d) C corresponds to 25% of the sum\n(e) B and D sum up 195 Instituto Tecnológico de Aeronáutica\nPrograma de Pós-Graduação em Engenharia de Infraestrutura Aeronáutica\nPrograma de Pós-Graduação em Engenharia Aeronáutica e Mecânica\nProva de Seleção – 1º semestre de 2022 – Questões de Matemática\n8 de novembro de 2021\nNome do Candidato\n\nObservações\n1. Duração da prova: 90 minutos (uma hora e meia)\n2. Não é permitido o uso de calculadoras ou outros dispositivos eletrônicos\n3. Cada pergunta admite uma única resposta\n4. Marque a alternativa que considerar correta no formulário Google enviado por e-mail\n\nQuestões em Inglês\n1. Two sequences of numbers a_i and b_i are defined by the recurrence relation\na_{i+1} = (4/5) a_i - (3/5) b_i (1)\nb_{i+1} = (3/5) a_i + (4/5) b_i (2)\nand by the initial values a_0 = 1 and b_0 = 0. Mark the wrong option:\n(a) a_i and b_i are bounded for any i ∈ ℕ\n(b) b_i is negative for some values of i ∈ ℕ\n(c) a_i = 1 and b_i = 0 for some i > 0 ∈ ℕ\n(d) a_i = (T_i - 24/25) b_i for any i ∈ ℕ\n(e) a_i = (4/5)a_{i-1} + (3/5)b_{i-1} for any i ∈ ℕ 8. If 0 ≤ x ≤ y ≤ 9, how many different integer values the ordered pair (x, y) can assume?\n(a) 100\n(b) 72\n(c) 66\n(d) 55\n(e) 45\n\n9. If a, b ∈ [0, π/2), in order to the following equation be true,\ntan(a + b) = [tan(a) + tan(b)]{1 + tan(a)tan(b) + [tan(a)tan(b)]² + ...}\nwhat statement describe the correct additional restrictions for a and b?\n(a) a - b < π/2\n(b) a + b < π/2\n(c) a = b\n(d) a ≠ b\n(e) This equation is analytic and is valid within the intervals specified without additional restrictions\n\n10. The number of real solutions of the equation cos(2θ) + 3sin(θ) - 2 = 0 in the interval [0, π] is\n(a) 0\n(b) 1\n(c) 2\n(d) 3\n(e) 4\n\n11. Let x = a and y = b be the real solutions for the following system of equations:\n{ log(x) - log(y) = 1\n x² - 198y = 4.\nSo, the value of a/5 + b is\n(a) 3 or -3\n(b) 6\n(c) 6 or -0.06\n(d) 9\n(e) 9 or -9 12. Let ABC be an isosceles triangle which has BC as its base and the sides AB and AC are\nopposite to 75° angles, as shown in Figure 2. Let P be a point inside the triangle such\nthat the sum of the distances PA + PB + PC has minimum value. One can say that\n(a) P is the circumcenter of ABC\n(b) P is the incenter of ABC\n(c) P is in the midpoint of BC\n(d) P is over the line segment between the incenter and the orthocenter of ABC\n(e) P is over the line segment between the barycenter and the incenter of ABC\n\n13. Figure 3 shows a cylinder with height 2L and radius R, such that L > 2R, and there\nare two spheres with radius R inside the cylinder, tangent to the top and bottom of\nthe cylinder. A plane which intersect the cylinder and is tangent to the two spheres will\ndefine a conic section with area\n(a) πR(L - R)\n(b) πR(L + R)\n(c) πR + L + R\n(d) πR L\n(e) πR²L - R²/2 14. Ten coins are stacked, some with heads up, other with tails up. How many ways of\nstacking these coins are there, in order to exactly six coins be with heads up?\n(a) 176\n(b) 210\n(c) 386\n(d) 596\n(e) 638\n\n15. Figure 4 shows a triangle ABC, where H is its orthocenter and I, J and K are the feet\nof the three altitudes from A, B and C, respectively. Mark the wrong statement about\nthis figure:\n(a) Quadrilaterals AJHK, BKHI and CIHJ can be inscribed in circles\n(b) AI, BJ and CK are bisectors of the angles JTK, KJI and TKJ, respectively\n(c) KAH = KJH\n(d) H is the circumcenter of triangle IJK\n(e) Among all the triangles with vertices over the sides of triangle ABC, IJK is the triangle of smallest perimeter\n\n16. I have three times the age you had by the time I had double of the age that you have\nnow. When you reach the age I have now, our ages will sum 69 years. Hence, I am\n(a) 24 years old\n(b) 27 years old\n(c) 32 years old\n(d) 35 years old\n(e) 42 years old